IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 7, JULY 2010 691

Local Sorting for Adaptive Signal RegularizationVittoria Bruni, Daniela De Canditiis, and Domenico Vitulano, Member, IEEE

Abstract—This letter investigates the possibility of removingnoise in correspondence to jump discontinuities using the sortedcopy of the signal. It will be proved that sorting makes noisepredictable so that it can be reproduced and subtracted from thesorted noisy signal. It will be also shown that the proposed methodcan substitute for the edge preserving term into an anisotropicdiffusion scheme, gaining in terms of mean square error, edgepreservation and computational effort.

Index Terms—Anisotropic smoothing, denoising, sorting.

I. INTRODUCTION

D ENOISING is one of the most investigated problems ofsignal and image processing. Its goal is to recover the

original information from its noisy version . It is some-what difficult to provide an exhaustive review of the approachesproposed in literature: linear and nonlinear filtering (such asthresholding [1]–[4], frequency band filtering [5], averaging [6],[7]), pde-based regularization methods [8], [9], minimization ofenergy functionals [10], rank order statistics based filtering [11],Bayesian minimization [12] and so on. Nevertheless, the maindrawback of many regularization schemes is that they tend tosuppress noise as well as original information. This is undesir-able, especially in correspondence to edge curves (images) orsingularity points (1D signals). This is the reason why severalschemes have been specifically designed to preserve signal sin-gularities. One of the most powerful is anisotropic smoothing,that inhibits smoothing around discontinuities [8]. Edge preser-vation is attained through an iterative procedure, where each it-eration corresponds to a shrinkage of a one level wavelet sub-band and the shrinkage function is equal to the conductance termof the anisotropic diffusion (see [13] and [14] for details).

This letter has a twofold objective. First, it presents anew nonlinear regularization scheme able to recover step (orstep-like) discontinuities by means of signal sorting. In partic-ular, it is proved that sorting makes noise predictable and thenit can be simply reproduced and subtracted from the sortednoisy copy of the signal around discontinuity points. Second,it uses the former result to modify the edge-preserving term ofan anisotropic diffusion scheme. More precisely, the proposedapproach performs a simple linear smoothing in correspon-dence to smooth regions, while it recovers discontinuities usingthe sorted copy of the noisy signal. The conductance term ofthe anisotropic diffusion balances the linear smoothing and

Manuscript received February 17, 2010; revised April 27, 2010; acceptedMay 04, 2010. Date of current version June 17, 2010. The associate editor coor-dinating the review of this manuscript and approving it for publication was Dr.Lap-Pui Chau.

The authors are with the Istituto per le Applicazioni del Calcolo“M. Picone”, C.N.R., 00185 Rome, Italy (e-mail: {bruni@iac.rm.cnr.it;danielad@iac.rm.cnr.it; vitulano@iac.rm.cnr.it).

Digital Object Identifier 10.1109/LSP.2010.2051616

the discontinuity recovery. Experimental results on some testsignals show the potential of the proposed approach in terms ofsignal to noise ratio (SNR) and required computational effort.

II. THE PROPOSED DENOISING MODEL

Let , , be a zero mean white Gaussian noisewith variance . Its sorted copy obeys the following rule

(1)

where is a sorted copy of the same noise with unitary variance.Specifically, , where is the in-verse of the distribution function of a standard Gaussian vari-able ([15], [16]).

Let now be a piecewise constant signal whose support is, i.e.

(2)

where is the number of constant pieces, is the unitaryfunction in the interval , ,and . Its sorted copy is still a piecewise constant func-tion , where s are set in increasing order. More precisely,let be the sorted sequence ,then

(3)

with and isthe size of , i.e. .

Let now be a noisy version of , i.e.with zero-mean white Gaussian noise with variance . If

, with high probability, willbe sorted independently in each . Hence, the sorted copy of ,i.e. , becomes

(4)

where is the function in (1) with , thathas been shifted by the quantity and scaledby the factor , as it is depicted in Fig. 1. Infact, , since is constant.Inserting (3) in (4) it follows

(5)

Hence, the difference between the sorted noisy signal and thesorted clean one is a piecewise function, where each piecebehaves as the function in (1). It turns out that the estimation ofthe quantity in the second member of (5) enables to recover ,

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692 IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 7, JULY 2010

Fig. 1. Top: Piecewise constant signal � , its noisy copy �, zero mean whiteGaussian noise � . Bottom: Sorted piecewise constant signal � , sorted noisysignal � , sorted noise � . � is a piecewise signal where each piece is a scaledversion of � .

i.e. the original values of the clean function . To show this, wewill prove the following proposition.

Proposition 1: Let us consider the sequence of functions

(6)

where each is a realization of a zero-mean Gaussian noisewith variance and is a sorted piecewise constant function.Let

(7)

be the difference between two consecutive elements of that se-quence. If , then, withhigh probability:

(8)

and

(9)

with .Proof: The proof is done by induction and considering just

one piece of the function , since each piece is independentof the others with high probability. From (5) we have

. Let us prove the result for .

Since is a permutation of the sequence ,is a sum of two independent random variables , thenit is still Gaussian with zero mean and variance . Hence, itssorted copy obeys the rule in (1) with . As aresult

(10)

Equation (9) follows by simply subtracting from .Let us now suppose that the result is true for , i.e.

and .We want to prove the proposition for . Since

Fig. 2. Three subsequent functions of the sequence �� � (left) and the corre-sponding � (right), as in (7). � is the piecewise constant function in Fig. 1.� s are proportional to each other.

, by applying the same arguments

used for , it holds

and then . By subtractingfrom both members of previous equation, we have

.Proposition 1 says that noise over a piecewise constant func-

tion can be estimated by randomly reproducing another realiza-tion of the same kind of noise (using any standard random gen-erator), adding it to the current sorted noisy signal to get ,as in (6), and then computing the difference in (7). allows toderive , that can be finally subtracted from to get the esti-mation of . The procedure can be iterated times when-ever the hypothesis is fulfilled. For computa-tional purposes, a single iteration has been considered. Hence,if , we get

(11)

The main peculiarity of this approach is that it tries to esti-mate the noise instead of the underling function, as classicallydone in the current literature. In this way, noise can be simplyreproduced and subtracted from the noisy signal (see Fig. 2).On the contrary, the drawback is that Proposition 1 only pro-vides the actual values of the clean function , while it doesnot give information about their actual location in the time do-main. Nevertheless, in the presence of discontinuities that canbe approximated by step functions in a proper neighborhood ,the sign of the gradient eliminates the ambiguity between in-creasing and decreasing (original) signals. If the gradient is pos-itive, then the regularized function around discontinuity pointsis , otherwise the reversed one is consid-ered, i.e.: .

In the following, the aforementioned sorting based pro-cedure is embedded into an anisotropic diffusion scheme,by substituting the edge-preserving term. In particular, let

be the partial differential equationof the anisotropic diffusion, where is the scale variable and

is the conductance term that is a function depending onthe gradient of . A single iteration of its numericalsolution can be written as

(12)where and respectively are the solution after and

iterations, while the noisy signal is the initial condition.In [13] and [14], it has been proved that (12) is equivalent toiteratively shrinking wavelet details of the noisy signal throughthe function , i.e.

BRUNI et al.: LOCAL SORTING FOR ADAPTIVE SIGNAL REGULARIZATION 693

Fig. 3. Blocks signal: Visual comparison of the results in Table I for � � ���.

(13)where are the pairs of analysis and synthesisscaling and wavelet functions. The conductance term allowsto discriminate between regular and non regular regions. Infact, if , then (12) and (13) reduce to a simple linearsmoothing. On the contrary, if , smoothing is inhibitedin order to preserve discontinuity points. Accounting for theseobservations, the final solution of the iterative scheme in (12)and (13) can be written as follows:

(14)

where is a smoothing kernel and can be the above de-fined sorting based function that is able to preserve signal dis-continuities. In fact, if , (14) applies a linear smoothingto the noisy signal. On the contrary, if , (14) straightfor-wardly recovers signal discontinuities through the functionwithout iterations. To this aim, should rapidly change from1 to 0 whenever a discontinuity is recognized. In this letter, theWeickert function ([13], [14])is used, where is the least admissible variation for a step func-tion and .

A. Algorithm

1) Apply a linear smoothing to using a positive kernelof size . Let be the smoothed signal.

2) Compute and evaluate , with .Select points whose approaches 0. For each ofthem, select a neighborhood of size :

a) Apply the scheme in (6) and (7) for to get ,as in (11).

b) If , then , , otherwise.

3) Use (14) to get the denoised signal .

B. Complexity

If is the signal length and the support size of thesmoothing kernel , then the algorithm in Section II-A re-quires the following number of operations: step 1: ;

Fig. 4. Zoom of the last two results in Fig. 3.

step 2: ; step 2a: ; step 2b: ;step 3: 4N.

If is the number of the involved discontinuities, the totalamount of operations is

Although sorting is an expensive operation, the whole effortof the proposed scheme is competitive in terms of complexitywith respect to anisotropic diffusion. In fact, taking into account(13) and the Weickert function , anistropic diffusion requires

, where is the support size ofthe adopted smoothing kernels, while is the number of re-quired iterations. It is straightforward to see that the proposedscheme is convenient whenever

.Setting , , , and ,

we have .

III. SOME RESULTS AND COMMENTS

This section shows some results achieved on selected testsignals: Blocks, Piecewise polynomial, Bumps and Doppler

. In all tests, a Gaussian kernel has beenemployed in the linear smoothing process (step 1 of the Algo-rithm), while the region around a singularity point has beenset equal to 100. The proposed model has been compared with:Wiener filter, hard and soft thresholding, recursive median filterand the anisotropic smoothing with . As Figs. 3and 5 show, the proposed denoiser does not create undesirableartifacts in correspondence to discontinuity points, as it hap-pens for thresholding based approaches and recursive medianfilter—see, for example, Blocks signal. In particular, jumpdiscontinuities are not blurred and false spikes do not appear.On the other hand, residual noise is limited in correspondenceto smooth regions, in contrast with Wiener filtering. In orderto further show the ability of the proposed method in faithfullyreconstructing discontinuity points, Figs. 4 and 6 show a zoomof Blocks signal and two details of Cameraman image. Inthe latter case, the proposed method has been applied to the1D raster scan version of the image and the parameters havebeen set in order to get the same SNR value of the anisotropicdiffusion. It is worth observing that, even if the global SNRvalues of the restored images are quite the same, the proposedmethod outperforms anisotropic smoothing in correspondenceto edge curves from both a subjective (visual quality) and anobjective (SNR) point of view—the gain is a bit less than 1.5db on average. This result can be further improved if properspace filling curves are locally employed. Using local hori-zontal or vertical raster scan, SNR further increases of about0.4 db. The same evaluation remains valid for moderate levelsof noise since constraint used for (11) does notsignificantly influence the final result, as Table I shows. On the

694 IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 7, JULY 2010

Fig. 5. Left to right: Original, noisy and denoised signal using the proposedmodel. Top to bottom: Piecewise polynomial signal �� � ��, Bumps �� ���� and Doppler �� � ���.

Fig. 6. Cameraman image corrupted with Gaussian noise with standard devi-ation � � ��. The SNR values of the recovered images using anisotropicdiffusion and the proposed method respectively are 23.58 db and 23.70 db. Top:Zoom of Cameraman shoulder; bottom: zoom of Cameraman camera tripod.From left to right: Original, noisy, anisotropic smoothing, the proposed model.The gain in terms of local SNR (restricted to the two shown regions) of the pro-posed method with respect to anisotropic diffusion is 1.4 db on average.

TABLE IBlocks ������� � � �, Bumps ������� � � � and Piecewise

polynonial ������� � ���: SNR RESULTS (MEASURED IN DECIBEL)OF SOME STANDARD DENOISING SCHEMES AND THE PROPOSED ONE USING

DIFFERENT NOISE STANDARD DEVIATIONS �

Fig. 7. Blocks signal: Denoising results for high noise level �� � �� achievedby anisotropic smoothing (middle) and the proposed method (right). Incorrectlocations of the recovered values generate spurious spikes.

TABLE IICOMPUTING TIME FOR 1-D AND 2-D SIGNALS

contrary, for higher noise levels the final result degrades sincethe above assumption is no more valid—as shown in Fig. 7.

With regard to the computing time, the proposed approachis very competitive. In fact, it requires a linear filtering, whosecomplexity is comparable to thresholding and Wiener filteringoperations, along with the computational cost for the sortingoperation. Nevertheless, its cost is quite negligible with respectto the number of iterations required by anisotropic smoothing.Table II compares the computing time of thresholding basedapproaches, anisotropic smoothing, recursive median filteringand the proposed denoiser in case of both 1-D and 2-D signals.

Future research will be oriented to extend the proposed modelto a larger class of signals and higher levels of noise, trying tocorrectly predict locations of the actual values of the signal.

REFERENCES

[1] D. L. Donoho, “Denoising by soft thresholding,” IEEE Trans. Inf.Theory, vol. 41, no. 3, pp. 613–627, May 1995.

[2] L. Sendur and I. W. Selesnick, “Bivariate shrinkage with local varianceestimation,” IEEE Signal Process. Lett., vol. 9, no. 12, pp. 670–684,Dec. 2002.

[3] F. Luisier, T. Blu, and M. Unser, “A new SURE approach to imagedenoising: Interscale orthonormal wavelet thresholding,” IEEE Trans.Image Process., vol. 16, no. 3, Mar. 2007.

[4] C. B. Smith, S. Agaian, and D. Akopian, “A wavelet-denoising ap-proach using polynomial threshold operators,” IEEE Sig. Process. Lett.,vol. 15, pp. 906–909, 2008.

[5] A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise shape adaptiveDCT for high quality denoising and deblocking of grayscale and colorimages,” in Proc. SPIE—IS&T Electron. Imag., 2006, vol. 6064.

[6] A. Buades, B. Coll, and J. M. Morel, “A review of image denoisingalgorithms with a new one,” SIAM Multiscale Model. Simul., vol. 4,no. 2, pp. 490–530, 2005.

[7] M. Mahmoudi and G. Sapiro, “Fast image and video denoising via nonlocal means of similar neighborhoods,” IEEE Signal Process. Lett., vol.12, no. 12, pp. 839–842, Dec. 2005.

[8] P. Perona and J. Malik, “Scale-space and edge detection usinganisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., vol.12, pp. 629–639, Jul. 1990.

[9] V. Bruni, B. Piccoli, and D. Vitulano, “Fast computation method fortime scale signal denoising,” Signal Image Video Process., vol. 3, no.1, Feb. 2009.

[10] S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Variationalapproach for edge-preserving regularization using coupled PDE’s,”IEEE Trans. Image Process., vol. 7, no. 3, pp. 387–397, Mar. 1998.

[11] J. Nieweglowsk and T. G. Campbell, “Recursive image filters and spacefilling curves,” IEEE Winter Workshop Nonlinear Digital Signal Pro-cessing, Jan. 1993.

[12] A. Pizurica, W. Philips, I. Lemanhieu, and M. Acheroy, “A joint inter-and intrascale statistical model for Bayesian wavelet based image de-noising,” IEEE Trans. Image Process., vol. 11, no. 5, May 2002.

[13] A. C. Shih, H. M. Liao, and C. Lu, “A new iterated two-band diffusionequation: Theory and its application,” IEEE Trans. Image Process., vol.12, no. 4, pp. 466–476, Apr. 2003.

[14] P. Mrazek, J. Weickert, and G. Steidl, “Diffusion-inspired shrinkagefunctions and stability results for wavelet denoising,” Int. J. Comput.Vis., vol. 64, no. 2/3, pp. 171–186, 2005.

[15] P. J. S. G. Ferreira, “Sorting continuous-time signals and the analogmedian filter,” IEEE Signal Process Lett., vol. 7, pp. 281–283, Oct.2000.

[16] A. Papoulis and U. Pillai, Probability, Random Variables and Sto-chastic Processes. New York: McGraw-Hill, 1965.