Edge Structure Preserving 3-D Image Denoising
Applications of Mathematics and Computer EngineeringEdge Structure Preserving 3-D Image DenoisingPeihua QiuUniversity of MinnesotaSchool of StatisticsMinneapolis, MN 55455USAqiu@stat.umn.eduPartha Sarathi MukherjeeUniversity of MinnesotaSchool of StatisticsMinneapolis, MN 55455USApsm@stat.umn.eduAbstract: In various applications, including magnetic resonance imaging (MRI) and functional MRI (fMRI), 3D images get increasingly popular. To improve reliability of subsequent image analyses, 3-D image denoisingis often a necessary pre-processing step, which is the focus of the current paper. In the literature, most existingimage denoising procedures are for 2-D images. Their direct extensions to 3-D cases generally can not handle 3-Dimages efficiently, because the structure of a typical 3-D image is substantially more complicated than that of atypical 2-D image. For instance, edge locations are surfaces in 3-D cases, which would be much more challengingto handle, compared to edge curves in 2-D cases. In this paper, we propose a novel 3-D image denoising procedure,by approximating the edge surfaces properly, using local smoothing and nonparametric regression methods. Oneimportant feature of this method is its ability to preserve edges and major edge structures (e.g., intersections of twoedge surfaces and pointed corners). Numerical studies show that it works well in various applications.Key–Words: Edge-preserving image restoration, jump regression analysis, surface estimation.1 Introductionmuch more complicated structures (e.g., intersectionsof two edge surfaces, pointed corners, and so forth),compared to edge curves in 2-D cases.In the literature, a number of 3-D image denoisingprocedures have been developed. Some of them areconstructed after properly generalizing and/or modifying their 2-D counterparts. For instance, 3-D image denoising based on minimization of total variation(TV) has gained certain popularity in the literature(e.g., [4]), and the TV approach is initially suggestedfor denoising 2-D images (e.g. [12]). MATLAB programs for 3-D image denoising using anisotropic diffusion have also been developed (e.g., [6]). Otherexisting 3-D image denoising procedures include theones based on 3-D wavelet transformations (e.g., [1]),non-local means (e.g., [2]), distance-weighted Wienerfiltering (e.g. [7]), and so forth.Besides noise removal, another important requirement for image denoising procedures is that theyshould preserve important image structures, such asedges and major edge features (e.g., intersections oftwo edge surfaces, pointed corners, and so forth).Most existing image denoising procedures mentionedabove can preserve parts of the edges where the curvature of edge surfaces is relatively small, but wouldusually blur or round certain edge features at placeswhere the curvature of edge surfaces is relativelylarge. In our opinion, edge features corresponding torelatively large curvature of the edge surfaces are anIn real life, most objects are three-dimensional (3D). Due to limitation of image acquisition techniques,people traditionally acquire a set of two-dimensional(2-D) images from a sequence of slices of a 3-D object (e.g., a patient’s head), and then the 3-D objectis reconstructed from the 2-D images, to study thebiological mechanism of the object. The related research area is termed 3-D image reconstruction in theliterature (e.g., Chapters 11 and 12 in [13]). 3-D image reconstruction is technically challenging, and thereconstructed 3-D image often contains a substantialamount of estimation error. This limitation is recentlylifted in certain applications, including magnetic resonance imaging (MRI) and functional MRI (fMRI). Wecan now acquire 3-D images directly in these applications, although observed 3-D images often containnoise due to hardware imperfection and other reasons.Noise removal is important for the reliability ofsubsequent image analyses, and is often one major focus during the pre-processing stage in image processing. However, most existing image denoising methodsare for analyzing 2-D images (e.g., [5],[8],[9],[11]).Their direct extensions to 3-D cases generally can nothandle 3-D images efficiently, because the structure ofa typical 3-D image is often substantially more complicated than that of a typical 2-D image. For instance,edge locations are surfaces in 3-D cases, which haveISBN: 978-960-474-270-7111
Applications of Mathematics and Computer Engineeringunit ball. The solution to (a, b, c, d) of (2) is deb y, z)),b(x, y, z), bb(x, y, z), cb(x, y, z), d(x,noted as (aand their expressions are given in the appendix. Then,b(x, y, z) is the LLK estimator of f (x, y, z), andab y, z)) are LLK estimators(bb(x, y, z), cb(x, y, z), d(x,′′of (fx (x, y, z), fy (x, y, z), fz′ (x, y, z)).bThegradientvectorβ(x,y, z) Tbb(b(x, y, z), cb(x, y, z), d(x, y, z))providesanestimate of the direction that f increases the fastest.Let us consider a plane that passes (x, y, z) andbis orthogonal to β(x,y, z). Then, this plan di vides O (x, y, z) into two halves O1 (x, y, z) andO2 (x, y, z), as demonstrated in Fig. 1. In O1 (x, y, z)and O2 (x, y, z), we compute weighted averages ofthe observed image intensities, respectively, with theweights determined by K. The weighted averages areb1 (x, y, z) and ab2 (x, y, z). Then, (x, y, z)denoted as ais flagged as an edge voxel ifimportant component of the image under study, because they often represent major characteristics of theimage objects and are easier to capture our visual attention, compared to places on the edge surfaces withrelatively small curvature. Therefore, they should bewell preserved during image denoising.In this paper, we propose a new 3-D image denoising procedure which can preserve edges and major edge features well. Our procedure is based on local approximation of the edge surfaces, using certainjump regression analysis (JRA) methodologies [10].Our proposed procedure is described in detail in Section 2. Some numerical examples are presented inSection 3 to evaluate its performance. Several remarksconclude the article in Section 4.2 Proposed MethodOur proposed 3-D image denoising procedure can bebriefly described as follows. In a neighborhood of agiven voxel, the edge surfaces are first approximatedby a surface template, and then the true image intensity at the given voxel is estimated by a weighted average of the observed image intensities whose voxelsare located on the same side of the surface template asthe given voxel. Details of the procedure are describedin several parts below.b1 (x, y, z) ab2 (x, y, z) Tn , awhere Tn is a threshold. In the case when thereare no edge voxels in O (x, y, z), it can be checkedb1 (x, y, z) ab2 (x, y, z) is distributed approxithat aPb 2 Kh2 (xi , yj , zk ), where σb2 mately as N (0, 4σnPn1b(xi , yj , zk ))2 . Therefore, a reai,j,k 1(ξijk aNsonable choice of Tn isbTn 2Z1 αn σ2.1 Identification of Edge Voxelsξijk f (xi , yj , zk ) εijk , for i, j, k 1, 2, . . . , n,(1)where {(xi , yj , zk ) (i/n, j/n, k/n), i, j, k 1, 2, . . . , n} are the equally spaced voxels in the design space Ω [0, 1] [0, 1] [0, 1], {εijk } are i.i.d.random errors with mean 0 and unknown varianceσ 2 , f (x, y, z) is an unknown image intensity function, and N n3 is the total number of voxels. Ata given voxel (x, y, z) Ω, to know whether it is anedge voxel, let us consider its spherical neighborhoodO (x, y, z) with radius h n . In O (x, y, z), a 3-D planeis fitted using the local linear kernel (LLK) smoothing[3] as follows.a,b,c,dnXrXKh2 n (xi , yj , zk ),where Z1 αn is the (1 αn )th quantile of the standardnormal distribution, and αn is a pre-specified significance level. In this paper, we choose αn 0.0001.Assume that an observed 3-D image follows the regression modelmin(3)Figure 1: The spherical neighborhood O (x, y, z) isdivided into two halves O1 (x, y, z) and O2 (x, y, z) bya plane passing the center (x, y, z) and perpendicularbto the estimated gradient β(x,y, z).{ξijk [a b(xi x) i,j,k 1c(yj y) d(zk z)]}2 Kh n (xi , yj , zk ),(2)2.2 Local Approximation of Edge Surfacesyj y zk zK( xhi x , h , h ),nnnandwhere Kh n (xi , yj , zk ) K is a 3-D density kernel function defined in aISBN: 978-960-474-270-7To estimate f at a given voxel (x, y, z), we suggestfirst approximating the edge surfaces in its spherical112
Applications of Mathematics and Computer Engineeringcross each other and form a subspace in O(x, y, z)containing s are used for approximating the edge surfaces in O(x, y, z).To approximate the edge surfaces by a cone (i.e.,the third surface template), we need to specify its central axis, vertex position, and the angle between thecentral axis and any generatrix. Assume that the direction of the central axis is d (1, d2 , d3 )T . Sincethe angle between this direction and the normal direction at any point on the cone is a constant, d2and d3 can be estimated by minimizing the weightedsample variance of the inner products between d andb , l 1, 2, . . . , m}. Simple calculations show that{βlFigure 2: Three basic surface templates.neighborhood O(x, y, z) with radius hn by a surfacetemplate, where hn may be different from h n usedin (2). To this end, the three basic surface templatesshown in Fig. 2 are considered. Among them, theplane is used for approximating planar parts of theedge surfaces, the intersection of two half planes is forapproximating ridges/roofs of the edge surfaces, andthe cone is for approximating pointed corners. Othersurface templates can be added to this list; but, wethink that the above three templates can describe mostparts of the edge surfaces well.To approximate the edge surfaces in O(x, y, z)by the three basic surface templates, let {sl , l 1, 2, . . . , m} be the detected edge voxels in O(x, y, z),b , l 1, 2, . . . , m} be the corresponding estimated{βlgradient directions (with unit lengths) at these edgevoxels by (2), and (Ψ23 Ψ13 Ψ33 Ψ12 )/ Ψ22 Ψ33 Ψ223d3 (Ψ12 Ψ23 Ψ22 Ψ13 )/ Ψ22 Ψ33 Ψ223 , where Ψj1 j2 is the (j1 , j2 )th component of theb , l weighted sample covariance matrix of {βl1, 2, . . . , m}, for j1 , j2 1, 2, 3. To specify the location of the central axis, let us consider a spheree h . The plane Pe passing seO(x,y, z) of radius hnnewith the normal direction of d would divide O(x,y, z)into two parts. Weighted centers of the detected edgevoxels in the two parts are then calculated, and theone closer to Pe is denoted as s (c x , c y , c z ). Then,the line passing s along the direction d is definedto be the central axis of the cone. In this paper, wee 3h . As a matter of fact, selection ofchoose hnne does not have much effect on the final results. Afhnter the central axis is determined, the angle betweenthe central axis and any generatrix can be easily estimated by the weighted average of the angles betweenb , l 1, 2, . . . , m}. The location of the verd and {βltex (vx , vy , vz ) of the cone can be estimated by minimizing the weighted orthogonal distance between thecone and the detected edge voxels in O(x, y, z).In practice, we need to choose one of the threeestimated surface templates based on observed image intensities for approximating the edge surfaces inO(x, y, z). For that purpose, we suggest the following two-step algorithm. Let RSS1 , RSS2 , and RSS3be the RSS values of the three estimated surface templates, respectively. Then,b , w2 βb , . . . , wm βb ) G (w1 β12mb , w2 βb , . . . , wm βb )T ,(w1 β12mb1 (sl ) ab2 (sl ) Tn which are all poswhere wl aitive at detected edge voxels {sl , l 1, 2, . . . , m}(cf., expression (3)). Therefore, G is a weighted secb }, and the weightsond moment from origin of {βlare determined by the significance of individual detected edge voxels. The eigenvalues of G are denotedas λ1 λ2 λ3 , and the corresponding eigenvectors with unit lengths are v1 , v2 , and v3 . Then, ifb s are the same (i.e., the underlying edge surfaceall βlis a plane), G would have a rank of 1 and v3 wouldbe the normal direction of the edge plane, and viceversa. Therefore, to approximate the edge surfaces inO(x, y, z) by the first surface template, a reasonablesolution is the plane that passes the weighted centers of {sl } with the weights {wl } and has the normaldirection of v3 .To approximate the edge surfaces by the secondsurface template, we proceed in two steps. First, {sl }are divided into two groups by a plane that passes s along the directions of v1 and β where β denotesb , l 1, 2, . . . , m} withthe weighted average of {βlthe weights {wl , l 1, 2, . . . , m}. Second, with eachgroup of the detected edge voxels, find an approximation plane in the same way as the above description about edge surface approximation by the first surface template. Then, the two resulting half planes thatISBN: 978-960-474-270-7 d2 (i) the estimated cone (i.e., the third template) is selected if RSS3 is the smallest one among RSS1 ,RSS2 , and RSS3 ;(ii) otherwise, the estimated plane (i.e., the first template) is selected ifF (x, y, z) 113(RSS1 (x, y, z) RSS2 (x, y, z))/3RSS2 (x, y, z)/(m 6) χ23,1 α̃ ,
Applications of Mathematics and Computer Engineeringand the estimated crossing half-planes (i.e., thesecond template) is selected if F (x, y, z) χ23,1 α̃ , where χ23,1 α̃ is the (1 α̃)th quantileof the χ23 distribution.of detected edge voxels is small, because it is constructed from all observations in O(x, y, z), instead offrom part observations. Considering the fact that mostvoxels in an image are not edge voxels, these benefitsshould be substantial.In the proposed denoising procedure, there aretwo parameters h n and hn involved. We suggestchoosing them by the following cross-validation (CV)procedure:In step (ii), we have used the statistical result thatF (x, y, z) is distributed asymptotically as χ23 when mincreases [?], which is true here because the first template is a special case of the second template and thesecond template has three more parameters than thefirst template. In this paper, we fix α̃ 0.01.CV (h n , hn ) n 21 X b i, j, k (xi , yj , zk ) ,ξ fijkn3i,j,k 1(4)2.3 3-D Image Denoisingwhere fb i, j, k (xi , yj , zk ) is the estimate off (xi , yj , zk ) when the (i, j, k)-th voxel (xi , yj , zk )is excluded from all subsequent steps of the proposed image denoising procedure after edge detection. Then, h n and hn are chosen by minimizingCV (h n , hn ).After determining the surface template for approximating the edge surfaces in O(x, y, z), O(x, y, z) isdivided into two parts by the surface template, in caseswhen the first template is selected, or when the second or third template is selected and their ridge orvertex is contained in O(x, y, z). In cases when thesecond or third template is selected and their ridge orvertex is outside O(x, y, z), O(x, y, z) could be divided into three parts. In any case, the part containing the voxel (x, y, z) is denoted as U (x, y, z). Then,f (x, y, z) can be estimated by the solution to a of theminimization problem (2), after Kh n (xi , yj , zk ) is replaced by I((xi , yj , zk ) U (x, y, z))Khn (xi , yj , zk ),where I(·) is the indicator function and it equals 1 ifits argument is “true” and 0 otherwise. From (2) andthe expressions given in the appendix for its solutions,we can see that fb(x, y, z) is a weighted average of theobserved image intensities whose voxels are locatedon the same side of the estimated surface template inO(x, y, z) as the given voxel (x, y, z). Intuitively, aslong as the estimated surface template approximatesthe underlying edge surfaces well, fb(x, y, z) wouldpreserve edges and major edge features well.For a real image, there are regions where f issmooth. In these regions, the number of detectededge voxels should be small. So, before estimating f , we suggest counting the number of detectededge voxels in O(x, y, z). If the number is so small(e.g., smaller than (nhn )2 ) that a potential edge surface in O(x, y, z) is unlikely, then f (x, y, z) can beestimated simply by the conventional LLK estimatorconstructed from all observations in O(x, y, z). Thedenoising procedure described in the previous paragraph is used only when the number of detected edgevoxels in O(x, y, z) is relatively large. To do so, thereare at least two benefits. One is that much computation is saved, because the conventional LLK estimatoris much easier to compute, compared to the proposeddenoising procedure based on local edge surface approximation. The other benefit is that the estimatedf would be more accurate in cases when the numberISBN: 978-960-474-270-73 Numerical StudyIn this section, we present some numerical examples to evaluate the performance of the proposed 3D image denoising procedure (denoted as NEW), incomparison with three existing procedures, including the ones based on total variation [4] (denoted asTV), anisotropic diffusion [6] (denoted as AD), andoptimized non-local means [2] (denoted as ONLM).The procedure TV has a regularization parameter involved, the procedure AD is an iterative algorithmand contains two parameters, i.e., the diffusion parameter and the number of iterations, and the procedure ONLM has two bandwidth parameters to choose.To evaluate the performance of a denoised image fb,a standard statistical criterion is the mean integratedsquared error (MISE), defined as MISE(fb, f ) R1R1R12b0 0 0 [f (x, y, z) f (x, y, z)] dxdydz, which isestimated by the sample mean ofISE(fb, f ) nnni21 XXXhbf(x,y,z) f(x,y,z)ijkijkn3 i 1 j 1k 1based on 100 replications.A 3-D MRI image of a human brain is used hereas a test image, which has 128 128 52 voxels. Itsimage intensity levels range from 0 to 809. A demonstration of the 3-D image and its three slices are shownin Fig. 3. Then, i.i.d. random noise from the distribution N (0, σ 2 ) is added to the image, and σ is chosen tobe 80, 100, or 120. With each σ value, the parametersof the four denoising procedures are chosen such thattheir estimated MISE values based on 100 replicationsreach the minimum. For the proposed denoising procedure, we also consider choosing its parameters by114
Applications of Mathematics and Computer EngineeringFigure 3: A demonstration of a 3-D image and three 2-D slices.density curve of the true image, shown by the thicksolid curve in the plot, than the density curves of thethree competing procedures.NEW-CV0.015the CV procedure (4). The corresponding method isdenoted as NEW-CV.The estimated MISE values, their standard errors, and the procedure parameter values of the related methods are presented in the first three columnsof Table 1. From the table, we can see that the proposed procedure NEW outperforms all three competing methods in all cases, and the outperformance isstatistically significant because the difference of theMISE values of the procedure NEW and any one of itspeers is larger than 2 times of the sum of the two corresponding standard errors. Also, when its parametersare chosen by the CV procedure (cf., NEW-CV), itsperformance gets slightly worse, compared to the procedure NEW with the parameters chosen by minimizing MISE. But, NEW-CV still outperforms all threecompeting methods in all cases by a reasonably bigmargin.To investigate the denoised images, the densitycurves of the image intensities of the denoised imageswhen σ 100 are shown in Fig. 4. Because the density curves of the procedures NEW and NEW-CV arealmost identical, only the one of NEW-CV is visiblein the plot. From the plot, it can be seen that the density curves of NEW and NEW-CV are closer to theISBN: 978-960-474-270-7TRUTHTVADONLMNEWNEW CV0.010NEWσ 1201649.1 (4.9).00191916.9 (6.3)900,11364.0 (6.2)25,11189.2 (4.1).0188,.01881189.2 (4.1).0188,.01880.005ONLMσ 1001401.1 (3.9).00251525.0 (4.8)700,11127.5 (4.6)16,11022.7 (3.1).0188,.01881029.0 (2.9).0172,.01880.000ADσ 801119.3 (2.9).00301157.8 (3.5)475,1884.8 (3.4)12,1837.3 (2.4).0172,.0156850.4 (2.4).0117,.0156DensityMethodTV0.020Table 1: In each entry, the first line presents the estimated MISE value and their standard errors (in parenthesis), the second line presents the searched procedure parameter values.050100150200Image intensity valuesFigure 4: Density curves of the image intensities ofthe true image (i.e., the 3rd one in Fig. 3) and thedenoised images when σ 100At the end of this section, we consider adding theRician noise to the 3-D test image. By adding the Rician noise, the observed imageq intensity at the voxel(x, y, z) is generated by[f (x, y, z) ǫ1 ]2 ǫ22 ,where ǫ1 and ǫ2 are i.i.d. noise from the distributionN (0, σ̃ 2 ). As in the previous examples, we considercases when σ̃ 80, 100, and 120. For each denoising procedure, we use the bias correction method proposed in [?] to remove estimation bias. Let f (x, y, z)be the intensity of the denoised image by a given denoising procedure.q Then, its bias-corrected versionis defined to be f 2 (x, y, z) 2σ̃ 2 . In practice, σ̃is often unknown and should be estimated from theobservedimage. In this example, it is estimated byps/2, where s is the sample standard deviation of thesquared observed intensities at the first 16 16 13115
Applications of Mathematics and Computer Engineeringvoxels. The calculated MISE values based on 100replications of all five procedures are presented in Fig.5. Again, the MISE curve of the procedure NEW isnot visible in the plot because it is overlapped withthe one of NEW-CV. From the plot, it can be seenthat procedures NEW and NEW-CV outperform allthree competing procedures in all cases except thecase when σ̃ 80. In that case, the procedure ONLMis the best.[2][3][4][5]1500MISE2000TVADONLMNEWNEW CV1000[6][7]8090100110120 σFigure 5: MISE values of various methods based on100 replications.[8]4 Conclusion[9]We have presented a 3-D image denoising procedurewhich can preserve edges and major edge structureswell. Numerical examples show that it performs favorably compare to several existing methods in various cases. However, 3-D image denoising is a challenging problem. The current version of the proposedprocedure may not be able to preserve edges aroundplaces where three or more edge surfaces cross orwhere jump magnitudes in image intensity are closeto zero. To solve these problems, it might help by including more surface templates in the denoising procedure than the ones shown in Fig. 2 when locallyapproximating the edge surfaces. Such potential improvements are left to our future research.[10][11][12][13]Acknowledgements: This research was supported inpart by an NSF grant.References:[1] P. Coupe, P. Hellier, S. Prima, C. Kervrann andC. Barillot, 3D wavelet subbands mixing for im-ISBN: 978-960-474-270-7116age denoising, International Journal of Biomedical Imaging 2008, 2008, pp. 1–11.P. Coupe, P. Yger, S. Prima, P. Hellier, C.Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-Dmagnetic resonance images, IEEE Transactionson Medical Imaging 27, 2008, pp. 425–441.J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, Chapman & Hall,London 1996P. Getreuer, tvdenoise.m (a MATLAB ,2007.I. Gijbel, A. Lambert and P. Qiu, Edgepreserving image denoising and estimation ofdiscontinuous surfaces, IEEE Transactions onPattern Analysis and Machine Intelligence 28,2006, pp. . Lu, C. Jui-Hsi, G. Han, L. Li and Z. Liang, A3D distance-weighted Wiener filter for Poissonnoise reduction in sinogram space for SPECTimaging, SPIE proceedings series 4320, 2001,pp. 905-913.P. Qiu, Discontinuous regression surfaces fitting,The Annals of Statistics 26, 1998, pp. 2218–2245.P. Qiu, The local piecewisely linear kernelsmoothing procedure for fitting jump regressionsurfaces, Technometrics 46, 2004, pp. 87–98.P. Qiu, Image Processing and Jump RegressionAnalysis, John Wiley, New York 2005.P. Qiu, Jump surface estimation, edge detection,and image restoration, Journal of the AmericanStatistical Association 102, 2007, pp. 745–756.L. Rudin, S. Osher and E. Fatemi, Nonlineartotal variation based noise removal algorithms,IEEE Transactions on Pattern Analysis and Machine Intelligence 14, 1992, pp. 259–268.M. Sonka, V. Hlavac and R. Boyle, Image Processing, Analysis, and Machine Vision (3rd ed),Thomson Learning, Toronto 2008.
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A Robust Method for Edge-Preserving Image Smoothing 391 In this paper, we propose a generalized edge-preserving smoothing model derived from robust statistical methods [10]-[14], by exploiting the fact that the basic common principle behind most edge-preserving smoothi
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