Review Article Fast Transforms in Image Processing ...downloads.hindawi.com/archive/2014/276241.pdfReview Article Fast Transforms in Image Processing: Compression, Restoration, and

Review ArticleFast Transforms in Image ProcessingCompression Restoration and Resampling

Leonid P Yaroslavsky

Department of Physical Electronics School of Electrical Engineering Tel Aviv University 69978 Tel Aviv Israel

Correspondence should be addressed to Leonid P Yaroslavsky yaroengtauacil

Received 4 March 2014 Accepted 19 May 2014 Published 6 July 2014

Academic Editor George E Tsekouras

Copyright copy 2014 Leonid P Yaroslavsky This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Transform image processing methods are methods that work in domains of image transforms such as Discrete Fourier DiscreteCosine Wavelet and alike They proved to be very efficient in image compression in image restoration in image resampling andin geometrical transformations and can be traced back to early 1970s The paper reviews these methods with emphasis on theircomparison and relationships from the very first steps of transform image compression methods to adaptive and local adaptivefilters for image restoration and up to ldquocompressive sensingrdquo methods that gained popularity in last few years References are madeto both first publications of the corresponding results and more recent and more easily available ones The review has a tutorialcharacter and purpose

1 Introduction Why TransformsWhich Transforms

It will not be an exaggeration to assert that digital imageprocessing came into being with introduction in 1965 byCooley and Tukey of the Fast Fourier Transform algorithm(FFT [1]) for computing the Discrete Fourier Transform(DFT) This publication immediately resulted in impetuousgrowth of activity in all branches of digital signal and imageprocessing and their applications

The second wave in this process was inspired by theintroduction into communication engineering and digitalimage processing in the 1970s of Walsh-Hadamard trans-form and Haar transform [2] and the development of a largefamily of fast transforms with FFT-type algorithms [3ndash5]WhereasWalsh-Hadamard andHaar transformshave alreadybeen known in mathematics other transforms for instancequite popular at the time Slant Transform [6] were beinginvented ldquofrom scratchrdquoThis development wasmainly drivenby the needs of data compression though the usefulnessof transform domain processing for image restoration andenhancement was also recognized very soon [3] This periodended up with the introduction of the Discrete Cosine

Transform (DCT [7 8]) which was soon widely recognizedas the best choice among all available at the time transformsand resulted in JPEG and MPEG standards for image audioand video compression

The third large wave of activities in transforms for signaland image processing was caused by the introduction in the1980s of a family of transforms that was coined the nameldquowavelet transformrdquo [9] The main motivation was achievinga better local representation of signals and images in contrastto the ldquoglobalrdquo representation that is characteristic to DiscreteFourier DCT Walsh-Hadamard and other fast transformsavailable at the time During 1980sndash1990s a large variety ofdiscrete wavelet transforms were suggested [10] for solvingvarious tasks in signal and in image processing

Presently fast transforms with FFT-type fast algorithmsand wavelet transforms constitute the basic instrumentationtool of digital image processing

The main distinctive feature of transforms that makesthem so efficient in digital image processing is their energycompaction capability In regular image representations inform of sets of ordered pixels some pixels for instance thosethat belong to object borders are more important than theothers and there are always some pixels in each particular

Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2014 Article ID 276241 23 pageshttpdxdoiorg1011552014276241

2 Advances in Electrical Engineering

Table 1 Register of relevant fast transforms their main characteristic features and application areas

Relevance to imagingoptics Main characteristic features Main application areas

Discrete FourierTransforms

Represents opticalFourier Transform

Cyclic shift invariance vulnerableto boundary effects

(i) Analysis of periodicities(ii) Fast convolution andcorrelation(iii) Fast and accurate imageresampling(iv) Image compression(v) Image denoising anddeblurring(vi) Numerical reconstruction ofholograms

Discrete CosineTransform


Cyclic shift invariance (withdouble cycle)virtually not sensitive toboundary effects

Discrete FresnelTransforms

Represent optical FresnelTransform Computable through DFTDCT Numerical reconstruction of

holograms

Walsh-HadamardTransform No direct relevance

Binary basis functionsProvides piece-wise constantseparable image band-limitedapproximations

(i) Image compression(marginal)(ii) Coded aperture imaging

Haar Transform andother Discrete WaveletTransforms

Subband decomposition

Binary basis functionsThe fastest algorithmMultiresolutionShift invariance in eachparticular scale

(i) Signalimage wideband noisedenoising(ii) Image compression

image that are of no such importance and can be dropped outfrom image representation and restored from the remainingldquoimportantrdquo pixels But the problem is that one never knowsin advance which pixels in the image are ldquoimportantrdquo andwhich are not

The situation is totally different in image representationin transform domain For orthogonal transforms that featuregood energy compaction capability a lion share of totalimage ldquoenergyrdquo (sum of squared transform coefficients) isconcentrated in a small fraction of transform coefficientswhich indices are as a rule known in advance for the giventype of images or can be easily detected It is this feature oftransforms that is called their energy compaction capability Itallows replacing images with their ldquoband-limitedrdquo in terms ofa specific transform approximations that is approximationsdefined by a sufficiently small fraction of image transformcoefficients

The optimal transform with the best energy compactioncapability can be defined for groups of images of the sametype or ensembles of images that are subjects of process-ing Customarily the ldquoband-limitedrdquo image approximationaccuracy is evaluated in terms of the root mean squareapproximation error (RMSE) for the image ensemble Byvirtue of this the optimal transform that secures the leastband limited approximation error is defined by the ensemblecorrelation function that is image autocorrelation functionaveraged over the image ensemble For continuous (notsampled) signals this transform is called theKarhunen-LoeveTransform [11 12] Its discrete analog for sampled signals iscalled the Hotelling transform [13] and the result of applyingthis transform to sampled signals is called signal principalcomponent decomposition Karhunen-Loeve and Hotelling

transforms provide the theoretical lower bound for compact(in terms of the number of terms in signal decomposition)signal discrete representation for RMS criterion of signalapproximation

However being optimal in terms of the energy com-paction capability Karhunen-Loeve and Hotelling trans-forms have generally high computational complexity theper pixel number of operations required for their compu-tation is of the order of image size This is why for prac-tical needs only fast transforms that feature fast transformalgorithms with computational complexity of the order ofthe logarithm of image size are considered A register ofthe most relevant fast transforms their main characteristicfeatures and application areas is presented in Table 1 Theirmathematical definitions are given in Table 2

Different fast transforms have different energy com-paction capability for different types of images Figure 1illustrates the energy compaction capabilities of DiscreteFourier Discrete Cosine Walsh and Haar transforms ona particular test image It shows that for this image DCTdemonstrates the best energy compaction capability 95 ofthe image energy is contained in only 95of all DCT spectralcoefficients whereas for DFT Walsh and Haar transformsthese fractions are 11 15 and 98 correspondinglyNote also that though for Haar Transform the fractionof the ldquomeaningfulrdquo transform coefficients (98) is closeto that for DCT quality degradations of the band-limitedapproximation to the image are noticeably more severe

Experience shows that DCT is in most applicationsadvantageous with respect to other transforms in terms ofthe energy compaction capability This property of DCT hasa simple and intuitive explanation

Advances in Electrical Engineering 3

Table 2 Selected fast transforms most relevant for digital image processing and their computational complexity (in 1D denotations)

Name Definition

Computational complexity

(operations per signal sample)ldquoGlobalrdquoapplied to entiresignal of119873 samples

ldquoLocalrdquo appliedin mowingwindow of119882119878119911 pixels

Discrete Cosine Transform(DFT)

120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896cos(2120587119896 + 12

119873119903)

Real numberoperations119862119866

DCT log119873


DCT(ltWSz)

Discrete Fourier Transforms

Canonic DFT 120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896exp(1198942120587119896119903

119873) Complex

numberoperations119862119866

DFT log119873

Complexnumberoperations119862119871

DFT(ltWSz )General Shifted Scaled DFT

120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896exp [1198942120587 (119896 + 119906) (119903 + V)

120590119873]

119906 Vmdashshift parameters that indicate shifts 119906Δ119909 VΔ119891 of samplinglattices in signal and spectrum domains

120590mdashsampling scale parameter (Δ119909Δ119891 = 1120590119873)Discrete Fresnel Transforms(DFrT)

Canonic DFrT120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896exp [1198942120587(

119896120583 minus 119903120583

119873)] 120583 ge 1 Complex


DFrT log119873

na120583 = radic120582119885119873Δ1198912

Convolutional DFrT 120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896exp[1198942120587 (119896 minus 119903)

2

1205832119873] 120583 le 1

Walsh-Hadamard Transform

Hadamard Transform120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896had119896(119903)

had119896(119903) = (minus1)

sum119899minus1

119898=0119896119898119903119898

Additionoperations only119862119866

WHT log119873na

119896 =

119899minus1

sum

119898=0

1198961198982119898 119903 =

119899minus1

sum

119898=0

1199031198982119898119873 = 2

119899

Walsh Transform120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896wal119896(119903)

wal119896(119903) = (minus1)

sum119899minus1

119898=0119896119898119903119866119903119886119910119862119900119889119890

119899minus119898minus1

119903119866119903119886119910119862119900119889119890

119899minus119898minus1mdashbinary digits of Gray code of 119903 taken in bit reversal order

Haar Transform(the simplest discrete wavelettransform)

120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896har119896(119903)

har119896(119903) = 2

msb(minus1)119896119899minus1minusmsb120575 (119903mod 2msb minus lfloor2

1minusmsb119896rfloor)

Msbmdashindex of the most significant nonzero bit of binary code ofmsb lfloorsdotrfloormdashresidual of (sdot)


Haar

na

DCT of a discrete signal is essentially to the accuracy ofan unimportant exponential shift factor DFT for the samesignal extended outside its border to a double length bymeans of its mirrored in the order of samples copy [15 16]This way of signal extension has a profound implication onthe speed of decaying to zero of signal DCT spectra and isa paramount and fundamental reason of the good energycompaction capability of DCTDFT implies periodical exten-sion of signals which may cause severe discontinuities atsignal borders and thus may require intensive high frequencycomponents to reproduce them DCT also implies periodicalextension however extension of not the signal itself but of

its above-described extended copy Thanks to such an ldquoevenrdquoextension the extended signal has no discontinuities at itsborders as well as at borders of the initial signal Thereforesuch an extended signal is ldquosmoothrdquo and its DFT spectrumdecays to zero faster than DFT spectrum of the initial notextended signal

It is noteworthy to mention that introducing DCT in [7]was motivated not by the above reasoning but by the findingthat basis functions of DCT provide a good approximation tothe eigenvectors of the class of Toeplitz matrices with entries120583119896119897

= 120588|119896minus119897|

120588 = 09 119896 119897 = 0 1 that is DCT can beconsidered as a good approximation to the Karhunen-Loeve


Selected fragment 256 times 256

(a)

Band limited approximation DFT

(b)

Band limited approximation DCT

(c)

Band limited approximation Walsh

(d)

Band limited approximation Haar

(e)

100

100

10minus4

10minus3

10minus2

10minus1

Fraction of ordered spectral coefficients

DCTDFTWalsh

HaarEnergy threshold

Fraction of spectrum energy versus fractionof spectral coefficients that contain it

DCT = 0095 DFT = 011 Walsh = 015 Haar = 0098

fraction of spectral coecffients of total energy 095

(f)

Figure 1 Illustration of the energy compaction capability of Discrete Fourier Discrete Cosine Walsh and Haar transforms for a test imageshown in the upper left corner The rest of images show band-limited approximations of the test image in the domain of correspondingtransforms for the approximation error variance equal to 5 of image variance Graphs on the plot in the bottom right corner present energycontained in fractions of transform coefficients for different transforms


Transform for signals with the Toeplitz correlation matrixIn fact the above-mentioned fundamental reason for goodenergy compaction capability of DCT is hidden in the evensymmetry of the Toeplitz matrix

In the family of orthogonal transforms DFT and DCToccupy a special place DFT and DCT are two discreterepresentations of the Integral Fourier Transform whichplays a fundamental role in signal and image processing asa mathematical model of signal transformations and of wavepropagation in imaging systems [17 18] Thanks to this DFTand DCT are applicable in much wider range of applicationsthan the other fast transforms (see Table 1)

In conclusion of this section come back to the commonfeature of fast transform to the availability for all of thesetransform of fast algorithms of FFT type Initially fasttransforms were developed only for signals with the numberof samples equal to an integer power of two These are thefastest algorithms At present this limitation is overcome andfast transform algorithms exist for arbitrary number of signalsamples Note that 2D and 3D transforms for image and videoapplications are built as separable that is theywork separablyin each dimension Note that the transform separability isthe core idea for fast transform algorithms All fast transformalgorithms are based on representation of signal and trans-form spectrum sample indices as multidimensional that isrepresented by multiple digits numbers

The data on computational complexity of the transformfast algorithms are collected inTable 2These data aremore orless commonly known Not as widely known is the existenceof so-called pruned algorithms for the cases when transforminput data contain substantial fraction of zero samples andorone does not need to compute all transform coefficients [15ndash24] These algorithms are useful for instance in numericalreconstruction of holograms and in interpolation of sampleddata by means of transform zero padding [17 25 26]

In some applications it is advisable to apply transformslocally in running window rather than globally to entireimage frames For such application Discrete Cosine andDiscrete Fourier Transforms have an additional very usefulfeature Computing these transforms in running window canbe carried out recursively signal transform coefficients ateach window position can be found by quite simplemodifica-tion of the coefficients found at the previous window positionwith per pixel computational complexity proportional to thewindow size rather than to the product of the window sizeand its logarithm which is the computational complexity ofthe fast transforms [17 26ndash28]

The rest of the paper is arranged as follows In Section 2applications of fast transforms for image data compressionare reviewed In Section 3 we proceed to transformmethodsfor image restoration and enhancement In Section 4 weshow how DFT and DCT can be applied for image perfectthat is error less resampling And finally in Section 5 weaddress the usage of fast transforms for solving a specific taskof image resampling the task of image recovery from sparseand irregularly taken samples and in particular compressivesensing approach to this problem

2 Image Compression

21 Dilemma Compressive Discretization or CompressionIdeally image digitization that is converting continuoussignals from image sensors into digital signals should becarried out in such a way as to provide as compact imagedigital representation as possible provided that quality ofimage reproduction satisfies given requirements Due totechnical limitation image digitization is most frequentlyperformed in two steps discretization that is convertingimage sensor signal into a set of real numbers and scalarquantization that is rounding off of those numbers to a setof fixed quantized values [15 18]

Image discretization can in general be treated as obtain-ing coefficients of image signal expansion over a set ofdiscretization basis functions In order to make the set ofthese representation coefficients as compact as possible oneshould choose discretization basis functions that secure theleast number of the signal expansion coefficients sufficientfor image reconstruction with a given required quality Onecan call this general method of signal discretization ldquoCom-pressive discretizationrdquo because it secures the most compactdiscrete representation of signals which cannot be furthercompressed Note that this term should not be confused withterms ldquoCompressive sensingrdquo and ldquoCompressive samplingrdquo thatgained large popularity in recent years [29ndash36] We willdiscuss the ldquocompressive sensingrdquo approach later in Section 5

There are at least two examples of practical implemen-tation of the principle of general discretization by meansof measuring transform domain image coefficients CodedAperture Imaging and Magnetic Resonance Imaging (MRI)In coded aperture imaging images are sensed throughbinary masks that implement binary basis functions In MRIimaging coefficients of Fourier series expansion of sensordata are measured

However by virtue of historical reasons and of thetechnical tradition image discretization is most frequentlyin imaging engineering is implemented as image samplingat nodes of a uniform rectangular sampling lattice usingsampling basis functions which are formed fromone ldquomotherrdquofunction by its shifts by multiple of fixed intervals calledldquosamplingrdquo intervals

The theoretical foundation of image sampling is the sam-pling theorem The traditional formulation of the samplingtheorem states that signals with Fourier spectrum limitedwith bandwidth 119861 can be perfectly restored from theirsamples taken with sampling interval Δ119909 = 1119861 commonlycalled the Nyquist sampling interval [37ndash39]

In reality no continuous signal is band-limited and theimage sampling interval is defined not through specifying inone or another way of the image bandwidth but directly bythe requirement to reproduce smallest objects and borders oflarger objects present in images sufficiently well The selectedin this way image sampling interval Δ119909 specifies image baseband 119861 = 1Δ119909

Since small objects and object borders usually occupyrelatively small fraction of the image area vast portions ofimages are oversampled that is sampled with redundantlysmall sampling intervals Hence substantial compression of


Set ofreconstructedimage pixels

Fast orthogonaltransform

Shrinkage andscalar

quantization oftransformcoefficients

Entropyencoding ofquantizedtransform

coefficients

Set of imagepixels

Compressed bitstream

Compressed bitstream Inverse fast

orthogonaltransform

Dequantizationof transformcoefficients

Entropydecoding

Encoding

Image reconstruction

Figure 2 Flow diagrams of image transform coding and reconstruction

image sampled representation is possible It can be imple-mented by means of applying to the image primary andredundant sampled representation a discrete analog of thegeneral compressive discretization that is by means of imageexpansion over a properly chosen set of discrete optimalbases functions and limitation of the amount of the expansioncoefficients This is exactly what is done in all transformmethods of image compression

22 Transform Methods of Image Compression Needs ofimage compression were the primary motivations of digi-tal image processing Being started in 1950s from variouspredictive coding methods (a set of good representativepublications can be found in [40]) by the beginning of 1970simage compression research began concentrating mostly onwhat we call now ldquotransform codingrdquo [41 42] which literallyimplements the compressed discretization principle

The principle of image transform coding is illustrated bythe flow diagrams sketched in Figure 2

According to these diagrams set of image pixels is firstsubjected to a fast orthogonal transform Then low intensitytransform coefficients are discarded which substantiallyreduces the volume of data This is the main source of imagecompression Note that discarding certain image transformcoefficients means replacement of images by their band-limited in terms of the selected transform approximationsThe remaining coefficients are subjected one by one tooptimal nonuniform scalar quantization that minimizes theaverage number of transform coefficient quantization levelsFinally quantized transform coefficients are entropy encodedto minimize the average number of bits per coefficient Forimage reconstruction from the compressed bit stream itmustbe subjected to corresponding inverse transformations

In 1970s main activities of researches were aimed atinventing in addition to known at the time Discrete FourierWalsh-Hadamard andHaarTransforms new transforms thathave guaranteed fast transform algorithms and improvedenergy compaction capability [3ndash5]

This transform invention activity gradually faded afterintroduction in a short note of the Discrete Cosine trans-form [7] which as it was already mentioned was finallyrecognized as the most appropriate transform for image

compression due to its superior energy compaction capabilityand the availability of the fast algorithm But the true break-through happened when it was realized that much higherimage compression efficiency can be achieved if transformcoding is applied not to entire image frames but blockwise[41 42]

Images usually contain many objects and their globalspectra are a mixture of object spectra whereas spectra ofindividual image fragments or blocks are much more specificand this enables much easier separation of ldquoimportantrdquo(most intense) from ldquounimportantrdquo (least intense) spectralcomponents This is vividly illustrated in Figure 3 whereglobal and block wise DCT power spectra of a test imageare presented for comparison Although as one can see inthe figure spectral coefficients of DCT global spectrum aremore sparse (spectrum sparsity of this image ie fraction ofspectral coefficients that contain 95 of the total spectrumenergy is 0095) than on average those of image of 16 times 16

pixel fragments (average sparsity of spectra fragments is 014)spectra of individual fragments are certainly more specificand their components responsible for small objects and objectborders havemuch higher energy than in the global spectrumandwill not be lostwhen 95of themost intense componentsare selected

Blockwise image DCT transform compression proved tobe so successful that by 1992 it was put in the base ofthe image and video coding standards ldquoJPEGrdquo and ldquoMPEGrdquo[43ndash46] which eventually resulted in the revolution in theindustry of photographic and video cameras and led to theemergence of digital television as well [47]

Though the standard does not fix the size of blocksblocks of 8 times 8 pixels are used in most cases Obviously thecompression may in principle be more efficient if the blocksize is adaptively selected in different areas of images hencethe use of variable size of blocks was discussed in number ofpublications (see for instance [48])

3 Transform Domain Filters forImage Restoration and Enhancement

31 Transform Domain Scalar Empirical Wiener FiltersVery soon after the image transform compression methods


Test image

(a)

Test image global DCT spectrumSpectrum sparsity0095 at energy level 095

(b)

Fragmented input image WSzX lowast WSzY = 16 times 16

(c)

Abs (fragment spectra)05 fragment spectra sparsity 014

(d)

Figure 3 Comparison of global and local (blockwise) image DCT spectra

TransformModificationof transformcoefficients

Inversetransform

Inputimage

Outputimage

Figure 4 Flow diagram of transform domain filtering

emerged it was recognized that transforms represent a veryuseful tool for image restoration from distortions of imagesignals in imaging system and for image enhancement [4950] The principle is very simple for image improvementimage transform coefficients are modified in a certain wayand then the image is reconstructed by the inverse transformof modified coefficients (Figure 4) This way of processing iscalled transform domain filtering

For the implementation of modifications of image trans-form coefficients two options are usually considered

(i) Modification of absolute values of transform coeffi-cients by a nonlinear transfer function usually it is afunction that compresses the dynamic range of trans-form coefficients which redistributes the coefficientsrsquointensities in favor of less intensive coefficients andresults in contrast enhancement of image small detailsand edges

(ii) Multiplication of transform coefficients by scalarweight coefficients this processing is called transformdomain ldquoscalar filteringrdquo [49]

For defining optimal scalar filter coefficients a Wiener-Kolmogorov [51 52] approach of minimization of meansquared filtering error is used and thus filters implementempirical Wiener filtering principles [17 18 26]

Three modifications of the filters based on this principleare (i) proper Empirical Wiener Filters (ii) Signal SpectrumPreservation Filters and (iii) Rejecting Filters [18] For image


Before

(a)

After

(b)

Figure 5 Image cleaning from moire noise

denoising from additive signal independent noise and imagedeblurring they modify input image Fourier spectra 119878119901inpat each frequency component according to the equationscorrespondingly

Empirical Wiener Filter

119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)

Signal Spectrum Preservation Filter

119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)

where 119878119901out is Fourier spectrum of output image 119878119901noise ispower spectrum of additive noise assumed to be knownor to be empirically evaluated from the input noisy image[15ndash18] and ISFR is the imaging system frequency responseassumed to be known for instance from the imaging systemcertificate

As one can see in these equations all these filters elimi-nate image spectral components that are less intensive thanthose of noise and the remaining components are correctedby the ldquoinverserdquo filter with frequency response of 1ISFRDivision of image spectra by ISFR compensates image blur inthe imaging system due to imperfect ISFR that is it performs

image deblurring It is applicable in processing specificallyin domains of DFT or DCT which are as it was alreadymentioned two complementing discrete representations ofthe integral Fourier Transform

In addition empirical Wiener filter modifies image spec-trum through deamplification of image spectral componentsaccording to the level of noise in them spectrumpreservationfilter modifies magnitude of the image spectrum as well bymaking it equal to a square root of image power spectrumempirical estimate as a difference between power spectrum ofnoisy image and that of noise Rejecting filter does notmodifyremaining that is not rejected spectral components at allIn some publications image denoising using the spectrumpreservation filter is called ldquosoft thresholdingrdquo and rejectingfiltering is called ldquohard thresholdingrdquo [53 54]

Versions of these filters are filters that combine imagedenoising-deblurring and image enhancement by means of119875-lawnonlinearmodifications of the corrected signal spectralcomponents

119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)

where 119875 le 1 is a parameter that controls the degree ofspectrumdynamic range compression and by this the degreeof enhancement of low intensity image spectral components[18]

The described filters proved to be very efficient indenoising so-called narrow band noises that is noises suchas ldquomoire noiserdquo or ldquobanding noiserdquo that contain only fewnonzero components in its spectrum in the selected trans-form Some illustrative examples are given in Figures 5 and6

However for image cleaning from wideband or ldquowhiterdquonoise above transform domain filtering applied globally toentire image frames is not efficient In fact it can even worsen


Before

(a)

After

(b)

Figure 6 Cleaning aMars satellite image from banding noise bymeans of separable (row-wise and column-wise) high pass rejecting filteringFiducial marks seen in the left image were extracted from the image before the filtering [14]

the image visual quality As one can see from an illustrativeexample shown in Figure 7 filtered image loses its sharpnessand contains residual correlated noise which is visually moreannoying than the initial white noise

The reason for this inefficiency of global transformdomain filtering is the same as for the inefficiency ofthe global transform compression which was discussed inSection 22 in global image spectra low intensity imagecomponents are hidden behind wideband noise spectralcomponents And the solution of this problem is the samereplace global filtering by local fragment wise filtering Thisidea is implemented in local adaptive linear filters Beingapplied locally filters are becoming locally adaptive as theirfrequency responses are determined according to (1)ndash(4) bylocal spectra of image fragments within the filter windowLocal adaptive linear filters date back to mid-1980s and wererefined in subsequent years [16 18 26 55] As a theoreticalbase for the design of local adaptive filters local criteria ofprocessing quality were suggested [56]

It is instructive to note that this is not an accident that theevolution of human vision came up with a similar solution Itis well known that when viewing image human eyersquos opticalaxis permanently hops chaotically over the field of view andthat the human visual acuity is very nonuniformover the fieldof view The field of view of a man is about 30∘ Resolvingpower of manrsquos vision is about 11015840 However such a relativelyhigh resolving power is concentrated only within a smallfraction of the field of view that has size of about 2∘ (see forinstance [57]) that is the area of the acute vision is about115th of the field of view For images of 512 times 512 pixels thismeans window of roughly 35 times 35 pixels

Themost straightforwardway to implement local filteringis to do it in a hopping window just as human vision doesand this is exactly the way of processing implemented inthe transform image coding methods However ldquohopping

windowrdquo processing being very attractive from the com-putational complexity point of view suffers from ldquoblockingeffectsrdquo artificial discontinuities that may appear in the resultof processing at the edges of the hopping window Blockingeffects are artifacts characteristic for transform image codingThe desire to avoid them motivated the advancement ofldquoLappedrdquo transforms [58] which are applied blockwise withthe half of the block size overlap Obviously the ultimatesolution of the ldquoblocking effectsrdquo problem would be pixel bypixel processing in sliding window that scans image row-wisecolumnwise

Thus local adaptive linear filters work in a transformdomain in a sliding window and at each position of thewindow modify according to the type of the filter definedby (1)ndash(4) transform coefficients of the image fragmentwithin the window and then compute an estimate of thewindow central pixel bymeans of the inverse transform of themodified transform coefficients As an option accumulationof estimates of the all window pixels overlapping in theprocess of image scanning by the filter window is possibleas well [59] As for the transform for local adaptive filteringDCT proved to be the best choice in most cases Figures8 and 9 give examples of local adaptive filtering for imagedeblurring and enhancement

Local adaptive filters can work with color and multicom-ponent images and videos as well In this case filtering iscarried out in the correspondingmultidimensional transformdomains for instance domains of 3D (two spatial and onecolor component coordinates) spectra of color images or3D spectra of a sequence of video frames (two spatial andtime coordinates) In the latter case filter 3D window scansvideo frame sequence in both spatial and time coordinatesAs one can see from Figures 10 and 11 the availability ofthe additional third dimension substantially adds to the filterefficiency [18 55 60]


Test image

(a)

Noisy test image PSNR=10(noise St Dev = 25)

(b)

Filtered image

(c)

Difference between original test and filtered imagesStandard deviation 108

(d)

Figure 7 Denoising of a piece-wise test image using empirical Wiener filter applied globally to the entire image

Certain further improvement of image denoising capa-bility can be achieved if for each image fragment in a givenposition of the sliding window similar according to somecriterion image fragments over the whole image frame arefound and included in the stack of fragments to which the3D transform domain filtering is appliedThis approach in itssimplest formwas coined the name ldquononlocal meansrdquo filtering[61ndash63] In fact similarmethod has beenmuch earlier knownas the correlational averaging (see for instance [64]) Itassumes finding similar image fragments by their cross-correlation coefficient with a template object and averaging ofsimilar fragments that is leaving only the dc component ofthe 3D spectra in this dimension In [65] the simple averagingwas replaced by full scale 3D transform domain filteringOne should however take into account that the correlationalaveraging is prone to produce artifacts due to false imagefragments that may be erroneously taken to the stack justbecause of the presence of noise that has to be cleaned [66]

Obviously image restoration efficiency of the slidingwindow transform domain filtering will be higher if thewindow size is selected adaptively at each window positionTo this goal filtering can be for instance carried out inwindows of multiple sizes and at each window position thebest filtering result should be taken as the signal estimate atthis position using methods of statistical tests for instanceldquointersection of confidence intervalsmethodrdquo described in [67]Another option inmultiple window processing is combiningin a certain way (say by weighted summation or by taking amedian or in some other way) filtering results obtained fordifferent windows All this however increases correspond-ingly the computational complexity of the filtering whichmay become prohibitive especially in video processing

In conclusion of this section note that DFT and DCTspectra of image fragments in sliding window form theso called ldquotime-frequency representationrdquo of signals whichcan be traced back to 1930s-1940s to Gabor and Wigner


Before

(a)

After

(b)

Figure 8 Denoising and deblurring of a satellite image by means of local adaptive filtering Top row raw image and its fragments magnifiedfor better viewing (marked by arrows) bottom row resulting image and its corresponding magnified fragments

Before

(a)

After

(b)

Figure 9 Enhancement of an astronomical image by means of 119875th law modification of its spectrum


(a) (b) (c)

Figure 10 2D and 3D local adaptive filtering of a simulated video sequence (a) one of noisy frames of a test video sequence (image size256 times 256 pixels) (b) a result of 2D local adaptive empirical Wiener filtering (filter window 5 times 5 pixels) (c) a result of 3D local adaptiveempirical Wiener filtering (filter window 5 times 5 pixels times 5 frames)

(a) (b)

Figure 11 3D local adaptive empiricalWiener filtering for denoising and deblurring of a real thermal video sequence (a) a frame of the initialvideo sequence (b) a frame of a restored video sequence Filter window is (5 times 5 pixels) times 5 frames image size is 512 times 512 pixels

and works on ldquovisible speechrdquo [68ndash70] For 1D signals thedimensionality of the representation is 2D for images it is3D Such representation is very redundant and this opensan interesting option of applying to it image (for 1D signals)and video (for 2D signals) processing methods for signaldenoising and in general signal separation

32 Wavelet Shrinkage Filters for Image Denoising In 1990sa specific family of transform domain denoising filters theso-called wavelet shrinkage filters gained popularity afterpublications [53 54 71 72] The filters work in the domainof one of wavelet transforms and implement for imagedenoising the above-mentioned Signal Spectrum Preserva-tion Filter (2) and Rejecting Filter (3) except that they donot include the ldquoinverse filteringrdquo component (1ISFR) thatcorrects distortions of image spectra in imaging systemsAs it was already mentioned the filters were coined in

these publications the names ldquosoft thresholdingrdquo and ldquohardthresholdingrdquo filters correspondingly

Basis functions of wavelet transforms (wavelets) areformed by means of a combination of two methods of build-ing transform basis functions from one ldquomotherrdquo functionshifting and scaling Thanks to this wavelets combine localsensitivity of shift basis functions and global properties ofldquoscaledrdquo basis functions and feature such attractive propertiesas multiresolution and good localization in both signal andtransform domain Thus the wavelet shrinkage filters beingapplied to entire image frames possess both global adaptivityand local adaptivity in different scales and do not require forthe latter specification of the window size which is necessaryfor the sliding window filters

These filters did show a good denoising capability How-ever a comprehensive comparison of their denoising capabil-ity with that of sliding window DCT domain filters reported


in [59] showed that even when for particular test images thebest wavelet transform from a transform set [73] was selectedsliding window DCT domain filters demonstrated in mostcases better image denoising results

The capability of wavelets to represent images in differentscales can be exploited for improving the image denoisingperformance of both families of filters and for overcomingthe above-mentionedmain drawback of slidingwindowDCTdomain filters the fixed window size that might not beoptimal for different image fragments This can be achievedby means of incorporating sliding window DCT domain(SWTD) filters into the wavelet filtering structure throughreplacing softhard thresholding of image wavelet decompo-sition components in different scales by their filtering withSWTD filters working in the window of the smallest size 3times3pixels This idea was implemented in hybrid waveletSWTDfilters and proved to be fruitful in ([74] see also [55])

33 Local Adaptive Filtering and Wavelet Shrinkage Filteringas Processing of Image Subband Decompositions Obviouslysliding window transformdomain andwavelet processing arejust different implementation of the scalar linear filtering intransformdomainThismotivates their unified interpretationin order to gain a deeper insight into their similarities and dis-similaritiesThe common base for such unified interpretationis the notion of signal subband decomposition [75] One canshow that both filter families can be treated as special casesof signal subband decomposition by band-pass filters withpoint spreads functions defined by the corresponding basisfunctions of the transform [26 55] From the point of view ofsignal subband decomposition the main difference betweenslidingwindow transformdomain andwavelet signal analysisis arrangement of bands in the signal frequency range Whilefor sliding window transform domain filtering subbands areuniformly arranged with the signal base band subbandsof the wavelet filters are arranged in a logarithmic scaleHybrid waveletSWDCT filtering combines these two typesof subband arrangements ldquocoarserdquo subband decompositionin logarithmic scale provided by wavelet decomposition iscomplemented with ldquofinerdquo uniformly arranged sub-subbandswithin eachwavelet subband provided by the sliding windowDCT filtering of the wavelet subbands

It is curious to note that this ldquologarithmic coarsemdashuniform finerdquo subband arrangement resembles very muchthe arrangements of tones and semitones in music In Bachrsquosequal tempered scale octaves are arranged in a logarithmicscale and 12 semitones are equally spaced within octaves [76]

4 Image Resampling and BuildinglsquolsquoContinuousrsquorsquo Image Models

As it was indicated in the introductory section DiscreteFourier and Discrete Cosine Transforms occupy the uniqueposition among other orthogonal transforms They are twoversions of discrete representation of the integral FourierTransform the fundamental transform for describing thephysical reality Among applications specific for DFT andDCT are signal and image spectral analysis and analysis of

periodicities fast signal and image convolution and correla-tion and image resampling and building ldquocontinuousrdquo imagemodels [17 18 26]The latter is associated with the efficiencywith which DFT and DCT can be utilized for fast signalconvolution

Image resampling is a key operation in solving manydigital image processing tasks It assumes reconstructionof an approximation of the original nonsampled image bymeans of interpolation of available image samples to obtainsamples ldquoin-betweenrdquo the available ones The most feasibleis interpolation by means of digital filtering implemented asdigital convolution A number of convolutional interpolationmethods are known beginning from the simplest and theleast accurate nearest neighbor and linear (bilinear for 2Dcase) interpolations to more accurate cubic (bicubic for 2Dcase) and higher order splinemethods [77 78] In some appli-cations for instance in computer graphics and print art thesimplest nearest neighbor or linear (bilinear) interpolationmethods can provide satisfactory results In applications thatare more demanding in terms of the interpolation accuracyhigher order spline interpolation methods gained popularityThe interpolation accuracy of spline interpolation methodsgrows with the spline order However their computationalcomplexity grows as well

There exists a discrete signal interpolation method that iscapable given finite set of signal samples to secure virtuallyerror free interpolation of sampled signals that is themethodthat does not add to the signal any distortions additional tothe distortions caused by the signal samplingThis method isthe discrete sinc-interpolation [18 25] Interpolation kernel forthe discrete sinc-interpolation of sampled signals of 119873 sam-ples is the discrete sinc-function sincd119909 = sin119909119873 sin(119909119873)This function replaces for sampled signals with a finitenumber of samples the continuous sinc-function sinc119909 =

sin119909119909 which according to the sampling theory is the idealinterpolation kernel for reconstruction of continuous signalsfrom their samples provided that the number of samples isinfinitely large Note that the discrete sinc-function shouldnot be confused with a windowed or truncated sinc-functionwith a finite number of samples which is very commonlybelieved to be a good replacement in numerical interpolationof the continuous sinc-functionThewindowed sinc-functiondoes not secure interpolation error free signal numericalinterpolation whereas the discrete sinc-interpolation does

Fast Fourier transform and fast Discrete Cosine trans-form are the ideal computational means for implementingthe convolutional interpolation by the discrete sinc-functionOf these two transforms FFT has a substantial drawbackit implements the cyclic convolution with a period equal tothe number 119873 of signal samples which is potentially proneto causing heavy oscillations at signal borders because of adiscontinuity which is very probable between signal valuesat its opposite borders This drawback is to a very highdegree surmounted when signal convolution is implementedthrough DCT [18 26] As it was already indicated convolu-tion through DCT is also a cyclic convolution but with theperiod of 2119873 samples and of signals that are evenly extendedto double length by their inversed in the order of samplescopies This completely removes the danger of appearance


of heavy oscillations due to discontinuities at signal bordersthat are characteristic for discrete sinc-interpolation throughprocessing in DFT domain

Several methods of implementation of DFTDCT baseddiscrete sinc-interpolation The most straightforward one isDFTDCT spectrum zero padding Interpolated signal is gen-erated by applying inverse DFT (or correspondingly DCT)transform to its zero pad spectrum Padding DFTDCTspectra of signals of119873 samples with 119871(119873minus1) zeroes produces119871119873 samples of the discrete sinc interpolated initial signal withsubsampling rate 119871 that is with intersample distance 1119871 ofthe initial sampling interval The computational complexityof this method is 119874(log 119871119873) per output sample

The same discrete sinc interpolated 119871 times subsampledsignal can be obtained more efficiently computation-wiseby applying to the signal 119871 times the signal perfect shiftingfilter [25 79] with shifts by 119896119871 at each 119896th applicationand by subsequent combining the shifted signal copies Thecomputational complexity of this method is 119874(log119873) peroutput sample

The method is based on the property of Shifted DFT(SDFT [18 26 80]) if one computes SDFT of a signalwith some value of the shift parameter in signal domainand then computes inverse SDFT with another value of theshift parameter the result will be a discrete sinc interpolatedcopy of the signal shifted by the difference between thosevalues of the shift parameter In order to avoid objectionableoscillations at signal borders the DCT based version of thisfilter is recommended [81ndash83]

Image subsampling using the perfect shifting filter canbe employed for creating ldquocontinuousrdquo image models forsubsequent image arbitrary resampling with any given accu-racy [18 84] It can also be used for computing imagecorrelations and image spectral analysis with a subpixelaccuracy [18 26] Particular examples of using created in thisway continuous signal models for converting image spectrafrom polar coordinate system to Cartesian coordinate systemin the direct Fourier method for image reconstruction fromprojections and for converting (rebinning) of image fan beamprojections into parallel projections one can find in [18]

Except for creating ldquocontinuousrdquo image models the per-fect shifting filter is very well suited for image sheering in thethree-step method for fast image rotation by means of threesubsequent (horizontalverticalhorizontal) image sheerings[85]

Perfect interpolation capability of the discrete sinc-interpolation was demonstrated in a comprehensive com-parison of different interpolation methods in experimentswith multiple 360∘ rotations reported in [25] The resultsof experiments illustrated in Figure 12 clearly evidence thatin contrast to other methods including high order splineinterpolation ones [78 86 87] discrete sinc-interpolationdoes not introduce any appreciable distortions into therotated image

In some applications ldquoelasticrdquo or space variant imageresampling is required when shifts of pixel positions arespecified individually for each image pixel In these casesthe perfect shifting filter can be applied to image fragmentsin sliding window for evaluating interpolated value of the

window central pixel at each window position Typicalapplication examples are imitation of image retrieval throughturbulent media (for illustration see [88]) and stabilizationand superresolution of turbulent videos [89ndash91]

Being working in DCT transform domain ldquoelasticrdquoresampling in sliding window can be easily combined withabove-described local adaptive denoising and deblurring[25] Yet another additional option is making resamplingin sliding window adaptive to contents of individual imagefragments bymeans of switching between the perfect shiftingand another interpolation method better suited for specificimage fragments This might be useful in application toimages that contain very heterogeneous fragments such asphotographs and binary drawings and text [25]

A certain limitation of the perfect shifting filter in creatingldquocontinuousrdquo image models is its capability of subsamplingimages only with a rate expressed by an integer or a rationalnumber In some cases this might be inconvenient forinstance when the required resampling rate is a valuebetween one and two say 11 12 or alike For such casesthere exists the third method of signal resampling withdiscrete sinc-interpolation It is based on the general ShiftedScaled DFT which includes arbitrary analog shift and scaleparameters Using Shifted Scaled (ShSc) DFT one can applyto the imageDFT spectrum inverse ShScDFTwith the desiredscale parameter and obtain a correspondingly scaled discretesinc interpolated image [82 83] Being the discrete sinc-interpolation method this method also does not cause anyimage blurring which other interpolationmethods are proneto do Figure 13 illustrates this on an example of multipleiterative image zooming-in and zooming-out

The Shifted Scaled DFT can be presented as a convolutionand as such can be computed using Fast Fourier or FastCosine Transforms [18 26] Thus the computational com-plexity of the method is119874(120590 log119873) per output signal samplewhere 120590 is the scale parameter As in all other cases the use ofDCT guaranties the absence of severe boundary effects Thismethod is especially well suited for numerical reconstructionof digitally recorded color holograms when it is required torescale images reconstructed from holograms recorded withdifferent wave lengths of coherent illumination [82]

Among the image processing tasks which involve ldquocon-tinuousrdquo image models are also signal differentiation andintegration the fundamental tasks of numerical mathematicsthat date back to such classics of mathematics as Newtonand Leibnitz One can find standard numerical differentiationand integration recipes in numerous reference books forinstance [92 93] All of them are based on signal approxi-mation through Taylor expansion However according to thesampling theory approximation of sampled signal by Taylorexpansion is not optimal and the direct use of these methodsfor sampled signals may cause substantial errors in caseswhen signals contain substantial high frequency componentsIn order to secure sufficiently accurate signal differentiationand integration by means of those standard algorithms onemust substantially oversample such signals

The exact solution for the discrete representation ofsignal differentiation and integration provided by the sam-pling theory tells that given the signal sampling interval


Test image

(a)

Nearest neighbor interpolation(rotation by 1080

∘ in 60 steps)

(b)

Bilinear interpolation(rotation by 1080

∘ in 60 steps)

(c)

(rotation by 1080∘ in 60 steps)

Bicubic interpolation

(d)

Spline interpolation Mems531 ([78] [87])(rotation by 18000

∘ in 1000 steps)

(e)


Discrete sinc-interpolation

(f)

Figure 12 Discrete sinc-interpolation versus other interpolation methods results of multiple image rotations


[original image bilinear intrp bicubic intrp discr-sincintrp] Nit = 75

Figure 13 Discrete sinc-interpolation versus bilinear and bicubic interpolations in image iterative zoom-inzoom-out with the scaleparameterradic2

and signal sampling and reconstruction devices discretefrequency responses (in DFT domain) of digital filters forperfect differentiation and integrations should be corre-spondingly proportional and inversely proportional to thefrequency index [18 26] This result directly leads to fastdifferentiation and integration algorithms that work in DFTor DCT domains using corresponding fast transforms withcomputational complexity 119874(log119873) operation per signalsample for signals of119873 samples As in all cases realization ofthe integration and especially differentiation filters in DCTdomain is preferable because of much lower for DCT bordereffect artifacts associated with the finite number of signalsamples

The comprehensive comparison of the accuracy of stan-dard numerical differentiation and integration algorithmswith perfect DCT-based differentiation and integrationreported in [25 94] shows that the standard numericalalgorithms in reality work not with original signals but withtheir blurred to a certain degree copies This conclusion isillustrated in Figure 14 on an example of multiple alternativedifferentiations and integrations of a test rectangular impulse

Computational efficiency of the DFTDCT based inter-polation error free discrete sinc-interpolation algorithms isrooted in the use of fast Fourier and Fast DCT transformsPerhaps the best concluding remark for this discussion ofthe discrete sinc-interpolation DFTDCT domain methodswould be mentioning that a version of what we call now FastFourier Transform algorithm was invented more than 200hundred years ago byGauss just for the purpose of facilitatingnumerical interpolation of sampled data of astronomicalobservation [95]

5 Image Recovery from Sparse andIrregularly Sampled Data

51 Discrete Sampling Theorem Based Method As we men-tioned at the beginning of Section 2 image discretization isusually carried out by image sampling at nodes of a uniformrectangular sampling lattice In this section we address usingfast transforms for solving the problem closely associatedwith the general compressive discretization how and with


50 100 150 2000

02

04

06

08

1

DCT diffintegr 100 iterations

Sample index

Initial signalIterated dif-integr signal

(a)

D2trapezoidal differintegr 100 iterations

50 100 150 2000

02

04

06

08

1

Sample index


(b)

Figure 14 Comparison of results of iterative alternated 100 differentiations and integrations of a test rectangular signal using the DCT-basedalgorithms (left) and standard numerical algorithms (right differentiation filter with point spread function [minus05 0 05] and trapezoidal ruleintegration algorithm)

what accuracy can one recover an image from its sparse andirregularly taken samples

There are many applications where contrary to thecommon practice of uniform sampling sampled data arecollected in irregular fashion Because image display devicesas well as computer software for processing sampling dataassume using regular uniform rectangular sampling latticeone needs in all these cases to convert irregularly sampledimages to regularly sampled ones

Generally the corresponding regular sampling grid maycontainmore samples than it is available because coordinatesof positions of available samples might be known with aldquosubpixelrdquo accuracy that is with the accuracy (in units ofimage size) better than 1119870 where 119870 is the number ofavailable pixelsTherefore one can regard available119870 samplesas being sparsely placed at nodes of a denser sampling latticewith the total amount of nodes119873 gt 119870

The general framework for recovery of discrete signalsfrom a given set of their arbitrarily taken samples can beformulated as an approximation task in the assumption thatcontinuous signals are represented in computers by their119870 lt 119873 irregularly taken samples and it is believed thatif all 119873 samples in a regular sampling lattice were knownthey would be sufficient for representing those continuoussignals [18 96] The goal of the processing is generatingout of an incomplete set of 119870 samples a complete set of119873 signal samples in such a way as to secure the mostaccurate in terms of the reconstruction mean square error(MSE) approximation of the discrete signal which would beobtained if the continuous signal it is intended to representwere densely sampled at all119873 positions

Above-described discrete sinc-interpolation methodsprovide band-limited in terms of signal Fourier spectraapproximation of regularly sampled signals One can alsothink of signal band-limited approximation in terms of theirspectra in other transforms This approach is based on theDiscrete Sampling Theorem [18 96 97]

The meaning of the Discrete Sampling Theorem is verysimple Given 119870 samples of a signal one can in principlecompute 119870 certain coefficients of a certain signal transformand then reconstruct 119873 samples of a band-limited in thistransform approximation of the signal by means of inversetransform of those 119870 coefficients supplemented with 119873-119870 zero coefficients If the transform has the highest forthis particular type of signals energy compaction capabilityand selected were those nonzero transform coefficients thatconcentrate the highest for this particular transform signalenergy the obtained signal approximation will have the leastmean square approximation error If the signal is known tobe band-limited in the selected transform and the computednonzero coefficients corresponded to this band limitationsignal will be reconstructed precisely

The discrete sampling theorem is applicable to signalsof any dimensionality though the formulation of the signalband-limitedness depends on the signal dimensionality For2D images and such transforms as Discrete Fourier DiscreteCosine and Walsh Transforms the most simple is compactldquolow-passrdquo band-limitedness by a rectangle or by a circlesector

It was shown in ([96] see also [18 97]) that for 1Dsignal and such transforms as DFT and DCT signal recoveryfrom sparse samples is possible for arbitrary positions ofsparse samples For images signals the same is true if band-limitation conditions are separable over the image dimen-sions For nonseparable band-limitations such as circle orcircle sector this may not be true and certain redundancy inthe number of available samples might be required to secureexact recovery of band-limited images

As it was indicated the choice of the transform mustbe made on the base of the transform energy compactioncapability for each particular class of images to which theimage to be recovered is believed to belong The type of theband-limitation also must be based on a priori knowledgeregarding the class of images at handThe number of samplesto be recovered is a matter of a priori belief of how many


samples of a regular uniform sampling lattice would besufficient to represent the images for the end user

Implementation of signal recoveryapproximation fromsparse nonuniformly sampled data according to the dis-crete sampling theorem requires matrix inversion whichis generally a very computationally demanding procedureIn applications in which one can be satisfied with imagereconstruction with a certain limited accuracy one canapply to the reconstruction a simple iterative reconstructionalgorithm of the Gerchberg-Papoulis [98 99] type whichat each iteration step alternatively applies band limitationin spectral domain and then restores available pixels intheir given positions in the image obtained by the inversetransform

Among reported applications one can mention super-resolution from multiple video frames of turbulent videoand superresolution in computed tomography that makesuse of redundancy of object slice images in which usually asubstantial part of their area is an empty space surroundingthe object [96]

52 ldquoCompressive Sensingrdquo Promises and Limitations Thedescribed discrete sampling theorem based methods forimage recovery from sparse samples by means of theirband-limited approximation in a certain transform domainrequire explicit formulation of the desired band limitationin the selected transform domain While for 1D signal thisformulation is quite simple and requires for most frequentlyused low pass band limitation specification of only oneparameter signal bandwidth in 2D case formulation of thesignal band limitation requires specification of a 2D shape ofsignal band-limited spectrumThe simplest shapes rectangleand circle sector ones may only approximate with a certainredundancy real areas occupied by the most intensive imagespectral coefficients for particular images Figure 15 illustratesthis on an example of spectral binary mask that correspondsto 95 of the DCT spectral coefficients of the test imageshown in Figure 3 that contain 95 of the spectrum energyand a rectangular outline of this mask which includes 47 ofthe spectral coefficients

In the cases when exact character of spectrum bandlimitation is not known image recovery from sparse samplescan be achieved using the ldquocompressive sensingrdquo approachintroduced in [29ndash32] During last recent years this approachto handling sparse spectral image representation has obtainedconsiderable popularity [33ndash36]

The compressive sensing approach assumes obtaininga band-limited in a certain selected transform domainapproximation of images as well but it does not requireexplicit formulation of the image band-limitation andachieves image recovery from an incomplete set of samples bymeans of minimization of 1198710 norm of the image spectrum inthe selected transform (ie of the number of signal nonzerotransform coefficients) conditioned by preservation in therecovered image of its available pixels

However there is a price one should pay for the uncer-tainty regarding the band limitation the number 119872 ofsamples required for recovering 119873 signal samples by this

Spectrum mask spectrum sparsity 0095

Spectrum sparsity for the rectangle 047

Figure 15 Spectral binary mask (shown white with the dc compo-nent in the left upper corner) that indicates components of DCTspectrum of the image in Figure 3 which contain 95 of the totalspectrum energy and its rectangular outline

approach must be redundant with respect to the givennumber 119870 nonzero spectral coefficients 119872 = 119870 log119873[30] In real applications this might be a serious limitationFor instance for the test image of 256 times 256 pixels shownin Figure 3 spectrum sparsity (relative number of nonzerospectral components) on the energy level 95 is 0095whereas log119873 = log(256 times 256) = 16 which means thatthe ldquocompressive sensingrdquo approach requires in this casemoresignal samples (16 sdot 95 = 152) than it is required torecover

53 Discrete Signal Band-Limitedness and the Discrete Uncer-tainty Principle Signal band-limitedness plays an importantrole in dealing with both continuous signals and discrete(sampled) signals representing them It is well known thatcontinuous signals cannot be both strictly band-limited andhave strictly bounded support In fact continuous signalsneither are band-limited nor have strictly bounded supportThey can only be more or less densely concentrated insignal and spectral domains This property is mathematicallyformulated in the form of the ldquouncertainty principlerdquo [100]

119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864

are intervals in signal and their Fourierspectral domains that contain a sufficiently large fraction 120576119864

of signal energyIn distinction to that sampled signals that represent

continuous signals can be sharply bounded in both sig-nal and spectral domains This is quite obvious for somesignal spectral presentation such as Haar signal spectraHaar basis functions are examples of sampled functionssharply bounded in both signal and Haar spectral domainsBut it turns out that there exist signals that are sharplybounded in both signal and their spectral domains of DFT


(a) (b)

Figure 16 Space-limited image ldquoC Shannonrdquo (left) and its band-limited DFT spectrum (right shown centered at the signal dc component)

or DCT which are discrete representations of the Fourierintegral transform An example of a space-limitedband-limited image is shown in Figure 16 Such images can begenerated using the above-mentioned iterative Gershberg-Papoulis type algorithm which at each iteration applies toimages alternatively space limitation in image domain andband limitation in transform domain Such images are veryuseful as test images for testing image processing algorithms

In a similar way one can generate space-limited and band-limited analogs of the discrete sinc-function ie functionswhich form band-limited shift bases in the given spacelimits In [97] such functions were coined a name ldquosinc-letsrdquoFigure 17 shows an example of a sinc-let in its three positionswithin a limited interval and for comparison the discretesinc-function for the same band limitation

The following relationship between signal bounds insignal and DFT spectral domains can be derived

119873sign times 119873spect lt 119873 (6)

where 119873sign is the number of signal nonzero samples 119873spectis the number of its nonzero spectral samples and 119873 is thenumber of samples in the signal sampling latticeThis is whatone can call the discrete uncertainty principle

6 Conclusion

Two fundamental features of fast transforms the energycompaction capability and fast algorithms for transformcomputation make them perfect tool for image compres-sion restoration reconstruction and resampling Of all fasttransforms Discrete Fourier Transform and Discrete CosineTransform are themost important as they are complementingeach other discrete representations of the integral Fouriertransform one of the most fundamental mathematical mod-els for describing physical reality and in addition they enablefast digital convolution From these two transforms DCTis preferable in most applications thanks to its ability to

very substantially reduce processing artifacts associated withimage discontinuities at the borders

Modern tendency in the imaging engineering is compu-tational imaging Computer processing of sensor data enablessubstantial price reduction and sometimes even completeremoval of imaging optics and similar imaging hardwareIt also gives birth to numerous new imaging techniques inastrophysics biology industrial engineering remote sensingand other applications No doubts this area promises manynew achievements in the coming years And it is certain thatfast transformswill remain to be the indispensable tool in thisprocess

Nomenclature

119886119896 Image samples

120572119903 Image spectral coefficients

119896 = 0 119873 minus 1119903 = 0 119873 minus 1 pixel and correspondingly spectra

coefficient vertical and horizontalindices

119862(sdot) Various constants

Δ119909 Image sampling intervalsΔ119891 Sampling interval in the transform

domain119873 Total number of image samples120582 Wavelength of coherent radiation119885 Distance between object plane and

hologram

Abbreviations

DCT Discrete Cosine TransformDFT Discrete Fourier TransformDFrT Discrete Fresnel TransformPSNR Peak Signal-to-Noise Ratio ratio of signal

dynamic range to noise standard deviation


0

02

04

06

08

1Discrete sinc-function

Sinclet

Signalsupportinterval

Sample index200 250 300 350

Signal Nit = 1000 N= 512 Slim = 103

minus02

(a)

0

02

04

06

08

1

Spectrumof the sinclet

Bandwidth

220 240 260 280 300

fftshift (frequency index)

Spectrum Nit = 1000 N= 512 BlimDFT = 51

(b)


0

02

04

06

08

1



minus02

Discrete sinc-function

(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)

Figure 17 Examples of a ldquosinc-letrdquo (red plots) and for comparison of the discrete sinc-function for the same band limitation (blue plots) intheir three different positions within the interval of 103 samples of 512 samples (boxes andashc) Plots of their DFT spectra are shown in boxes(dndashf)


RMSE root mean square errorSNR Signal-to-Noise Ratio (ratio of signal and

noise variances)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J W Cooley and J W Tukey ldquoAn Algorithm for the MachineCalculation of Complex Fourier Seriesrdquo Mathematics of Com-putation vol 19 no 90 pp 297ndash301 1965

[2] H F Harmuth Transmission of Information by OrthogonalFunctions Springer Berlin Germany 1970

[3] H C Andrews Computer Techniques in Image ProcessingAcademic Press New York NY USA 1970

[4] H C Andrews ldquoTwo-dimensional Transformsrdquo in PictureProcessing and Digital Filtering T S Huang Ed Springer NewYork NY USA 1975

[5] N Ahmed and K R Rao Orthogonal Transforms for DigitalSignal Processing Springer Berlin Germany 1975

[6] W K Pratt W-H Chen and L R Welch ldquoSlant TransformImage Codingrdquo IEEE Transactions on Communications vol 22no 8 pp 1075ndash1093 1974

[7] N Ahmed N Natarajan and K R Rao ldquoDiscrete cosinetransformrdquo IEEE Transactions on Computers vol 23 no 1 pp90ndash93 1974

[8] K R Rao and P Yip Discrete Cosine Transform AlgorithmsAdvantages Applications Academic Press Boston Mass USA1990

[9] I Daubechies ldquoWhere do wavelets come from A personalpoint of viewrdquo Proceedings of the IEEE vol 84 no 4 pp 510ndash513 1996

[10] S Mallat A Wavelet Tour of Signal Processing Academic PressBoston Mass USA 3rd edition 2008

[11] H Karhunen Uber lineare Methoden in der Wahrscheinlichkeit-srechnung Series AI Annales AcademiaeligsimScientiarum Fen-nicaelig 1947

[12] M Loeve ldquoFonctions aleatoires de seconde ordrerdquo in ProcessesStochastiques et Movement Brownien P Levy Ed HermannParis France 1948

[13] H Hotelling ldquoAnalysis of a complex of statistical variables intoprincipal componentsrdquo Journal of Educational Psychology vol24 no 6 pp 417ndash441 1933

[14] T P Belokova M A Kronrod P A Chochia and LP Yaroslavakii ldquoDigital processing of martian surface pho-tographs frommars 4 andmars 5rdquo Space Research vol 13 no 6pp 898ndash906 1975

[15] L P Yaroslavsky Digital Picture Processing An IntroductionSpringer Berlin Germany 1985

[16] L Yaroslavsky Digital Signal Processing in Optics and Hologra-phy Radio i Svyazrsquo Moscow Russia 1987 (Russian)

[17] L YaroslavskyDigital Holography and Digital Image ProcessingPrinciples Methods Algorithms Kluwer Academic PublishersBoston Mass USA 2004

[18] L YaroslavskyTheoretical Foundations of Digital Imaging UsingMatlab Taylor amp Francis Boca Raton Fla USA 2013

[19] L Yaroslavsky ldquoIntroduction to Digital Holographyrdquo in DigitalSignal Processing in Experimental Research L Yaroslavsky and JAstola Eds vol 1 of Bentham Series of e-books Bentham 2009httpwwwbenthamsciencecomebooks9781608050796ind-exhtm

[20] L P Yaroslavskii and N S Merzlyakov Methods of DigitalHolography Consultance Bureau New York NY USA 1980

[21] L P Jaroslavski ldquoComments on FFT algorithm for both inputand output pruningrdquo IEEE Transactions on Acoustics Speechand Signal Processing vol 29 no 3 pp 448ndash449 1981

[22] J Markel ldquoFFT pruningrdquo IEEE Trans Audio Electroacoust vol19 no 4 pp 305ndash311 1971

[23] H V Sorensen and C S Burrus ldquoEfficient computation ofthe DFT with only a subset of input or output pointsrdquo IEEETransactions on Signal Processing vol 41 no 3 pp 1184ndash12001993

[24] R G Alves P L Osorio and M N S Swamy ldquoGeneral FFTpruning algorithmrdquo in Proceedings of the 43rd IEEE MidwestSymposium onCircuits and Systems vol 3 pp 1192ndash1195 August2000

[25] L Yaroslavsky ldquoFast discrete sinc-interpolation a gold standardfor image resamplingrdquo in Advances in Signal transforms Theoryand Applications J Astola and L Yaroslavsky Eds vol 7 ofEurasip Book Series on Signal Processing and Communicationspp 337ndash405 2007

[26] L Yaroslavky ldquoFast transforms in digital signal processingtheory algorithms applicationsrdquo in Digital Signal Processing inExperimental Research L Yaroslavsky and J Astola Eds vol2 of Bentham E-book Series 2011 httpwwwbenthamsciencecomebooks9781608052301indexhtm

[27] R Yu Vitkus and L P Yaroslavsky ldquoRecursive Algorithms forLocal Adaptive Linear Filtrationrdquo in Mathematical ResearchComputer Analysis of Images and Patterns L P Yaroslavsky ARosenfeld andWWilhelmi Eds pp 34ndash39 Academie VerlagBerlin Germany 1987 Band 40

[28] J Xi and J F Chicharo ldquoComputing running DCTs and DSTsbased on their second-order shift propertiesrdquo IEEETransactionson Circuits and Systems I FundamentalTheory andApplicationsvol 47 no 5 pp 779ndash783 2000

[29] E Candes ldquoCompressive samplingrdquo in Proceedings of theInternational Congress of Mathematicians Madrid Spain 2006

[30] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[31] E J Candes J K Romberg and T Tao ldquoStable signal recoveryfrom incomplete and inaccurate measurementsrdquo Communica-tions on Pure and Applied Mathematics vol 59 no 8 pp 1207ndash1223 2006

[32] E J Candes and M B Wakin ldquoAn introduction to compressivesampling A sensingsampling paradigm that goes against thecommon knowledge in data acquisitionrdquo IEEE Signal ProcessingMagazine vol 25 no 2 pp 21ndash30 2008

[33] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash124 2007

[34] M Elad Spatse and Redundant Representations FromTheory toApplications in Signal and Image Processing Springer NewYorkNY USA 2010

[35] R M Willet R F Mareia and J M Nichols ldquoCompressedsensing for practical optical imaging systems a tutorialrdquoOpticalEngineering vol 50 no 7 Article ID 072601 2011

[36] ldquoCompressive optical imaging architectures and algorithmsrdquoin Optical and Digital Image Processing Fundamentals and


Applications G Cristobal P Schelkens andHThienpont EdsWiley-VCH New York NY USA 2011

[37] J M Whittaker Interpolatory Function Theory CambridgeUniversity Press Cambridge UK 1935

[38] V A Kotelnikov ldquoon the carrying capacity of the ether andwire in telecommunicationsrdquo Material for the First All-UnionConference on Questions of Communication Izd Red UprSvyazi RKKA Moscow Russia 1933 (Russian)

[39] C Shannon ldquoA mathematical theory of communicationrdquo BellSystem Technical Journal vol 27 no 3 pp 379ndash423 623ndash6561948

[40] T S Huang and O J Tretiak Picture Bandwidth CompressionGordon and Breach New York NY USA 1972

[41] A Habibi and PWintz ldquoImage coding by linear transformationand block quantizationrdquo IEEE Transactions on Communica-tions vol 19 no 1 pp 50ndash62 1971

[42] T A Wintz ldquoTransform picture codingrdquo Proceedings of theIEEE vol 60 no 7 pp 809ndash823 1972

[43] W B Pennebaker and J L Mitchell JPEG Still Image DataCompression Standard Springer Berlin Germany 3rd edition1993

[44] I E G Richardson H264 and MPEG-4 Video CompressionJohn Wiley amp Sons Chichester UK 2003

[45] T Wiegand G J Sullivan G Bjoslashntegaard and A LuthraldquoOverview of the H264AVC video coding standardrdquo IEEETransactions on Circuits and Systems for Video Technology vol13 no 7 pp 560ndash576 2003

[46] P Schelkens A Skodras and T EbrahimiThe JPEG 2000 SuiteJohn Wiley amp Sons Chichester UK 2009

[47] I Pitas Digital Video and Television Ioannis Pitas 2013[48] D Vaisey and A Gersho ldquoVariable block-size image coding

acoustics speech and signal processingrdquo in Proceedings of theIEEE International Conference on (ICASSP rsquo87) vol 12 pp 1051ndash1054 1987

[49] W K Pratt ldquoGeneralized wiener filtering computation tech-niquesrdquo IEEE Transactions on Computers vol 21 pp 636ndash6411972

[50] H C Andrwes ldquoDigital computers and image processingrdquoEndeavour vol 31 no 113 1972

[51] N Wiener The Interpolation Extrapolation and Smoothing ofStationary Time Series Wiley New York NY USA 1949

[52] A N Kolmogorov ldquoSur lrsquointerpolation et extrapolation dessuites stationnairesrdquoComptes Rendus de lrsquoAcademie des Sciencesvol 208 pp 2043ndash2045 1939

[53] D L Donoho and J M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[54] D L Donoho ldquoDe-noising by soft-thresholdingrdquo IEEE Trans-actions on Information Theory vol 41 no 3 pp 613ndash627 1995

[55] L Yaroslavsky ldquoSpace variant and adaptive transform domainimage and video restoration methodsrdquo in Advances in Sig-nal Transforms Theory and Applications J Astola and LYaroslavsky Eds EURASIP Book Series on Signal Processingand Communications 2007

[56] L Yaroslavsky ldquoLocal criteria a unified approach to localadaptive linear and rank filtersonrdquo in Signal Recovery andSynthesis III Technical Digest Series 15 1989

[57] M D Levine Vision in Man and Machine McGraw-Hill NewYork NY USA 1985

[58] H SMalvar andD H Staelin ldquoLOT transform coding withoutblocking effectsrdquo IEEE Transactions on Acoustics Speech andSignal Processing vol v pp 553ndash559 1992

[59] R Oktem L Yaroslavsky K Egiazarian and J Astola ldquoTrans-form domain approaches for image denoisingrdquo Journal ofElectronic Imaging vol 11 no 2 pp 149ndash156 2002

[60] L P Yaroslavsky B Fishbain A Shteinman and S GepshteinldquoProcessing and fusion of thermal and video sequences forterrestrial long range observation systemsrdquo in Proceedings of the7th International Conference on Information Fusion (FUSIONrsquo04) pp 848ndash855 Stockholm Sweden July 2004

[61] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[62] A Buades B Coll and J-M Morel ldquoA non-local algorithm forimage denoisingrdquo in Proceedings of the IEEE Computer SocietyConference on Computer Vision and Pattern Recognition (CVPRrsquo05) pp 60ndash65 June 2005

[63] J M Morel and A Buades Capo Non Local Image Processinghttpwwwfarmanens-cachanfr6 JeanMichel Morelpdf

[64] S E Hecht and J J Vidal ldquogeneration of ECG prototype wave-forms by piecewise correlational averagingrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 2 no 5 pp415ndash420 1980

[65] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[66] L Yaroslavsky and M Eden ldquoCorrelational accumulation as amethod for signal restorationrdquo Signal Processing vol 39 no 1-2pp 89ndash106 1994

[67] V Katkovnik K Egiazarian and J Astola ldquoAdaptive varyingwindow methods in signal and image processingrdquo in Advancesin Signal Transforms Theory and Applications J Astola andL Yaroslavsky Eds vol 7 of Eurasip Book Series on SignalProcessing and Communications pp 241ndash284 2007

[68] D Gabor ldquoTheory of Communicationrdquo Journal of IEEE vol 93pp 429ndash457 1946

[69] E Wigner ldquoOn the quantum correction for thermodynamicequilibriumrdquo Physical Review vol 40 no 5 pp 749ndash759 1932

[70] R K Potter G Koppand and H C Green Visible Speech VanNostrand New York NY USA 1947

[71] D L Donoho ldquoNonlinear wavelet methods for recovery ofsignals densities and spectra from indirect and noisy datardquoProceedings of Symposia in Applied Mathematics AmericanMathematical Society vol 47 pp 173ndash205 1993

[72] R R Coifman and D L Donoho ldquoTranslation- Invariant De-Noisingrdquo in Wavelets and Statistics A Antoniadis Ed vol103 of Lecture Notes in Statistics pp 125ndash150 Springer BerlinGermany 1995

[73] ftpftpcisupennedupubeeromatlabPyrToolstargz[74] B Z Shaick L Ridel and L Yaroslavsky ldquoA hybrid transform

method for image denoisingrdquo in Proceedings of the 10th Euro-pean Signal Processing Conference (EUSIPCO rsquo00) M Gabboujand P Kuosmanen Eds Tampere Finland September 2000

[75] M Vetterli and J Kovacevic Wavelets and Sub-Band CodingPrentice Hall Englewood Cliffs NJ USA 1995

[76] J BackusTheAcoustical Foundations of Music Norton and CoNew York NY USA 1969

[77] M Unser A Aldroubi and M Eden ldquoPolynomial spline signalapproximations Filter design and asymptotic equivalence withShannons sampling theoremrdquo IEEE Transactions on Informa-tion Theory vol 38 no 1 pp 95ndash103 1992


[78] A Gotchev ldquoImage interpolation by optimized spline-basedkernelsrdquo in Advances in Signal transforms Theory and Applica-tions J Astola and L Yaroslavsky Eds vol 7 of Eurasip BookSeries on Signal Processing and Communications pp 285ndash3352007

[79] L P Yaroslavsky ldquoEfficient algorithm for discrete sine interpo-lationrdquo Applied Optics vol 36 no 2 pp 460ndash463 1997

[80] L P Yaroslavsky ldquoShifted discrete fourier transformsrdquo inDigitalSignal Processing V Cappellini and A G Constantinides Edspp 69ndash74 Avademic Press London UK 1980

[81] L Yaroslavsky ldquoBoundary effect free and adaptive discrete sig-nal sinc-interpolation algorithms for signal and image resam-plingrdquo Applied Optics vol 42 no 20 pp 4166ndash4175 2003

[82] L Bilevich and L Yaroslavsky ldquoFast DCT-based image convolu-tion algorithms and application to image resampling and holo-gram reconstructionrdquo in Real-Time Image and Video ProcessingProceedings of SPIE Brussels Belgium April 2010

[83] L Bilevich and L Yaroslavsky ldquoFast DCT-based algorithmfor signal and image accurate scalingrdquo in Image ProcessingAlgorithms and Systems XI vol 8655 of Proceedings of SPIEFebruary 2013

[84] L P Yaroslavsky ldquoLinking analog and digital image processingrdquoin Optical and Digital Image Processing Fundamentals andApplications G Cristobal P Schelkens andHThienpont Edspp 397ndash418 Wiley-VCH New York NY USA 2011

[85] M Unser P Thevenaz and L Yaroslavsky ldquoConvolution-basedinterpolation for fast high-quality rotation of imagesrdquo IEEETransactions on Image Processing vol 4 no 10 pp 1371ndash13811995

[86] P Thevenaz EPFLSTIIOALIB Bldg +BM-Ecublens 4137Station 17 CH-1015 Lausanne VD Switzerland httpbigwwwepflchthevenaz

[87] A Gotchev ldquoSpline and wavelet based techniques for signaland image processingrdquo Tampere University of TechnologyPublication 429 Tampere Finland TTY-Paino 2003

[88] httpwwwengtauacilsimyaroEtudesPDFMonaLisa-320x256CurvMirravi

[89] L Yaroslavsky B Fishbain G Shabat and I Ideses ldquoSuperreso-lution in turbulent videos making profit from damagerdquo OpticsLetters vol 32 no 21 pp 3038ndash3040 2007

[90] B Fishbain I A Ideses G Shabat B G Salomon and L PYaroslavsky ldquoSuperresolution in color videos acquired throughturbulentmediardquoOptics Letters vol 34 no 5 pp 587ndash589 2009

[91] B Fishbain L P Yaroslavsky and I A Ideses ldquoReal-timestabilization of long range observation system turbulent videordquoJournal of Real-Time Image Processing vol 2 no 1 pp 11ndash222007

[92] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes the Art of Scientific Computing CambridgeUniversity Press Cambridge UK 1987

[93] J H Mathew and K D Fink Numerical Methods UsingMATLAB Prentice-Hall Englewood Cliffs NJ USA 1999

[94] L P Yaroslavsky A Moreno and J Campos ldquoFrequencyresponses and resolving power of numerical integration ofsampled datardquoOptics Express vol 13 no 8 pp 2892ndash2905 2005

[95] C F Gauss ldquoNachclass theoria interpolationis methodo novatractatardquo IEEE ASSP Magazine vol 1 no 4 pp 14ndash81 1984

[96] L F Yaroslavsky G Shabat B G Salomon I A Ideses and BFishbain ldquoNonuniform sampling image recovery from sparsedata and the discrete sampling theoremrdquo Journal of the OpticalSociety of America vol 26 no 3 pp 566ndash575 2009

[97] L P Yaroslavsky ldquoImage recovery from sparse samples discretesampling theorem and sharply bounded band-limited discretesignalsrdquo inAdvances in Imaging and Electron Physics P HawkesEd vol 167 chapter 5 pp 295ndash331 Academic Press 2011

[98] R W Gerchberg and W O Saxton ldquoPractical algorithm forthe determination of phase from image and diffraction planepicturesrdquo Optik vol 35 no 2 pp 237ndash250 1972

[99] A Papoulis ldquoNew algorithm in spectral analysis and band-limited extrapolationrdquo IEEE Trans Circuits Syst vol 22 no 9pp 735ndash742 1975

[100] D Gabor ldquoTheory of communicationrdquo Journal of IEEE vol 93pp 429ndash457 1946

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Table 1 Register of relevant fast transforms their main characteristic features and application areas

Relevance to imagingoptics Main characteristic features Main application areas

Discrete FourierTransforms


Cyclic shift invariance vulnerableto boundary effects

(i) Analysis of periodicities(ii) Fast convolution andcorrelation(iii) Fast and accurate imageresampling(iv) Image compression(v) Image denoising anddeblurring(vi) Numerical reconstruction ofholograms

Discrete CosineTransform


Cyclic shift invariance (withdouble cycle)virtually not sensitive toboundary effects

Discrete FresnelTransforms

Represent optical FresnelTransform Computable through DFTDCT Numerical reconstruction of

holograms

Walsh-HadamardTransform No direct relevance

Binary basis functionsProvides piece-wise constantseparable image band-limitedapproximations

(i) Image compression(marginal)(ii) Coded aperture imaging

Haar Transform andother Discrete WaveletTransforms

Subband decomposition

Binary basis functionsThe fastest algorithmMultiresolutionShift invariance in eachparticular scale

(i) Signalimage wideband noisedenoising(ii) Image compression

image that are of no such importance and can be dropped outfrom image representation and restored from the remainingldquoimportantrdquo pixels But the problem is that one never knowsin advance which pixels in the image are ldquoimportantrdquo andwhich are not

The situation is totally different in image representationin transform domain For orthogonal transforms that featuregood energy compaction capability a lion share of totalimage ldquoenergyrdquo (sum of squared transform coefficients) isconcentrated in a small fraction of transform coefficientswhich indices are as a rule known in advance for the giventype of images or can be easily detected It is this feature oftransforms that is called their energy compaction capability Itallows replacing images with their ldquoband-limitedrdquo in terms ofa specific transform approximations that is approximationsdefined by a sufficiently small fraction of image transformcoefficients

The optimal transform with the best energy compactioncapability can be defined for groups of images of the sametype or ensembles of images that are subjects of process-ing Customarily the ldquoband-limitedrdquo image approximationaccuracy is evaluated in terms of the root mean squareapproximation error (RMSE) for the image ensemble Byvirtue of this the optimal transform that secures the leastband limited approximation error is defined by the ensemblecorrelation function that is image autocorrelation functionaveraged over the image ensemble For continuous (notsampled) signals this transform is called theKarhunen-LoeveTransform [11 12] Its discrete analog for sampled signals iscalled the Hotelling transform [13] and the result of applyingthis transform to sampled signals is called signal principalcomponent decomposition Karhunen-Loeve and Hotelling

transforms provide the theoretical lower bound for compact(in terms of the number of terms in signal decomposition)signal discrete representation for RMS criterion of signalapproximation

However being optimal in terms of the energy com-paction capability Karhunen-Loeve and Hotelling trans-forms have generally high computational complexity theper pixel number of operations required for their compu-tation is of the order of image size This is why for prac-tical needs only fast transforms that feature fast transformalgorithms with computational complexity of the order ofthe logarithm of image size are considered A register ofthe most relevant fast transforms their main characteristicfeatures and application areas is presented in Table 1 Theirmathematical definitions are given in Table 2

Different fast transforms have different energy com-paction capability for different types of images Figure 1illustrates the energy compaction capabilities of DiscreteFourier Discrete Cosine Walsh and Haar transforms ona particular test image It shows that for this image DCTdemonstrates the best energy compaction capability 95 ofthe image energy is contained in only 95of all DCT spectralcoefficients whereas for DFT Walsh and Haar transformsthese fractions are 11 15 and 98 correspondinglyNote also that though for Haar Transform the fractionof the ldquomeaningfulrdquo transform coefficients (98) is closeto that for DCT quality degradations of the band-limitedapproximation to the image are noticeably more severe

Experience shows that DCT is in most applicationsadvantageous with respect to other transforms in terms ofthe energy compaction capability This property of DCT hasa simple and intuitive explanation



Name Definition





120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896cos(2120587119896 + 12

119873119903)


DCT log119873


DCT(ltWSz)



1

radic119873

119873minus1

sum

119896=0

119886119896exp(1198942120587119896119903

119873) Complex


DFT log119873



120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896exp [1198942120587 (119896 + 119906) (119903 + V)

120590119873]




1

radic119873

119873minus1

sum

119896=0

119886119896exp [1198942120587(

119896120583 minus 119903120583

119873)] 120583 ge 1 Complex


DFrT log119873

na120583 = radic120582119885119873Δ1198912


1

radic119873

119873minus1

sum

119896=0

119886119896exp[1198942120587 (119896 minus 119903)

2

1205832119873] 120583 le 1



1

radic119873

119873minus1

sum

119896=0

119886119896had119896(119903)

had119896(119903) = (minus1)

sum119899minus1

119898=0119896119898119903119898


WHT log119873na

119896 =

119899minus1

sum

119898=0

1198961198982119898 119903 =

119899minus1

sum

119898=0

1199031198982119898119873 = 2

119899


1

radic119873

119873minus1

sum

119896=0

119886119896wal119896(119903)

wal119896(119903) = (minus1)

sum119899minus1

119898=0119896119898119903119866119903119886119910119862119900119889119890


119903119866119903119886119910119862119900119889119890



120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896har119896(119903)

har119896(119903) = 2





Haar

na




= 120588|119896minus119897|




(a)


(b)


(c)


(d)


(e)

100

100

10minus4

10minus3

10minus2

10minus1


DCTDFTWalsh





(f)









2 Image Compression











Shrinkage andscalar



coefficients

Set of imagepixels



orthogonaltransform


Entropydecoding

Encoding

















Test image

(a)


(b)


(c)


(d)



Inversetransform

Inputimage

Outputimage









Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Name Definition





120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896cos(2120587119896 + 12

119873119903)


DCT log119873


DCT(ltWSz)



1

radic119873

119873minus1

sum

119896=0

119886119896exp(1198942120587119896119903

119873) Complex


DFT log119873



120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896exp [1198942120587 (119896 + 119906) (119903 + V)

120590119873]




1

radic119873

119873minus1

sum

119896=0

119886119896exp [1198942120587(

119896120583 minus 119903120583

119873)] 120583 ge 1 Complex


DFrT log119873

na120583 = radic120582119885119873Δ1198912


1

radic119873

119873minus1

sum

119896=0

119886119896exp[1198942120587 (119896 minus 119903)

2

1205832119873] 120583 le 1



1

radic119873

119873minus1

sum

119896=0

119886119896had119896(119903)

had119896(119903) = (minus1)

sum119899minus1

119898=0119896119898119903119898


WHT log119873na

119896 =

119899minus1

sum

119898=0

1198961198982119898 119903 =

119899minus1

sum

119898=0

1199031198982119898119873 = 2

119899


1

radic119873

119873minus1

sum

119896=0

119886119896wal119896(119903)

wal119896(119903) = (minus1)

sum119899minus1

119898=0119896119898119903119866119903119886119910119862119900119889119890


119903119866119903119886119910119862119900119889119890



120572119903=

1

radic119873

119873minus1

sum

119896=0

119886119896har119896(119903)

har119896(119903) = 2





Haar

na




= 120588|119896minus119897|




(a)


(b)


(c)


(d)


(e)

100

100

10minus4

10minus3

10minus2

10minus1


DCTDFTWalsh





(f)









2 Image Compression











Shrinkage andscalar



coefficients

Set of imagepixels



orthogonaltransform


Entropydecoding

Encoding

















Test image

(a)


(b)


(c)


(d)



Inversetransform

Inputimage

Outputimage









Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(a)


(b)


(c)


(d)


(e)

100

100

10minus4

10minus3

10minus2

10minus1


DCTDFTWalsh





(f)









2 Image Compression











Shrinkage andscalar



coefficients

Set of imagepixels



orthogonaltransform


Entropydecoding

Encoding

















Test image

(a)


(b)


(c)


(d)



Inversetransform

Inputimage

Outputimage









Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















2 Image Compression











Shrinkage andscalar



coefficients

Set of imagepixels



orthogonaltransform


Entropydecoding

Encoding

















Test image

(a)


(b)


(c)


(d)



Inversetransform

Inputimage

Outputimage









Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Shrinkage andscalar



coefficients

Set of imagepixels



orthogonaltransform


Entropydecoding

Encoding

















Test image

(a)


(b)


(c)


(d)



Inversetransform

Inputimage

Outputimage









Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Test image

(a)


(b)


(c)


(d)



Inversetransform

Inputimage

Outputimage









Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Before

(a)

After

(b)




119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)119878119901inp (1)


119878119901out =1

ISFRmax(0

10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

minus 119878119901noise10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2)

12

119878119901inp (2)

Rejecting Filter

119878119901out =

119878119901inp

ISFR if 10038161003816100381610038161003816119878119901inp

10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(3)






119878119901out =

1

ISFR10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

(1minus119875)

119878119901inp if 10038161003816100381610038161003816119878119901inp10038161003816100381610038161003816

2

gt 119878119901noise

0 otherwise(4)





Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Before

(a)

After

(b)










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Test image

(a)


(b)

Filtered image

(c)


(d)






Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Before

(a)

After

(b)


Before

(a)

After

(b)



(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b) (c)


(a) (b)



































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





































Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Test image

(a)


∘ in 60 steps)

(b)


∘ in 60 steps)

(c)



(d)


∘ in 1000 steps)

(e)



(f)











50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 100 150 2000

02

04

06

08

1


Sample index


(a)


50 100 150 2000

02

04

06

08

1

Sample index


(b)
























119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















119883120576119864times 119861120576119864

gt 1 (5)

where 119883120576119864

and 119861120576119864





(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)







6 Conclusion




Nomenclature








hologram

Abbreviations




0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08


Sinclet




minus02

(a)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(b)


0

02

04

06

08

1



minus02


(c)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(d)


Sinclet


0

02

04

06

08

1



minus02

(e)

0

02

04

06

08

1


Bandwidth

220 240 260 280 300



(f)




noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











noise variances)



References










































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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