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SIViPDOI 10.1007/s11760-014-0625-8
ORIGINAL PAPER
A novel image compression method for medical images usinggeometrical regularity of image structure
Sujitha Juliet · Elijah Blessing Rajsingh ·Kirubakaran Ezra
Received: 12 February 2013 / Revised: 29 January 2014 / Accepted: 29 January 2014© Springer-Verlag London 2014
Abstract This paper proposes a novel medical image com-pression method using sparse representation approach thatexploits the geometrical regularity of image structure. Thegeometric flow represents the direction in which the imagegray levels have regular variations. The wavelet decomposi-tion of geometric regularized data results in less number ofsignificant coefficients. The directions of regularity are rep-resented using two-dimensional vector field, and the approx-imation of these directions is obtained using spline repre-sentation. The directional decomposition of the image alongwith geometric flow is further improved by bandelet basis.The bandelet coefficients are encoded using Set Partitioningin Hierarchical Trees encoder, followed by global threshold-ing with fixed encoding. Experimental results demonstratethat the proposed method provides significant improvementin compression performance over state-of-the-art compres-sion methods.
Keywords Medical image compression · Geometricregularity · Bandelet construction · Spline approximation ·SPIHT
S. Juliet (B)Department of Information Technology, Karunya University,Coimbatore, Indiae-mail: [email protected]
E. B. RajsinghSchool of Computer Science and Technology, Karunya University,Coimbatore, Indiae-mail: [email protected]
K. EzraDepartment of Outsourcing, Bharat Heavy Electricals Limited,Trichy, Indiae-mail: [email protected]
1 Introduction
With an increase in population, providing basic health care tothe people, especially in rural areas, is one of the major chal-lenges facing humankind. Telemedicine is a good solutionto bridge the physical distance between patients in remoteareas and medical specialists around the world. It generallyrefers to the delivery of clinical care and the exchange ofhealth care information across distances. In telemedicine,medical images are transferred over the network, causinga high storage cost and heavy increase in network traffic dur-ing transmission. Therefore, compression of medical imagesis essential in order to reduce the storage and bandwidthrequirements [1,2]. Apart from preserving vital informationin the medical images, high compression ratio and the abilityto decode the compressed images at various qualities are themajor concerns in medical image compression [3].
Many advanced image compression methods have beendeveloped in response to the increasing demands for med-ical images. Among the existing methods, much interest hasbeen focused on resolving 2D singularities and attaining thedesirable characteristics such as high PSNR, but little workhas been done on sparse image representation that exploitsthe geometric regularity of edges in images. Even thoughthe recent constructions in image analysis community suchas Curvelets [4] and Contourlets [5] are able to capture thegeometric regularity, none of these methods is able to con-struct orthogonal bases of regular functions which are highlydesirable for compression. Image representation in separableorthonormal bases, such as wavelets, cannot take advantageof the geometric regularity of image structures. Integratingthe geometric regularity in image representation is a majorchallenge to improve state-of-the-art image compression.
Grounded on this motivation, this paper constructs sparsegeometric representation for medical image compression that
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decomposes the input medical image over a basis of ban-delets adapted to image geometry. The bandelet constructionis orthogonal, and the corresponding basis functions are reg-ular. The geometric flow regions are segmented into smallnumber of dyadic squares, and the quadtree-based segmen-tation algorithm is computed in order to optimize the bitrateand compression error. The resultant bandelet coefficientsare encoded using SPIHT encoder and global thresholdingwith fixed encoding to improve further the compression per-formance.
Performances of the proposed method are evaluated andcompared with conventional and state-of-the art compres-sion methods, such as discrete cosine transform (DCT), Haarwavelet, contourlet, and curvelet- and JPEG-based compres-sion on a set of medical images. Experimental results demon-strate that the proposed method outperforms the existingmethods in terms of peak signal-to-noise ratio (PSNR), struc-tural similarity (SSIM) index, and compression ratio. Therest of the paper is organized as follows. A brief reviewof existing image compression methods is given in Sect. 2.Section 3 deals with bandelets along geometric flows, andSect. 4 describes the proposed medical image compressionusing geometric regularity. Performance evaluations are pre-sented in Sect. 5, and finally, the conclusions are given inSect. 6.
2 Review of literature
In recent years, there has been abundant interest in wavelet-based methods for the compression of images. In one-dimensional functions, wavelet-based methods are optimal torepresent piecewise-smooth signals with point singularities[6]. Although wavelets provide scalability in quality and res-olution, they fail to capture the geometric regularity along thesingularities, because of their isotropic support. In 1999, Can-des and Donoho introduced curvelet transform [4] to repre-sent 2-D piecewise-smooth functions with smooth curve dis-continuities at an optimal rate. The curvelet transform is char-acterized by its specific anisotropic support, which obeys theparabolic scaling law width = length2. Although curveletsare desired for image compression [7,8] and are able to cap-ture the geometric regularity, they are unable to constructorthogonal bases of regular functions. The discretization ofcurvelet transform turns out to be challenging, and the result-ing algorithm is highly complicated.
Do and Vetterli have proposed contourlet transform [5,9]as an alternative to 2-D curvelet transform. This transformis directly constructed in discrete domain, and hence, thereis no need for transformation from continuous time-spacedomain. It provides directional analysis similar to curvelettransform with reduced redundancy. Its implementation isbased on pyramidal band-pass decomposition of the image
followed by multiresolution directional filtering stage. Eventhough contourlet transform is suited for image compression[10,11] because of its lower redundancy and less complex-ity, its aggressive subsampling can lead to artifacts in signalreconstruction.
Wedgelet transform developed by Donoho [12] is shownto be optimal for piecewise-constant images with regularedge discontinuities. It consists of constant-valued squares,bisected by straight lines, and spanning many sizes andlocations. This elegant structure has been adopted by manyresearchers, leading to wedgeprints [13], platelets [14], andhigher-dimensional wedgelets named as surflets [15]. How-ever, it is not able to construct orthogonal bases of regu-lar functions. Similar to wedgelet transform, Candes andDonoho introduced ridgelet transform as a multidimensionalextension of the wavelet transform [16]. This transform isproven to be optimal for piecewise-smooth functions withplane discontinuities. However, the basic Ridgelet transformis unsuitable for natural signals due to its lack of localiza-tion. Moreover, the development of discrete versions of theridgelet transforms leads to challenging algorithmic imple-mentations [17].
Ripplet transform proposed by Xu et al. [18] provideshierarchical representation of images by representing sin-gularities along arbitrarily shaped curves. When the rippletfunction intersects with curves in images, the correspondingcoefficients will have large magnitude and the coefficientsdecay rapidly along the direction of singularity. Multireso-lution, good localization, and high directionality propertiesof ripplet transform lead to efficient compression of medicalimages [19]. Directionlet transform proposed by Velisavl-jevic et al. [20] is a discrete transform which constructsanisotropic wavelets based on local image directionality, uti-lizing a specialized directional grouping of the grid points forits implementation. The shearlet transform [21] is also pro-posed as an alternative to curvelets, which utilizes structuredshear operations rather than rotations to control orientation.However, it cannot be applied to images of less regularity[22].
DCT is possibly the most popular transform used incompression of images in standards like Joint PhotographicExperts Group (JPEG) [23,24]. Singh et al. [25] proposed aDCT-based compression method, in which the input imageis split into smaller blocks and each block is classified basedon an adaptive threshold value of variance. Chen [26] alsohas presented a DCT-based subband decomposition methodfor the compression of medical images. This approach usestransform function to DCT coefficients to concentrate sig-nal energy and a modified SPIHT algorithm to organize dataand entropy encoding. Ponomarenko et al. [27] proposed anadaptive JPEG-based image compression method to achieveperceptually lossless images. Ansari et al. [24] and Kim [28]have evaluated the performance of JPEG compression for the
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compression of medical images. However, DCT-based JPEGhas blocking artifacts at low bitrates. Using DCT, when morecompression is required, discontinuities will be created atblock boundaries [29]. Reducing the blocking artifacts usingany smoothing filter would sacrifice the detailed informa-tion in the image. Minasyan et al. [30] demonstrated theperformance of Haar wavelet transform for image compres-sion. The implementation of Haar wavelet involves averag-ing and differencing terms, storing detail coefficients, elim-inating data, and reconstructing the matrix similar to initialmatrix. Even though it has less computational complexity,Haar wavelet does not provide sparse representation of edgesin images.
Most of the traditional image representation methods usedonly the edges of images to describe their geometric reg-ularities. Pennec and Mallat introduced geometric flow torepresent the image regularity and presented the first gen-eration bandelet transform in 2000 [31]. This fixed scaletransform is used directly on the image, but produces block-ing artifacts due to piecewise-constant nature of basis func-tions. The second generation bandelet transform introducedin 2005 [32] is an orthogonal multiscale transform thatexploits the geometric regularity of the image. This rep-resentation decomposes the image over a set of bandeletbasis and represents sharp image transitions such as edgesefficiently. Many researchers have demonstrated the effi-ciency of bandelet transform for the compression of syn-thetic aperture radar (SAR) images [33,34], SAR imagedespeckling [35], image fusion [36], and video compression[37].
Several encoding methods have been proposed and repor-ted in the literature for image compression. These coders aredeveloped to improve the quality of images at high compres-sion rates. The effectiveness of wavelet-based image cod-ing was first demonstrated by Shapiro’s Embedded ZerotreeWavelet (EZW) [38], and it is the first subband coding algo-rithm by zerotree. Later, research by Said and Pearlmanon Set Partitioning in Hierarchical Trees encoder (SPIHT)[39,40] improved upon EZW coding and applied success-fully to both lossy and lossless compression of images.SPIHT is a tree-based embedded coder, which uses pro-gressive transmission by coding the bit planes in decreas-ing order. This algorithm exploits the dependencies betweenlocation and value of the coefficients across subbands. TheSet Partitioned Embedded block coder (SPECK) algorithm[41] is a block-based image coding algorithm which employsrecursive set-partitioning procedure to sort subsets of waveletcoefficients by maximum magnitude with respect to inte-ger powers of two thresholds. The Embedded Block Codingwith Optimized Truncation (EBCOT) algorithm [42] is alsoa block-based coding algorithm which processes the codeblock by bit-plane-by-bit plane. But this algorithm is morecomplicated and time-consuming [43].
Binary set splitting with K-d trees (BISK) coder [44] per-forms bitplane coding in which significant coefficients arelocated by recursive partitioning of the dataset. This coder ismainly used as shape-adaptive coding of ocean-temperaturedata. Global thresholding with fixed encoding method is athreshold-based encoder which calculates the threshold with-out pretreatment to histogram [45]. By varying the threshold,the visual quality of the image can be improved. Althoughseveral encoders have been reported for image compression,SPIHT is considered as an efficient encoder for image com-pression due to its salient features viz. intensive progressivecapability, SNR scalability, and low computational complex-ity [46].
3 Bandelets along geometric flows
The bandelet transform, introduced by Pennec and Mallat[32], takes advantage of geometric regularity of image struc-ture for efficient sparse image representation. It decomposesthe image along multiscale vectors that are elongated in thedirection of geometric flow. The geometric flow shows thelocal directions in which the image gray levels have regu-lar variations. Bandelet construction is orthogonal, and thecorresponding basis functions are regular. Sect. 3.1 explainsthe concept of geometric regularity flow, and Sect. 3.2 dealswith the optimization of the geometric flow to the precisionof bandelet image approximations.
3.1 Geometric regularity flow
Instead of describing the geometry of images through edges,the image geometry is characterized with geometric flow ofvectors. These vectors give the local directions in which theimage has regular variations. The geometric flow around aregion can be represented using a two-dimensional vectorfield �v(x, y) that shows the direction in which the imagehas regular variations in the neighborhood of each (x, y).If the image intensity is uniformly regular in the neighbor-hood, then this direction is not uniquely defined. In order todefine the direction uniquely, a global regularity conditionis imposed on the flow. To construct orthogonal bases withthe resulting flow, the regularity condition imposes that theflow is either parallel vertically, and hence, �v(x, y) = �v(x)or parallel horizontally, then �v(x, y) = �v(y).
In order to maintain flexibility, the image is partitionedinto squares of varying dyadic sizes, and this parallel con-dition is imposed by determining the image sample valuesalong the flow lines, in each square region. Each regionincludes at most one contour, and in each region, the flowis parallel to the tangents of the contour curve. The flowcurves show the direction on which the pixels change theleast.
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3.2 Optimized geometry for image approximation
For image approximation, the best geometry is the one thatleads to an approximation f A from A elements that mini-mizes the approximation error ‖ f − f A‖. In bandelet approx-imation, the A elements include bandelet coefficients used tocompute f A, as well as the elements that specify the imagepartition and the geometric flow in each square region. Inorder to represent the image partition with few elements andto determine an optimal partition, the image is partitionedinto squares of varying dyadic sizes. A dyadic square imagepartition is obtained by successive subdivisions of squareregions into four subsquares of smaller width. The imagepartition into dyadic squares is represented by quadtree. Fig-ure 1 shows an example of a dyadic square image partitionwith quadtree.
One of the important requirements for flow representationis that it should be compact, such that the overhead introducedto compression will be minimum [37]. In order to representthe flow with few parameters and to obtain smooth flow, theflow directions are approximated with the translated B-splinefunctions of first degree B(u) as,
Fig. 1 Quadtree of dyadic square image partition
c′[u] =2k−1∑
n=1
αn B(2−lu − n) (1)
where αn(n = 1 . . . 2l) is the nth spline coefficient dilatedby a scale factor 2l , 2k is the width of the square, and u is anindex u ∈ {x, y}. The B-spline function B(u) is formulatedas
B(u) ={
1 − |u|, if |u| < 10, otherwise
}
With this representation, the geometric regularity flow canbe recovered by storing only the spline control point values.
4 Proposed methodology
Figure 2 shows the block diagram of the proposed med-ical image compression using geometric regularity. Theinput medical image f (x, y) is first decomposed throughtwo-dimensional biorthogonal wavelet transform to obtainwavelet decomposition subbands. Each level of the wavelettransform decomposes its input into four spatial frequencysubbands. The approximation low-pass subband,a j , is acoarser version of the original image, while the other sub-bands, d H
j , dVj , d D
j represent the high frequency details inthe horizontal, vertical, and diagonal directions, respectively.The subbands are given as
a j (x, y) = 〈 f (x, y) , φ j (x)φ j (y) 〉d H
j (x, y) = ⟨f (x, y), ψ j (x)φ j (y)
⟩
dVj (x, y) = ⟨
f (x, y), φ j (x)ψ j (y)⟩
d Dj (x, y) = ⟨
f (x, y), ψ j (x)ψ j (y)⟩
(2)
where φ j (x, y) represents the scaling function and ψ j (x, y)denotes the wavelet function. j represents the scale factor.The decomposition is iterated on the approximation low-passsubband, which contains most of the energy [47]. The result-ing wavelet transformed image is partitioned into squares of
1D DWT Construction of geometric
flow
Inputimage
Computation of warped wavelet
coefficients
Global thresholding
Compressedimage
Extraction of dyadic squares
2D Wavelettransform
SPIHT encoder
Fig. 2 Proposed block diagram for medical image compression using geometric regularity
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varying dyadic sizes. A dyadic square S is obtained by recur-sively splitting the transformed image into four subsquares ofequal size 2k . The image partition into dyadic squares is rep-resented by quad tree. A square subdivided into four smallersquares corresponds to a node having four children in thequad tree.
For each square region Si , the geometric flow is con-structed by determining the image sample values along theflow lines. The geometry of images is characterized by afield of vectors, and the geometric flow shows the directionsin which the image gray levels have regular variations. In adiscrete framework, the geometric flow in a region Si is avector field �v(x, y) defined over the image sampling grid.If the geometric flow is parallel to the vertical direction,then �v(x, y) = �v(x). The flow vectors are normalized as�v(x) = (1, c′(x)) for a fixed integer y and a varying integerx , where c′(x) measures an average relative displacementof the image gray levels in Si along the line x with respectto x − 1. In order to convert the flow directions into actualgeometric regularity, the flow lines need to be computed. Aflow line is an integral curve, whose tangents are parallelto �v(x). It can be used to warp the wavelet basis along thedirections of regularity. Since the flow is parallel vertically,the discretized flow line in Si is a set of points of coordinates(x, y + c(x)) ∈ Si for a fixed y and a varying x , with
ci (x) =x∑
p=ai
c′i (p) (3)
and ai = minx {(x, y) ∈ Si }. The coordinates of flow linesare stored in a sampling grid array, defined as Gi (x, y) =(x, y + ci (x)) if (x, y + ci (x)) ∈ Si , and is null otherwise.
If the geometric flow is parallel horizontally, then �v(x, y)= �v(y) = (c′(y), 1). Each flow line is defined by (x +c(y), y) for a fixed x and varying y, where ci (y) is stilldefined by Eq. (3) with ai = miny{(x, y) ∈ Si }. Thecoordinates of these flow lines are stored in Gi (x, y) =(x + ci (y), y) if (x + ci (y), y) ∈ Si and is null other-wise. When Gi (x, y) is a noninteger value, the data have tobe carefully sampled at that point. In the proposed method,the nearest-neighbor interpolation technique is used to pre-vent loss of data during sampling. In order to represent theflow with few parameters and to obtain smooth flow, theflow directions are approximated with the translated B-splinefunctions as given in Eq. (1).
In each dyadic region, the wavelets are matched to theimage flow, in such a way that the wavelet coefficients essen-tially ‘wrap-around’ the edges rather than cross them. Thisprocess significantly reduces the number of large waveletcoefficients and leads to higher compression rate. The warpedwavelet coefficients are calculated with subband filtering thatgoes across the boundaries of image partition. The subbandfiltering can be implemented by the lifting scheme [48],
which requires the right and left neighbors of a point beknown. The wavelet coefficients are computed with a filterbank that convolves the image rows and columns with a pairof perfect reconstruction filters, together with subsampling.These wavelet coefficients are inner products of f (x, y)witha basis of discrete separable wavelets
⎧⎪⎨
⎪⎩
φm1j (x)ψ
m2j (y),
ψm1j (x)φm2
j (y),
ψm1j (x)ψm2
j (y)
⎫⎪⎬
⎪⎭j,m1,m2
(4)
The separable wavelets in Eq. (4) are warped alongthe flow lines with an operator W , which is defined asW ( f (x, y)) = f (x, y + c(x)) for the vertical parallel flow.By construction of the flow, the image gray level has regu-lar variations along these flow lines, and hence, the warpedimage W ( f (x, y)) is regular along the horizontal lines forfixed y and varying x . As a consequence, if ψ(x, y) is awavelet function having several vanishing moments along xfor each fixed y, then the inner product 〈 Wf,ψ〉 = 〈 f,W ∗ψ〉has a small amplitude. The warping operator W is an orthog-onal operator since its adjoint is equal to its inverse.
W ∗ f (x, y) = W −1 f (x, y) = f (x, y − c(x)) (5)
Since W ∗ = W −1, the warped wavelet basis is obtainedby W −1 to each separable wavelet basis
⎧⎪⎨
⎪⎩
φm1j (x) ψm2
j (y − c(x)) ,
ψm1j (x) φm2
j (y − c(x)) ,
ψm1j (x) ψm2
j (y − c(x))
⎫⎪⎬
⎪⎭j,m1,m2
(6)
The warped wavelet basis for the horizontal parallel flow canbe written in the same manner using the warping operatorW ( f (x, y)) = f (x + c(y)) and is given as
⎧⎪⎨
⎪⎩
φm1j (x − c(y))ψm2
j (y),
ψm1j (x − c(y))φm2
j (y),
ψm1j (x − c(y))ψm2
j (y)
⎫⎪⎬
⎪⎭j,m1,m2
(7)
After warping the wavelet basis, the next step is to con-struct bandelets by applying a bandeletization procedure. Thewarped wavelet coefficient 〈 f,W −1ψ〉 is small ifψ(x, y)hasvanishing moments along x for each y. Since the waveletfunctionψ(x, y) consists of high-pass filters and has vanish-ing moments at lower resolutions, this is valid forψ (x, y) =ψ
m1j (x)φm2
j (y) and also for ψ (x, y) = ψm1j (x)ψm2
j (y),
but not for ψ (x, y) = φm1j (x)ψ
m2j (y). Since the scal-
ing function φ(x, y) consists of low-pass filters and doesnot have vanishing moment at lower resolutions, it cannottake advantage of the regularity along the flow lines for
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ψ (x, y) = φm1j (x)ψ
m2j (y). In order to solve this prob-
lem, the warped wavelet basis is bandeletized by replac-
ing {φm1j (x)ψ
m2j (y)} with
{ψ
m1l (x)ψm2
j (y)}
for l > j . It is
implemented with a simple discrete wavelet transform. Thefunctions ψm1
l (x)ψm2j (y) are called bandelets because their
support is parallel to the flow lines and is more elongated inthe direction of the geometric flow. Inserting these bandeletsin warped wavelet basis yields bandelet basis and is written as
⎧⎪⎨
⎪⎩
ψm1l (x)ψm2
j (y − c(x)),
ψm1j (x)φm2
j (y − c(x)),
ψm1j (x)ψm2
j (y − c(x))
⎫⎪⎬
⎪⎭j,l> j,m1,m2
(8)
A similar process is followed for bandeletization, if theflow is parallel in the horizontal direction. The bandeleti-zation is done by replacing the family of scaling functions{φ
m2j (y)
}
j,m2by an equivalent family of wavelet functions
{ψ
m2l (y)
}l,m2
at lower resolutions. Inserting the bandelets inthe warped wavelet basis yields a bandelet basis and is givenas
⎧⎪⎨
⎪⎩
φm1j (x − c(y))ψm2
j (y),
ψm1j (x − c(y))ψm2
l (y),
ψm1j (x − c(y))ψm2
j (y)
⎫⎪⎬
⎪⎭j,l> j,m1,m2
(9)
The resulting bandelet coefficients are computed fromwarped wavelets with 1D discrete wavelet transform alongthe geometric flow curves. Since bandelet transform is aredundant transform, there is a need for layout of squaresthat forms the best segmentation of each scale. Such seg-mentation is represented as quadtree. It now remains to buildthe best quadtree in an optimal manner using Lagrangianoptimization on the quantized geometry and bandelet coeffi-cients. The distortion rate can be decomposed into
D + λR =∑
i
Di + λRi (10)
where Di = ‖ f − f A‖2 is the Euclidean norm restricted tothe region Si of image partition. Ri denotes the bit cost ofthe number of bits to code the geometric flow and the ban-delet coefficients in each square at scale 2l , and λ = 3/28 isthe lagrange multiplier in the cost function [49]. The mini-mization of the distortion rate (Eq. 10) starts with computingthe compression cost for each dyadic square in the quadtree.The cost may be minimized by a horizontal or vertical par-allel flow or by no flow at all. If the flow exists, the optimaldirection scale parameter 2l(1 ≤ 2l ≤ 2k) in Eq. (1) is foundby trying all possible values of l and the smallest compressioncost is selected. In the end, the flow that results in minimumcost determines the flow type.
The optimal segmentation is found by split/merge algo-rithm starting from the leaf nodes of the quad tree. At eachlevel, four child nodes are merged into a single node if theircumulative cost is greater than the cost of the parent. Other-wise, they stay split. This merging can be formulated as
Di + λRi <
4∑
n=1
Di,n + λRi,n (11)
where Di,n is the child of node Di . Continuing this algorithmuntil the top of the tree concludes an optimal segmentationwhich minimizes the overall distortion rate.
Finally, the resultant bandelet coefficients are encodedusing SPIHT algorithm which encodes coefficients by spatialorientation tree. This algorithm orders the bandelet coeffi-cients according to the significance as in Eq. (12) and storesthe information in three separate lists: list of insignificantsets (LIS), list of insignificant pixels (LIP), and list of sig-nificant pixels (LSP). After the initialization, this algorithmtakes two stages for each level of threshold: sorting stageand refinement stage. During the sorting stage, the pixels inLIP are tested using significance test and those that becomesignificant are moved to LSP. The sets are sequentially eval-uated following the LIS order, and when the set is found tobe significant, it is removed from the list and partitioned. Thenew subsets with more than one element are added back toLIS, while the single-coordinate sets are added to the end ofLIP or LSP, depending upon whether they are insignificantor significant, respectively.
sn(um) ={
1, max |Bx,y | ≥ 2n
0, otherwise(12)
where sn(um) is the significance of a set of coordinates andBx,y represents bandelet coefficients at coordinates (x, y).
LSP now contains the coordinates of the pixels that arevisited in the refinement stage, which outputs the nth mostsignificant bit of sn(um). The value of n is decreased by 1, andthe sorting and refinement stages are repeated. The resultantcoefficients are further encoded using global thresholdingwith fixed encoding.The proposed algorithm is formulated as follows:
1. Decompose the input medical image f (x , y) through twodimensional wavelet transform to obtain wavelet decom-position of subbands.
a j (x, y) = 〈 f (x, y) , φ j (x)φ j (y) 〉d H
j (x, y) = ⟨f (x, y), ψ j (x)φ j (y)
⟩
dVj (x, y) = ⟨
f (x, y), φ j (x)ψ j (y)⟩
d Dj (x, y) = ⟨
f (x, y), ψ j (x)ψ j (y)⟩
2. Obtain dyadic squares S of constant size 2k by recursivelysegmenting the wavelet transformed image.
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Fig. 3 Examples of test images used for evaluation. a Sagittal view ofT1 WI-MRI Lumbar spine, b Sagittal section of MRI knee, c Axial sec-tion of CT abdomen, d CT scan of chest (left side) pulmonary arteries,e Axial section of T2 Weighted MRI brain, f CV junction CT brain, gAxial section Haste-T2WI MRI abdomen, h Axial section of MRI brain,
i Axial section of CT abdomen with oral contrast, j Sagittal section ofT1 WI MRI brain, k Coronal section of T1WI-MRI chest, l Axial sec-tion of CT skull, m Coronal section of T1 WI-MRI leg, n Axial sectionof T2 WI-MRI brain, o Sagittal section of MRI cerebral
3. For each square region Si , construct geometric flow bydetermining the image sample values along the flow lines.The geometric flow in a region Si is a vector field �v(x, y)defined over the image sampling grid. If the geometricflow is parallel in the vertical direction, then �v(x, y) =�v(x) and if the flow is parallel in the horizontal direction,then �v(x, y) = �v(y).
4. Warp the discrete wavelets along the flow lines with anoperator W and compute the warped wavelet coefficients.For the vertical parallel flow, the warped wavelet coeffi-cients are obtained as
⎧⎪⎨
⎪⎩
φm1j (x) ψm2
j (y − c(x)) ,
ψm1j (x) φm2
j (y − c(x)) ,
ψm1j (x) ψm2
j (y − c(x))
⎫⎪⎬
⎪⎭j,m1,m2
and for the horizontal parallel flow, the warped waveletcoefficients are given as
⎧⎪⎨
⎪⎩
φm1j (x − c(y))ψm2
j (y),
ψm1j (x − c(y))φm2
j (y),
ψm1j (x − c(y))ψm2
j (y)
⎫⎪⎬
⎪⎭j,m1,m2
5. Bandeletize the warped wavelet basis by replacing
{φm1j (x)ψ
m2j (y)} with
{ψ
m1l (x)ψm2
j (y)}
through 1D dis-
crete wavelet transform.6. Encode the resultant bandelet coefficients using SPIHT
encoder followed by global thresholding with fixedencoding.
7. Measure the resulting image quality in terms of PSNR,SSIM and Compression ratio.
5 Performance evaluations
The performances of the proposed method (Bandelet-SPIHT)are evaluated on various medical images of size (256X256,8bits per pixel), and the quality of the compressed imageshas been assessed in terms of PSNR (dB), SSIM, compres-sion ratio, and computational complexity. The efficiency ofthe proposed method is evaluated on comparison with DCT[26], Haar wavelet [30], contourlet [10], curvelet [7]-basedcompression methods, encoded using SPIHT encoder, andJPEG compression [23]. Figure 3 shows the sample of med-ical images used for evaluation. Since MRI (magnetic reso-
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nance imaging) and CT (computed tomography) images pro-vide detailed visuals and greater contrast between differenttissues of the body, these anatomical images are used forevaluation. In order to implement the proposed method, theimage processing toolbox of the MATLAB software is used.
The following subsections present thorough experimentalinvestigations of the overall behavior of the proposed methodusing geometric regularity. Section 5.1 analyses the imagequality of the proposed and existing methods in terms ofPSNR, and Sect. 5.2 evaluates the image quality in terms ofSSIM. In Sect. 5.3, the compression ratio achieved by theproposed and existing methods are tabulated and analyzed.Finally, Sect. 5.4 describes the computational complexity ofthe proposed and existing methods.
5.1 Evaluation of image quality based on PSNR
The major design objective of image compression methodis to obtain the best visual quality of images with minimumutilization of bits. PSNR is one of the most adequate para-meters to measure the quality of compression. If the PSNRvalues are higher, the quality of compression is better andvice versa. It is defined as
PSNR = 10 × log10
(2552/MSE
)(13)
MSE in (13) represents the mean-square error and is definedas
MSE = 1
M×N×
⎡
⎣M−1∑
x=0
N−1∑
y=0
( f (x, y)− F (x, y))2
⎤
⎦ (14)
where M × N represent the size of the image, f (x, y)denotes original image, and F(x, y) denotes compressedimage. Bitrate (bpp) is defined as the ratio of the size ofthe compressed image in bits to the total number of pixels.
Figure 4a–e shows the results obtained for the proposedand existing methods in terms of PSNR (dB) achieved for dif-ferent medical images at various bitrates, and Fig. 5 shows thePSNR (dB) achieved at 1.5 bpp. The proposed method pro-vides efficient sparse representation by capturing the geomet-rics in images. However, in contourlet-based method, aggres-sive subsampling can lead to artifacts in signal reconstruc-tion. DCT and JPEG methods also suffer from blocking arti-facts caused by discontinuities. Since the bandelet functionsare regular and introduce no blocking artifacts, the proposedmethod achieves high PSNR compared to existing methods.
5.2 Evaluation of image quality based on SSIM
The structural similarity (SSIM) index is an objective imagequality metric used to measure the similarity between two
images based on the characteristics of human visual system.It measures the structural similarity rather than error visibilitybetween two images. SSIM is defined as
SSI M(x, y) = (2μxμy + C1)(2σxy + C2)
(μ2x + μ2
y + C1)(σ 2x + σ 2
y + C2)(15)
where x and y are spatial patches. μx and μy are the meanintensity values, and σ 2
x and σ 2y are standard deviation of x
and y, respectively. C1 and C2 are constants. In Eq. (15),SSIM(x, y) is equal to unity if and only if x = y.
Figure 6a–e shows the SSIM values of the proposed andreference methods at various bitrates. From Fig. 6a–e, it isclear that the proposed method yields better SSIM value(close to 1) than reference methods. This is due to the factthat the proposed method provides multiscale representationof the image geometry and represents sharp image transitionssuch as edges efficiently.
5.3 Evaluation of compression ratio
Compression ratio enumerates the minimization in imagerepresentation size produced by the compression algorithm.It is defined as the ratio of the number of bits in theoriginal image to that of the compressed image. Table 1shows the compression ratio and bitrates (bpp) of sam-ple medical images using the proposed and existing meth-ods.
The proposed method outperforms other methods on thecompression of MRI images such as sagittal section of T1WI-MRI lumbar spine, axial section of MRI brain, coronalsection of T1WI-MRI chest, axial section of T2 WI-MRI,and sagittal section of MRI cerebral. This is due to the factthe proposed method uses splines to approximate the direc-tion of regularity. The wavelet decomposition of geometricregularized input image results in a less number of signifi-cant coefficients, which yields higher compression ratio com-pared to existing methods. However, for CT images such asaxial section of CT abdomen with oral contrast, axial sectionof CT abdomen, and axial section of CT skull, contourlet-based method performs good because it deals effectively withpiecewise-smooth images with smooth contours. Contourletmethod provides high compression ratio for coronal sectionof T1 WI-MRI leg also. Similarly, for CT scan of chest (leftside) pulmonary arteries, curvelet-based method performswell because it approximates images from coarse to fine reso-lutions. From Table 1, it is observed that the proposed methodoutperforms DCT by 9.63 %, Haar wavelet by 9.43 %, con-tourlet by 3.09 %, curvelet by 5.14 %, and JPEG by 15.41 %in terms of compression ratio.
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(a)
0
10
20
30
40
50
60
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
PS
NR
(dB
)
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
(b)
0
10
20
30
40
50
60
PS
NR
(dB
)
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
(c)
0
10
20
30
40
50
60
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
PS
NR
(dB
)
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
(d)
0
10
20
30
40
50
60
PS
NR
(dB
)
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
(e)
0
10
20
30
40
50
60
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
PS
NR
(dB
)
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
Fig. 4 PSNR (dB) achieved for test images using the proposed and existing methods. a Sagittal section of MRI Cerebral, b Sagittal section of T1WI MRI brain, c Axial section of CT skull, d coronal section of T1 WI-MRI leg, e CV junction of CT brain
Fig. 5 PSNR (dB) achieved fortest images using differentmethods at 1.5 bpp
0
10
20
30
40
50
60
PSN
R(d
B)
Filesize(KB)
DCT-SPIHT
Haar-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
Bandelet-SPIHT
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(a)
0
0.2
0.4
0.6
0.8
1
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
SS
IM
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
(b)
0
0.2
0.4
0.6
0.8
1
SS
IM
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
(c)
0
0.2
0.4
0.6
0.8
1
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
SS
IM
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
(d)
0
0.2
0.4
0.6
0.8
1
SS
IM
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
(e)
0
0.2
0.4
0.6
0.8
1
0.07 0.09 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
SS
IM
Bitrate(bpp)
Bandelet-SPIHT
DCT-SPIHT
HAAR-SPIHT
Contourlet-SPIHT
Curvelet-SPIHT
JPEG
Fig. 6 SSIM values for test images at various bitrates using the pro-posed and existing methods. a Axial section of CT abdomen with oralcontrast, b Sagittal section of MRI Cerebral, c Axial section of T2
Weighted MRI brain, d CT scan of chest (left side) pulmonary arteries,e Sagittal section of T1 WI-MRI Lumbar spine
5.4 Computational complexity considerations
The performance evaluation section is concluded with a briefdiscussion regarding the complexity of the proposed andexisting methods. The proposed method uses adaptive seg-mentation and a local geometric flow, and well suited to cap-ture the anisotropic regularity of edge structures. However,the best visual quality is achieved at the expense of substantialincrease in the computational complexity. Since an exhaus-tive search algorithm is required for the partition of imagesand the searching of best directions on each scale, the pro-posed method requires O(n2(log2n)2) operations for n × nimage. The implementation of contourlet transform is basedon pyramidal band-pass decomposition of the image fol-lowed by a multiresolution directional filtering stage. Sincethis transform is directly constructed in discrete domain,there is no need for transformation from continuous time-space domain which leads to less complexity of O(n) for
n × n image. Similarly, Haar wavelet is also computation-ally attractive because it has the complexity of O(log2(n)).The computational complexity of DCT and JPEG methods isO(nlog(n)) and O(n2), respectively. Curvelet methods runin O(n2logn) flops for n × n image.
6 Conclusions
A novel medical image compression method with sparse rep-resentation is proposed. The main focus of the proposedmethod is to provide high-quality compressed images bydecomposing the input image over a basis of bandeletsadapted to image geometry and to achieve high compressionratio. Bandelet functions are regular and hence introduce noblocking artifacts. The wavelet decomposition of geomet-ric regularized data results in a less number of significantcoefficients, thus yielding high compression performance.
123
SIViP
Tabl
e1
Com
pres
sion
ratio
san
dbi
trat
es(b
pp)
ofsa
mpl
em
edic
alim
ages
usin
gth
epr
opos
edan
dex
istin
gm
etho
ds
Med
ical
imag
es25
6×
256
:8bp
pC
ompr
essi
onm
etho
ds
DC
T-SP
IHT
Haa
r-SP
IHT
Con
tour
let-
SPIH
TC
urve
let-
SPIH
TJP
EG
Ban
dele
t-SP
IHT
(Pro
pose
d)
Sagi
ttals
ectio
nof
T1
WI-
MR
IL
umba
rsp
ine
8.76
:1(0
.91
bpp)
9.62
:1(0
.83
bpp)
9.96
:1(0
.80
bpp)
9.53
:1(0
.84
bpp)
8.21
:1(0
.97
bpp)
10.3
7:1
(0.7
7bp
p)
Cor
onal
sect
ion
ofT
1W
I-M
RI
leg
12.0
7:1
(0.6
6bp
p)10
.29:
1(0
.78
bpp)
12.2
3:1
(0.6
5bp
p)11
.14:
1(0
.72
bpp)
10.8
5:1
(0.7
4bp
p)12
.22:
1(0
.65
bpp)
Axi
alse
ctio
nof
MR
Ibr
ain
10.3
6:1
(0.7
7bp
p)10
.57:
1(0
.76
bpp)
11.0
5:1
(0.7
2bp
p)10
.89:
1(0
.73
bpp)
9.82
:1(0
.81
bpp)
11.3
8:1
(0.7
0bp
p)
Cor
onal
sect
ion
ofT
1WI-
MR
Ich
est
9.83
:1(0
.81
bpp)
9.41
:1(0
.85
bpp)
9.76
:1(0
.82
bpp)
10.3
1:1
(0.7
8bp
p)9.
27:1
(0.8
6bp
p)10
.86:
1(0
.74
bpp)
Axi
alse
ctio
nof
T2
WI-
MR
Ibr
ain
12.1
3(0
.65
bpp)
10.4
5(0
.76
bpp)
11.3
2(0
.71
bpp)
12.1
6(0
.65
bpp)
10.4
8(0
.76
bpp)
12.3
6(0
.65
bpp)
Sagi
ttals
ectio
nof
MR
Ice
rebr
al9.
7:1
(0.8
2bp
p)10
.22:
1(0
.78
bpp)
9.73
:1(0
.82
bpp)
9.24
:1(0
.87
bpp)
9.48
:1(0
.84
bpp)
10.4
5:1
(0.7
7bp
p)
Axi
alse
ctio
nof
CT
abdo
men
with
oral
cont
rast
8.22
:1(0
.97
bpp)
9.73
:1(0
.82
bpp)
9.92
:1(0
.81
bpp)
9.41
:1(0
.85
bpp)
8.86
:1(0
.90
bpp)
9.89
:1(0
.81
bpp)
Axi
alse
ctio
nof
CT
abdo
men
9.56
:1(0
.83
bpp)
10.2
1:1
(0.7
8bp
p)10
.33:
1(0
.77
bpp)
9.78
:1(0
.82
bpp)
9.47
:1(0
.84
bpp)
10.1
4:1
(0.7
9bp
p)
Axi
alse
ctio
nof
CT
skul
l9.
22:1
(0.8
7bp
p)9.
37:1
(0.8
5bp
p)9.
84:1
(0.8
1bp
p)9.
61:1
(0.8
3bp
p)8.
23:1
(0.9
7bp
p)9.
79:1
(0.8
2bp
p)
CT
scan
ofch
est(
left
side
)pu
lmon
ary
arte
ries
8.51
:1(0
.94
bpp)
8.67
:1(0
.92
bpp)
10.4
6:1
(0.7
6bp
p)10
.49:
1(0
.76
bpp)
8.77
:1(0
.92
bpp)
10.3
8:1
(0.7
7bp
p)
Bol
dva
lues
indi
cate
the
high
com
pres
sion
ratio
sac
hiev
edby
the
com
pres
sion
met
hods
123
SIViP
Experimental results demonstrate that besides achieving highPSNR, the proposed method outperforms DCT by 9.63 %,Haar wavelet by 9.43 %, contourlet by 3.09 %, curvelet by5.14 %, and JPEG by 15.41 % in terms of compression ratio.
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