ieee transactions on medical imaging, vol. 27, no. 4, …read.pudn.com/downloads675/doc/2731726/an...

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 27, NO. 4, APRIL 2008 425

An Optimized Blockwise Nonlocal Means DenoisingFilter for 3-D Magnetic Resonance Images

Pierrick Coupé*, Pierre Yger, Sylvain Prima, Pierre Hellier, Charles Kervrann, and Christian Barillot

Abstract—A critical issue in image restoration is the problem ofnoise removal while keeping the integrity of relevant image infor-mation. Denoising is a crucial step to increase image quality andto improve the performance of all the tasks needed for quantita-tive imaging analysis. The method proposed in this paper is basedon a 3-D optimized blockwise version of the nonlocal (NL)-meansfilter (Buades, et al., 2005). The NL-means filter uses the redun-dancy of information in the image under study to remove the noise.The performance of the NL-means filter has been already demon-strated for 2-D images, but reducing the computational burden isa critical aspect to extend the method to 3-D images. To overcomethis problem, we propose improvements to reduce the computa-tional complexity. These different improvements allow to drasti-cally divide the computational time while preserving the perfor-mances of the NL-means filter. A fully automated and optimizedversion of the NL-means filter is then presented. Our contribu-tions to the NL-means filter are: 1) an automatic tuning of thesmoothing parameter; 2) a selection of the most relevant voxels;3) a blockwise implementation; and 4) a parallelized computation.Quantitative validation was carried out on synthetic datasets gen-erated with BrainWeb (Collins, et al., 1998). The results show thatour optimized NL-means filter outperforms the classical imple-mentation of the NL-means filter, as well as two other classical de-noising methods [anisotropic diffusion (Perona and Malik, 1990)]and total variation minimization process (Rudin, et al., 1992) interms of accuracy (measured by the peak signal-to-noise ratio) withlow computation time. Finally, qualitative results on real data arepresented.

Index Terms—Image denoising, image enhancement, nonlocalmeans filter.

Manuscript received May 22, 2007; revised July 31, 2007. Asterisk indicatescorresponding author.

*P. Coupé is with University of Rennes, I-CNRS UMR 6074, IRISA, F-35042Rennes, France, with INRIA, VisAGeS U746 Unit/Project, IRISA, F-35042Rennes, France, and also with INSERM, VisAGeS U746 Unit/Project, IRISA,F-35042 Rennes, France.

P. Yger is with University of Rennes, I-CNRS UMR 6074, IRISA, F-35042Rennes, France, with INRIA, VisAGeS U746 Unit/Project, IRISA, F-35042Rennes, France, with INSERM, VisAGeS U746 Unit/Project, IRISA, F-35042Rennes, France, and also with ENS, 94235 Cachan, France.

S. Prima, P. Hellier, and C. Barillot are with University of Rennes, I-CNRSUMR 6074, IRISA, F-35042 Rennes, France, with INRIA, VisAGeS U746 Unit/Project, IRISA, F-35042 Rennes, France, and also with INSERM, VisAGeSU746 Unit/Project, IRISA, F-35042 Rennes, France.

C. Kervrann is with INRA, UR341 Mathématiques et Informatique Ap-pliquées, F-78352 Jouy en Josas, France and with INRIA, VISTA Project-team,IRISA, F-35042 Rennes, France.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2007.906087

I. INTRODUCTION

QUANTITATIVE imaging involves image processingworkflows (registration, segmentation, visualization,etc.) with increasing complexity and sensitivity to pos-

sible image artifacts. As a consequence, image processingprocedures often require to remove image artifacts beforehandin order to make quantitative postprocessing more robust andefficient. A critical issue concerns the problem of noise removalwhile keeping the integrity of relevant image information.This is particularly true for ultrasound images or magneticresonance images (MRI) in presence of small structures withsignals barely detectable above the noise level. In addition, aconstant evolution of quantitative medical imaging is to processalways larger cohorts of 3-D data in order to find significantdiscriminants for a given pathology (e.g., see [5]). In thiscontext, complex automatic image processing workflows arerequired [6] since human interpretation of images is no longerfeasible. For effectiveness, these workflows have to be robustto a wide range of image qualities and parameter-free (or atleast using auto-tuned parameters). This paper focuses on thesecritical aspects by introducing a new restoration scheme in thecontext of 3-D medical imaging. The nonlocal (NL) means filterwas originally introduced by Buades et al. [1] for 2-D imagedenoising. The adaptation of this filter we propose for 3-Dimages is based on: 1) an automatic tuning of the smoothingparameter; 2) a selection of the most relevant voxels for theNL-means computation; 3) a blockwise implementation; and4) a parallelized computation. These different contributionsallow to make the adapted filter fully automated and above allto overcome the main limitation of the classical NL-means: thecomputational burden.

Section II gives a short overview of the literature on imagerestoration. Section III presents the proposed method with de-tails about our contributions. Sections IV–VI show 1) the impactof our adaptations compared to the classical NL-means imple-mentation and 2) a comparison with respect to other well-estab-lished denoising methods on Gaussian and Rician noise. Bothvalidation experiments are performed on a phantom data set ina quantitative way. Section VII shows results on real data suchas 3 T T1-weighted (T1-w) MRI and T2-weighted (T2-w) MRIof a patient with multiple sclerosis (MS) lesions. In Section VIII,we propose a discussion on the applicability and the further im-provements of the NL-means filter in the context of 3-D medicalimaging.

II. STATE-OF-THE-ART

A. General Overview

Many methods have been proposed for edge-preservingimage denoising. Some popular approaches include Bayesian

0278-0062/$25.00 © 2008 IEEE

426 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 27, NO. 4, APRIL 2008

approaches [7], PDE-based approaches [3], [4], [8], [9], ro-bust and regression estimation [10], adaptive smoothing [11],wavelet-based methods [12]–[14], bilateral filtering [15]–[17],local mode filtering [18], and hybrid approaches [19]–[21].

Strong theoretical links exist between most of these tech-niques, as recently shown for local mode filtering [18], bilateralfiltering, anisotropic diffusion (AD) and robust estimation [17],[22] and adaptive smoothing [23], AD, and total variation (TV)minimization scheme [24].

More recently, some promising methods have been proposedfor improved image denoising, based on statistical averagingschemes enhanced via incorporating a variable spatial neigh-borhood scheme [25]–[29]. Other approaches consist in mod-eling NL pairwise interactions from training data [30] or a li-brary of natural image patches [31], [32]. The idea is to improvethe traditional Markov random field (MRF) models by learningpotential functions from examples and extended neighborhoodsfor computer vision applications [30]–[32]. Awate and Whitakerproposed another nonparametric patch-based method relying oncomparisons between probability density functions [33].

Some of these techniques, generally developed for 2-D im-ages, have often been extended to 3-D medical data, especiallyto MR images: AD [34], [35], TV [36], bilateral filtering andvariants [37], wavelet-based filtering [38]–[42], and hybrid ap-proaches [43], [44].

B. Introduction of the NL-Means Filter

Most of the denoising methods restore the intensity value ofeach image voxel by averaging in some way the intensities of its(spatially) neighboring voxels. The basic and intuitive approachis to replace the value of the voxel by the average of the voxelsin its neighborhood (so-called box filtering [45]). In practice,this filter has been shown to be outperformed by the Gaussianfilter, which consists in weighting each voxel in the neighbor-hood according to its distance to the voxel under study. Bothfilters can be iterated until the desired amount of smoothingis reached. Such data-independent approaches can be imple-mented very efficiently. Their major drawback is that they blurthe structures of interest in the image (e.g., edges or small struc-tures and textures).

This has naturally led to data-dependent approaches, whichaim at eliminating (or reducing the influence of) the neighboringvoxels dissimilar to the voxel under study. Simple order statisticoperators can be used for this purpose, such as the median filter,leading to a simple generalization of the box filter. More so-phisticated approaches, based on image derivatives have beensuccessfully proposed for many applications, such as adaptivesmoothing [11] and AD [3]. Neighborhood filters [46], [47] andvariants [15], [16], have been also proposed and consist in av-eraging input data over the image voxels that are spatially closeto the voxel under study and with similar gray-level values.

All these techniques rely on the idea that the restored value ofa voxel should only depend on the voxels in its spatial neighbor-hood that belong to the same population, that is the same imagecontext. This has been termed by Elad as the locally adaptiverecovery paradigm [17]. Another approach has been recentlyproposed, that has shown very promising results. It is based onthe idea that any natural image has redundancy, and that any

voxel of the image has similar voxels that are not necessarilylocated in a spatial neighborhood. First introduced by Buadeset al. in [1], the NL-means filter is based on this redundancyproperty of periodic images, textured images, or natural imagesto remove noise. In this approach, the weight involving voxelsin the average, does not depend on their spatial proximity tothe current voxel but is based on the intensity similarity of theirneighborhoods with the neighborhood of the voxel under study,as in patched-based approaches. In other words, the NL-meansfilter can be viewed as an extreme case of neighborhood filterswith infinite spatial kernel and where the similarity of the neigh-borhood intensities is substituted to the point-wise similarity ofgray levels as in commonly-used bilateral filtering. This new NLrecovery paradigm allows to combine the two most importantattributes of a denoising algorithm: edge preservation and noiseremoval.

III. METHODS

In the following, we introduce the notations.Image where represents the grid of the image,considered as cubic for the sake of simplicity andwithout loss of generality .

For the original voxelwise NL-means approach:

Intensity observed at voxel .

Cubic search volumecentered on voxel of size

, .

Cubic local neighborhood ofof size ,

.

Vector containing theintensities of .

Restored value of voxel .

Weight of voxel whenrestoring

For the blockwise NL-means approach.

Block centered on of size, .

Vector containing the intensities of theblock .

Vector containing the restored value of .

Vector containing the intensities of .

Restored value of voxel .

Weight of block when restoring theblock .

Blocks centered on voxels th

and represents the distance between thecenters of the blocks .

COUPÉ et al.: AN OPTIMIZED BLOCKWISE NONLOCAL MEANS DENOISING FILTER 427

Fig. 1. (a) Classical voxelwise NL-means filter: 2-D illustration of the NL-means principle. Restored value of voxel x (in red) is the weighted average of allintensities of voxels x in the search volume V , based on the similarity of their intensity neighborhoods u(N ) and u(N ). In this example, we set d = 1 andM = 8. (b) Blockwise NL-means filter: 2-D illustration of the blockwise NL-means principle. Restored value of the block B is the weighted average of all theblocks B in the search volume V . In this example, we set � = 1 and M = 8.

A. NL Means Filter

In the classical formulation of the NL-means filter, the re-stored intensity of the voxel , is the weighted av-erage of all the voxel intensities in the image defined as

(1)

where is the intensity of voxel and is theweight assigned to in the restoration of voxel . Moreprecisely, the weight quantifies the similarity of the local neigh-borhoods and of the voxels and under the assump-tions that and [cf.Fig. 1(a)]. The classical definition of the NL-means filter con-siders that each voxel can be linked to all the others, but forpractical computational reasons the number of voxels taken intoaccount in the weighted average can be limited to the so-called“search volume” of size , centered at the currentvoxel .

For each voxel in , the Gaussian-weighted Euclideandistance defined in [1], is computed between and

. This distance is a classical norm convolved with aGaussian kernel of standard deviation , and measures the dis-tance between neighborhood intensities. Given this distance,

is computed as follows:

(2)

where is a normalization constant ensuring that, and acts as a smoothing parameter

controlling the decay of the exponential function. When isvery high, all the voxels in will have the same weight

with respect to the voxel . The restored valuewill be then approximately the average of the

intensity values of the voxels in leading to strong smoothingof the image. When is very low, the decay of the exponential

function will be strong, thus only few voxels in withvery similar to will have a significant weight. The

restored value will tend to be the weighted averageof some voxels with a similar neighborhood to current voxelleading to a weak smoothing of the image. In Section III-B-1,a tradeoff has then to be found, and we propose a method toautomatically estimate the optimal value of .

In [1], Buades et al. show that, for 2-D natural images,the NL-means filter outperforms state-of-the-art denoisingmethods such as the Rudin–Osher–Fatemi TV minimiza-tion scheme [4], the Perona–Malik AD [3], or translationinvariant wavelet thresholding [48]. Nevertheless, the maindrawback of the NL-means filter is the computational burdendue to its complexity, especially for 3-D images. Indeed, foreach voxel of the volume, distances between the intensityneighborhoods and for all the voxels con-tained in need to be computed. Let denote the sizeof the 3-D image, then the complexity of the filter is in theorder of . For a 3-D MRI of size

with the smallest possible value for and( and ), the computational time reaches up

to 6 h on a 3-GHz CPU. This time is far beyond a reasonableduration expected for a denoising filter in a medical practice.For this reason, we propose several adaptations to reduce thecomputational burden which are detailed in Section III-B. Wealso show that these adaptations improve the quality of thedenoising compared to the classical implementation.

B. Improvements of the NL-Means Filter

1) Automatic Tuning of the Smoothing Parameter : Ac-cording to [1], the smoothing parameter depends on the stan-dard deviation of the noise , and typically a good choice for2-D images is . Equation (2) shows that also needsto take into account , if we want a filter independent of theneighborhood size. Indeed, the norm increasing with ,

needs also to be increased to obtain an equivalent filter. The


Fig. 2. (a) Noisy image with 9% of Gaussian noise (see Section IV). (b) Map of the mean of u(N ) denoted u(N ). (c) Map of the variance of u(N ) denotedVar(u(N )). In these examples, we set N = 5 � 5 � 5 voxels.

automatic tuning of the smoothing parameter comes to deter-mine the relationship where is a constant.Let us show how we can estimate this relationship.

1) In case of an additive white Gaussian noise, the standarddeviation of noise can be estimated via pseudo-resid-uals as defined in [49], [50]. For each voxel of thevolume , let us define

(3)

being the six-neighborhood at voxel and the con-stant is used to ensure that in homo-geneous areas. Thus, the standard deviation of noiseis computed as the least square estimator

(4)

2) Initially, the NL-means filter was defined with aGaussian-weighted Euclidean distance, definedin [1]. However, in order to make the filter independentof , to simplify the complexity of the problem, andto reduce the computational time, we used the classicalEuclidean distance normalized by the number ofelements:

(5)Finally, (2) becomes

(6)

where only the adjusting constant needs to be manu-ally tuned. In the case of Gaussian noise, is theoret-ically be close to 1 (see [51, p. 21] for theoretical jus-tification) if the estimation of the standard deviationof the noise is correct. The adjustment of will be dis-cussed in Section V-A.

2) Voxel Selection in the Search Volume: To deal with com-putational burden, Mahmoudi and Sapiro [52] recently proposed

a method to preselect a subset of the most relevant voxels into avoid useless weight computations. In other words, the

main idea is to select only the voxels in that will havethe highest weights in (1) without having to com-pute all the Euclidean distances between and . Apriori neglecting the voxels which are expected to have smallweights tends to speed up the filter and to improve the results(see Table II). In [52], this selection is based on the similarity ofthe mean value of and , and on the similarity of theaverage over the neighborhoods and of the gradient orien-tation at pixel and . Intuitively, similar neighborhoods havethe same mean and the same gradient orientation. The computa-tion of the gradient orientation is very sensitive to noise and thusrequires robust estimation techniques. This is too computation-ally expensive for medical applications. For this reason, in ourimplementation, the preselection of voxels in is based on themean and the variance of and which allows to de-crease the computational burden. Fig. 2 shows that the maps oflocal means and local variances are simple estimators allowingto discriminate different tissue classes and edges in images. Inthis way, the maps of local means and local variances are pre-computed in order to avoid repetitive calculations for the sameneighborhood. The selection tests can be expressed as follows:

if

and

otherwise.

(7)

where and represents, respectively, the meanand the variance of the local neighborhood of voxel . Assuggested in [52], with this kind of selection, the NL-meansfilter tends to better preserve the detailed regions while slightlyspoiling the denoising of the flat regions. Indeed, in flat regionsincreasing the number of voxels tends to improve denoising be-cause there are a large number of similar voxels. In more clut-tered regions, increasing the number of voxels tends to removethe details during smoothing because there are very few similarvoxels.

3) Blockwise Implementation: A blockwise implementationof the NL-means is developed as suggested in [1]. This approach


Fig. 3. Blockwise NL-means Filter. For each blockB centered on voxel x ,a NL-means like restoration is performed from blocks B . In this way, for avoxel x included in several blocks, several estimations are obtained. Restoredvalue of voxel x is the average of the different estimations stored in vectorA .In this example, � = 1, n = 2, and jA j = 3.

consists in: 1) dividing the volume into blocks with overlap-ping supports; 2) performing NL-means-like restoration of theseblocks; and 3) restoring the voxels values based on the restoredvalues of the blocks they belong to.

1) A partition of the volume into overlapping blocksof size is performed, such as ,under the constraint that the intersections betweenthe are nonempty (see Fig. 3). These blocksare centered on voxels which constitute a subsetof . The are equally distributed at positions

, where repre-sents the distance between the centers of . To ensurea global continuity in the denoised image, the supportoverlapping of blocks has to be nonempty: .

2) For each block , a NL-means-like restoration is per-formed as follows:

(8)

with

(9)

where is a normalization constant ensuring that[see Fig. 1(b)].

3) For a voxel included in several blocks , several esti-mations of the restored intensity are obtained indifferent (see Fig. 3). The estimations givenby different for a voxel are stored in avector . The final restored intensity of voxel is thendefined as

(10)

The main advantage of this approach is to significantly reducethe complexity of the algorithm. Indeed, for a volume of size

, the global complexity is. For instance, with , the complexity is divided by

a factor 8. The voxels selection principle can also be applied tothe blockwise implementation

if

and

otherwise(11)

where and represent respectively the meanand the variance of the intensity function, for the block cen-tered on the voxel .

4) Parallel Computation: Another way to reduce the compu-tational time is to distribute the operations on several processorsvia a cluster or a grid. The intrinsic nature of the NL-means filtermakes it perfectly suited for parallelization and multithreadingimplementation. One of the main advantage of this filter, whencompared to other methods such as AD or TV minimization, isthat the operations are performed without any iterative schemes.Thus, the parallelization of the NL-means filter is straightfor-ward to perform and very efficient. We divide the volume intosubvolumes, each of them being treated separately by one pro-cessor. A server with 8 Xeon processors at 3 GHz and an IntelPentium D CPU 3.40 GHz were used in our experiments.

IV. MATERIALS

A. BrainWeb Database

In order to evaluate the performances of the NL-means filteron 3-D MR images, tests were performed on the BrainWeb data-base1 [2]. Two images were simulated: T1-w MR image usingSFLASH sequence volume size and T2-wMR image with MS from SFLASH sequence volume size

. As reported previously, it is a known factthat the MR images are corrupted by a Rician noise [53], [54],which can be well approximated by a white Gaussian noise inhigh intensity areas, typically in brain tissues [38]. In order toverify if this approximation can be used for a NL-means baseddenoising, experiments are performed on phantom images withGaussian and Rician noise.

1) Gaussian Noise: A white Gaussian noise was added onthe “ground truth,” and the notations of BrainWeb are used: anoise of 3% is equivalent to , where is thevalue of the brightest tissue in the image (150 for T1-w and250 for T2-w). Several images were simulated to validate theperformances of the denoising on various images (see Fig. 4):

• T1-w MR images for four levels of noise 3%, 9%, 15%,and 21%.

• T2-w MR images with multiple sclerosis (MS) lesions forfour levels of noise 3%, 9%, 15%, and 21%.

T2-w images were used in order to show that our approach andits calibration are not specific to T1-w MRI sequences. More-over, the tests on T2-w MRI with MS show how the NL-means

1http://www.bic.mni.mcgill.ca/brainweb/.


Fig. 4. Synthetic data used for validation with Gaussian noise. Example of thebrainweb database. Top: T1-w images without any noise (a), and corrupted witha white Gaussian noise at 9% (b). Bottom: T2-w images with MS lesions withoutnoise (c), and corrupted with a white Gaussian noise at 9% (d).

filter could be useful in a pathological context due to its preser-vation of anatomic and pathologic structures.

2) Rician Noise: The Rician noise was built from whiteGaussian noise in the complex domain. First, two images arecomputed

• ,• ,

where is the “ground truth” and is the standard deviationof the added white Gaussian noise. Then, the noisy image iscomputed as

(12)

The notation 3% for the Rician noise means that the Gaussiannoise used in complex domain is equivalent to ,where is the value of the brightest tissue in the image (150for T1-w). According to the peak signal-to-noise ratio (PSNR)[see (13)] between “ground truth” and noisy images, for a samelevel of noise, the Rician noise is stronger than the Gaussiannoise (see Table I). Several images were simulated (see Fig. 5)

• T1-w MR images for four levels of noise 3%, 9%, 15%,and 21%.

B. Real Data

1) T1-w High Field MRI Data: To show the efficiency ofthe NL-means filter on real data, tests were performed on imageacquired with a high field MR system (Bruker 3 T). The dataused was a T1-w image.

TABLE IPSNR BETWEEN “GROUND TRUTH” AND NOISY IMAGES FOR GAUSSIAN

AND RICIAN NOISES. FOR A SAME LEVEL OF NOISE, THE RICIAN NOISE IS

STRONGER THAN THE GAUSSIAN NOISE

Fig. 5. Synthetic data used for validation with Rician noise. Example of theBrainweb database. T1-w images without any noise (a), and corrupted with aRician noise at 9% (b).

2) T2-w With Multiple Sclerosis Lesions: In a pathologicalcontext, the denoising step is crucial especially when the struc-tures of interest have a small size: the integrity of pathologicalstructures must be preserved by the denoising method. As saidearlier, one objective of denoising is to include such processingin complex medical imaging workflows. This kind of workflowis widely used to process large cohort of subjects in many neu-rological diseases such as MS lesions. The data used for MSlesions qualitative validation was a T2-w MR image from anaxial dual-echo, turbo spin-echo sequence (Philips 1.5 T).

V. VALIDATION ON A PHANTOM DATA SET

WITH ADDED GAUSSIAN NOISE

In the following, let us define the following.• NL-means is the standard voxelwise implementation with

automatic tuning of the smoothing parameter.• Optimized NL-means is a voxelwise implementation with

automatic tuning of the smoothing parameter, voxel selec-tion and multithreading.

• Blockwise NL-means is the standard blockwise im-plementation with automatic tuning of the smoothingparameter.

• Optimized Blockwise NL-means is a blockwise imple-mentation with automatic tuning of the smoothing param-eter, block selection and multithreading.

In this section, different aspects of NL-means filter implementa-tion were investigated. First, the impact of the automatic tuningof the filtering parameter (Section V-A) and the influenceof the size of the search volume and the neighborhood werestudied (Section V-B). Then, the impact of voxels selection andblockwise implementation is analyzed via the comparison ofthe NL-means, Optimized NL-means, Blockwise NL-means,and Optimized Blockwise NL-means filters (Sections V-C


and V-D). Finally, we compare the proposed OptimizedBlockwise NL-means filter with other well-established de-noising methods: AD filter [3] (implemented in VTK2) andRudin-Osher-Fatemi TV minimization process [4] (3-D exten-sion of the Megawave2 implementation3) (Section V-G). Thedifferent variants of the NL-means filter can be freely testedonline4.

In the following, several criteria are used to quantify theperformances of each method: the PSNR obtained for differentnoise levels, histogram comparisons between the denoisedimages and the “ground truth,” and finally visual assessment.For the sake of clarity, the PSNR and the histograms are es-timated only in a region of interest obtained by removing thebackground. For 8-bit encoded images, the PSNR is defined asfollows:

(13)

where the RMSE is the root mean square error estimated be-tween the ground truth and the denoised image.

In this section, the central parameters of interest are• defining the smoothing parameter

(see Section III-B-1);• related to ;• related to ;• and corresponding to the thresholds involving in the

voxel selection.In each experiment (Sections V-A–V-C), we let one pa-

rameter vary while remaining the others constant, with defaultvalues: , , , and , . Con-cerning the blockwise implementation the default parametersare and .

Our experiments have shown that all the versions of theNL-means filter (NL-means, Optimized NL-means, Block-wise NL-means, and Optimized Blockwise NL-means) tendto have a similar behavior with respect to the variation of theparameters. In that context, all the results are displayed withthe proposed Optimized Blockwise NL-means filter, evenif equivalent conclusions can be drawn with the NL-means,Optimized NL-means, and Blockwise NL-means filters.

Validation was performed on T1-w and T2-w MRI, but the re-sults concerning the study of the parameter influences are shownfor T1-w MRI only. The results on T2-w MRI are shown inSection V-F in order to underline that the parameters calibratedfor T1-w MRI work fine on T2-w MRI.

A. Influence of the Automatic Tuning of SmoothingParameter

Fig. 6 shows the influence of the automatic determinationof the smoothing parameter . As described inSection III-B-1, is a function of the global standard deviationof the noise in the volume esimated from pseudo-residuals[see (3) and (4)]. Here, allows to adjust the automatic estima-tion of in order to determine the optimal smoothing param-eter for each level of noise [see (6)]. For low levels

2http://www.vtk.org3http://www.cmla.ens-cachan.fr/Cmla/Megawave/index.html4http://www.irisa.fr/visages/benchmarks

Fig. 6. Calibration of the smoothing parameter h: Influence of the smoothingparameter 2��̂ jN j on the PSNR, according to � and for several levels ofnoise. For low levels of noise the best value of � is close to 0.5. For high levelsof noise this value is 1. The default value of � is set to 1, thus the estimation of his h = 2�̂ . These results are obtained with �̂ = 3:42% at 3%, �̂ = 7:93%

at 9%, �̂ = 12:72% at 15%, and �̂ = 17:44% at 21%.

of noise, the best value of is close to 0.5. For high levels ofnoise this value is 1. These results show that the estimation ofthe standard deviation of the noise is correctly performed bypseudo-residuals. These observations underline 1) how efficientthe automatic estimation of the smoothing parameter is, and2) how the NL-means can be used without manual parametertuning.

B. Influence of the Size of the Search Volume and theNeighborhood

Fig. 7 shows the influence of the size of the search volume andthe local neighborhood. Increasing the number of voxels in thesearch volume does not seem to affect the PSNR when isgreater than 5. Indeed, the theoretic definition of the NL-meansfilter states that the weighted average [see (1)] computed for therestoration of voxel should be performed on all voxels

. Practically, the limit prevents useless computations.Moreover, increasing degrades the denoising process. When

increases, the NL-means filter drastically slows down. That iswhy we have not investigated the impact of for .

C. Influence of the Voxel Selection

The selection of the voxels in the search volume isachieved by supposing that only voxels whose neighbor-hood is similar to the neighborhood of the voxel understudy could be considered [see (1)]. To do so, as definedin Section III-B-2, the weight is calculated onlyfor voxels such that: and

. The influence of thelimits and is studied in Fig. 8. In a first experiment,varies according to such as while .In a second experiment varies according to following

while .


Fig. 7. Influence of the size jV j = (2M +1) and jN j = (2d+1) for denoising: influence of the size of the search volume and the size of the neighborhoodon the PSNR, for several levels of noise. (a) Variation of the size M of the search volume V for d = 1. (b) Variation of the size d of the neighborhood N forM = 5. These results show that the limit M = 5 prevents useless computation. Moreover, increasing d degrades and drastically slows down the algorithm.

Fig. 8. Influence of the limits of the voxels selection. Influence of the limits � and � on the PSNR, for several level of noise. (a) � = 0:5, while � varies with . A restrictive selection based on the mean (low values of ) increases the PSNR. The optimal limits are obtained for � = 0:95. (b) � = 0:95 and � variesaccordingly to . A too restrictive selection (low values of ) degrades the PSNR. In addition, a too permissive selection (high values of ) does not increase thePSNR while concurrently increasing uselessly the computational burden. A good compromise is found by fixing � = 0:5.

Fig. 8(a) shows that a restrictive selection based on the mean(low values of ) increases the PSNR. In other words, thenumber of voxels taken into account in the weighted average isdrastically reduced, as well as the computational time (also seeTable II). The optimal limits were obtained for while

. Concerning the variance [Fig. 8(b)], we observethat a too restrictive selection degrades the PSNR. In addition,a too permissive selection does not increase the PSNR whileincreasing uselessly the computational burden. A compromisewas found by fixing . There is a clear dependencybetween the bounds for the mean and the variance. An optimaltradeoff was determined experimentally.

D. Influence of the Blockwise Implementation

Table II shows that the blockwise approach of the NL-meansfilter, with and without voxels selection [see (11)], allows todrastically reduce the computational time. With a distancebetween the block centers , the blockwise approachdivides this time by a factor (see Table II). However,computational time reduction needs to be balanced with aslight decrease of the PSNR [see Fig. 9(a)]. For the optimizedversions, the voxels/blocks selection in the search volumehas several impacts. First, by reducing the average number ofvoxels/blocks used in the weighted averages, this decreases


TABLE IICOMPARISON OF DIFFERENT IMPLEMENTATIONS OF THE NL-MEANS FILTERS IN TERMS OF COMPUTATIONAL TIME AND DENOISING QUALITY. TIME IS OBTAINED

WITH MULTITHREADING ON 8 CPUS AT 3 GHz AND INTEL PENTIUM D CPU 3.40 GHz AND WITHOUT MULTITHREADING ON 1 CPU AT 3 GHz. THESE RESULTS

ARE OBTAINED ON A T1-W BRAINWEB IMAGE WITH 9% OF GAUSSIAN NOISE (� = 13:5). PARAMETERS USED ARE THE DEFAULT PARAMETERS. AVERAGE

NUMBER OF VOXELS/BLOCKS USED IN V TO DENOISE u(x ) SHOWS THE IMPACT OF VOXELS/BLOCKS SELECTION. FOR THE NONOPTIMIZED IMPLEMENTATIONS

ALL THE VOXELS/BLOCKS IN V ARE TAKEN INTO ACCOUNT TO DENOISE u(x ). THUS, THE NUMBER OF VOXELS/BLOCKS USED ARE jV j = (2M +1) = 11 .FOR THE OPTIMIZED IMPLEMENTATIONS, THE VOXEL SELECTION ALLOWS TO DRASTICALLY REDUCE THIS NUMBER

Fig. 9. Impact of the blockwise implementation and voxels selection. Comparison of the different implementations of the NL-means filter, with � = 1. (a) OnT1-w images. For the Optimized Blockwise NL-means filter, as for the Optimized NL-means filter, the selection of voxels/blocks in the search volume improvesthe quality of denoising and decreases the computational burden (see Table II). Reduction of computational time brought by the blockwise approach needs to bebalanced with a slight decrease in quality of denoising. (b) On T2-w images with MS lesions. Same conclusions can be drawn for this kind of images. These resultssuggest that the parameters tuning determined experimentally on T1-w images are not T1-specific.

the computational time compared to the nonoptimized ver-sions (see Table II). Second, the selection of the most relevantvoxels/blocks increases the quality of denoising for all the noiselevels (see Fig. 9(a) and Table II).

E. Multithreading

As described in Section III-B-4, the multithreading in thecase of the NL-means filter is particularly adapted due itsnoniterative nature. For the classical pixelwise NL-meansimplementation, the parallelization allows to divide the compu-tational time by a factor close to the numbers of CPU. As eightprocessors were used for our experiments, the computationaltime with multithreading is about 8 times smaller (see Table II,

and ). For the blockwiseimplementations, the speedup is less ( and

). The difference of speedup between the clas-sical NL-means and the blockwise NL-means filters have twoorigins.

• First, in blockwise version several threads could write atthe same memory location (i.e., vector ) at the same

time. In multithreading programming, this kind of possi-bility requires a lock which protects the memory locationduring the writing. Unfortunately, to lock a memory loca-tion speeds down the computational process.

• Second, as the required computation time is shorter forthe blockwise than for the voxelwise implementation, therelative contribution of the nonmultithreaded operations inthe overall computation time (opening and closing of file,computation of the local maps, etc.) is much higher in theblockwise compared to the voxelwise implementation. Asa consequence, the speedup factor will be higher in thelatter.

In order to underline that the utilization of 8-CPUs is not re-quired by our filter, the denoising have been also performed ona more common architecture: a DualCore Intel Pentium D CPU3.40 GHz. The results show that our filter takes less than 3 minto denoise a volume voxels on this architecture.

To conclude, the different improvements included in the pro-posed Optimized Blockwise NL-means filter (i.e., blockwiseapproach and blocks selection) allow to speed up the denoising


Fig. 10. Comparison of the optimized and nonoptimized blockwise NL-means on T2-w images: NL-means restoration of T2-w Brainweb data with MS lesions.From left to right: “ground truth,” noised image at 9% of Gaussian noise, restored images by the Optimized Blockwise NL-means filter and by the BlockwiseNL-means filter. Optimized Blockwise NL-means filter preserves efficiently the contours of the MS lesions.

Fig. 11. Result for AD filter and TV minimization on phantom images with Gaussian noise at 9%. For AD filter K varies from 0.05 to 1 with step of 0.05 and thenumber of iterations varies from 1 to 10. For TV minimization � varies from 0.01 to 1 with step of 0.01 and the number of iterations varies from 1 to 10.

procedure, compared to NL-means filter, by a factor of 66 on1 Xeon at 3 GHz, 44 on 8 Xeon at 3 GHz and 31 on a DualCoreat 3.40 GHz.

F. Optimized Blockwise NL-Means Filter on T2-w MRIWith MS

Fig. 9(b) presents the results obtained by the differentNL-means filter versions on T2-w MRI with MS lesions. Theoptimal parameters (i.e., the default parameters described inSection V), experimentally determined on T1-w MRI, and the

automatic tuning of were used on T2-w MRI. The OptimizedNL-means and Optimized Blockwise NL-means filters out-perform the NL-means and Blockwise NL-means filters alsoon T2-w MRI. The most important difference between theoptimized and nonoptimized versions are observed on T2-wMRI, which could be explained by the higher level of noise inthe simulated T2-w MRI compared to T1-w MRI. Actually, thevariance of noise varies with respect to the highest intensitytissues which is 150 in T1-w and 250 in T2-w. For 9% the vari-ance of noise is 13.5 in T1-w images and is 22.5 in T2-w images


Fig. 12. Comparison between AD, TV, and Optimized Blockwise NL-means denoising: the PSNR values and histograms for different denoising methods onBrainWeb at 9% of Gaussian noise. (a) PSNR experiment shows that the Optimized Blockwise NL-means filter outperforms the well-established TV minimizationprocess and the AD approach. (b) Contrary to others methods, the NL-means based restoration clearly distinguishes the three main peaks representing the whitematter, the gray matter, and the cerebrospinal fluid. Sharpness of the peaks shows how the Optimized Blockwise NL-means filter increases the contrast betweendenoised biological structures.

because the highest tissue intensity is superior in T2-w images.These results suggest that the parameters experimentally tunedon T1-w images can be used for T2-w images. Fig. 10 showsan example of denoising obtained by the optimized blockwiseNL-means and the blockwise NL-means filters. The MS lesionsare visually more preserved with the optimized version; thiswas confirmed by an experienced MRI reader.

G. Comparison With Other Denoising Methods

1) Focus on Two Classical Denoising Approaches:a) AD filtering: As reported in Section II, the AD filter

was introduced to overcome the blurring effect of the Gaussiansmoothing approach. First introduced by Perona and Malik [3],in this approach the image is only convolved in the directionorthogonal to the gradient of the image which ensures the preser-vation of edges. The iterative denoising process of initial image

can be expressed as

(14)

where is the image gradient at voxel and iteration ,is the partial temporal deviation of and

(15)

where is the diffusivity parameter. The AD filter method pro-duces a good preservation of edges [34], [35]. Nonetheless, themain disadvantage of AD filter is to poorly denoise the constantregions (see Fig. 13).

b) TV minimization scheme: The difficult task to preserveedges while correctly denoising constant areas has been ad-

dressed also by Rudin, Osher, and Fatemi. They proposed tominimize the TV norm subject to noise constraints [4], that is

(16)

subject to

(17)where is the original noisy image, is the restored imageand the standard deviation of the noise. In this model, theTV minimization tends to smooth inside the image structureswhile keeping the integrity of boundaries. The TV minimizationscheme can be expressed as an unconstrained problem

(18)where is a Lagrange multiplier which controls the balancebetween the TV norm and the fidelity term. Thus, acts as thefiltering parameter. Indeed, for high values for the fidelityterm is encouraged. For small values for the regularity termis desired. In practice, the TV minimization scheme tends toremove texture and small image structures, as seen in Fig. 13[36]. To solve this problem, iterative TV schemes have beenrecently developed [55], [56].

2) Quantitative and Qualitative Comparison: The main dif-ficulty to achieve this comparison is related to the tuning ofsmoothing parameters in order to obtain the best results for ADfilter and TV minimization scheme. In order not to penalize ADfilter and TV minimization scheme, an exhaustive search for


TABLE IIICOMPARISON OF HISTOGRAMS OBTAINED WITH THE THREE DIFFERENT METHODS AT 9% OF GAUSSIAN NOISE. THIS TABLE PRESENTS (a) THE BHATTACHARYYA

COEFFICIENT COMPUTED BETWEEN THE HISTOGRAMS OF DENOISED IMAGES AND THE “GROUND TRUTH” ONE AND (b) THE MUTUAL INFORMATION COMPUTED

BETWEEN THE DENOISED IMAGES AND THE “GROUND TRUTH.” DISTANCE BETWEEN THE NOISY IMAGE AND THE “GROUND TRUTH” IS USED AS A REFERENCE.COMPARED TO AD FILTER AND TV MINIMIZATION, THE OPTIMIZED BLOCKWISE NL-MEANS FILTER ALLOWS TO

OBTAIN A DENOISED IMAGE WHOSE HISTOGRAM IS MORE CLOSER TO “GROUND TRUTH” HISTOGRAM

Fig. 13. Comparison with AD, TV, and NL-means denoising on synthetic T1-wimages. Top: zooms on T1-w BrainWeb images. Left: the “ground truth.” Right:the noisy images with 9% of Gaussian noise. Middle: the results of restora-tion obtained with the different methods and the images of the removed noise(i.e., the difference (centered on 128) between the noisy image and the denoisedimage. Bottom: the difference (centered on 128) between the denoised imageand the ground truth. Left: AD denoising. Left: AD denoising. Middle: TVminimization process. Right: Optimized Blockwise NL-means filter. NL-meansbased restoration better preserves the anatomical structure in the image while ef-ficiently removing the noise as it can be seen in the image of removed noise.

all parameters into a certain range. Then, the best results ob-tained with AD filter and TV minimization have been selected,whereas the fully automatic results have been mentioned for theNL-means filters. The results of the NL-means filters are not

Fig. 14. Comparison with AD, TV, and NL-means denoising on synthetic T1-wimages. Top: zooms on T1-w BrainWeb images. Left: the “ground truth.” Right:the noisy images with 21% of Gaussian noise. Middle: the results of restora-tion obtained with the different methods and the images of the removed noise(i.e., the difference (centered on 128) between the noisy image and the denoisedimage. Bottom: the difference (centered on 128) between the denoised imageand the ground truth. Left: AD denoising. Middle: TV minimization process.Right: Optimized Blockwise NL-means filter.

“optimal” due to the non perfect estimation of the noise stan-dard deviation. For AD filter, the parameter K varies from 0.05to 1 with a step of 0.05 and the number of iterations varies from1 to 10. For TV minimization, the parameter varies from 0.01to 1 with step of 0.01 and the number of iterations varies from1 to 10. The results obtained for 9% of Gaussian noise are pre-sented Fig. 11, but this screening has been performed for the


TABLE IVCOMPARISON OF DIFFERENT IMPLEMENTATIONS OF THE NL-MEANS FILTER IN TERMS OF COMPUTATIONAL TIME AND DENOISING QUALITY. THESE RESULTS ARE

OBTAINED ON A T1-W BRAINWEB IMAGE WITH 9% OF RICIAN NOISE ON A INTEL PENTIUM D CPU 3.40 GHz WITH 2 G OF RAM. THE PARAMETERS USED ARE

THE DEFAULT PARAMETERS. AVERAGE NUMBER OF VOXELS/BLOCKS USED IN V TO DENOISE x SHOWS THE IMPACT OF VOXELS/BLOCKS SELECTION. FOR

NONOPTIMIZED IMPLEMENTATIONS ALL THE VOXELS/BLOCKS IN V ARE TAKEN INTO ACCOUNT TO DENOISE x . THUS, THE NUMBER OF VOXELS/BLOCKS

USED ARE jV j = (2M + 1) = 11 . FOR OPTIMIZED IMPLEMENTATIONS, THE VOXEL SELECTION ALLOWS TO DRASTICALLY REDUCE THIS NUMBER

Fig. 15. Comparison with AD, TV, and Optimized Blockwise NL-means denoising. PSNR values and histograms for different denoising methods on BrainWebat 9% of Rician noise. (a) PSNR study shows that the Optimized Blockwise NL-means filter outperforms the well-established TV minimization process and theAD approach. (b) When the histograms are compared low values of intensity (<20) are incorrectly restored for all the filters; the Gaussian approximation is notappropriate in that case. Nevertheless, it seems the underlying assumption is well suited to high values (>60). Contrary to others methods, the NL-means basedrestoration clearly emphasizes the three main peaks representing the white matter, the gray matter, and the cerebrospinal fluid. Sharpness of the peaks shows howthe Optimized Blockwise NL-means filter increases the contrast between denoised biological structures.

four levels of noise. It is important to underline that the resultsgiving the best PSNR are used, but these results do not neces-sary give the best visual output. Indeed, the best PSNR for ADfilter is obtained for a visually under-smoothed image since thismethod tends to spoil the edges. To obtain a high PSNR, thedenoised image needs to balance edge preserving and noise re-moving. For AD filter, this tradeoff leads to inhomogeneities inflat areas in denoised image (see Fig. 13). For TV minimiza-tion, the best PSNR is obtained with a visually under-smoothedimage since noise is present in denoised image (see Fig. 13).

a) PSNR comparison: As presented in Fig. 12(a), ourblock optimized NL-means filter produces the best PSNRvalues whatever the noise level. On average, a gain of 1.85 dBis measured compared to TV and AD methods. The PSNRvalue between the noisy image and the ground truth is called“no processing” and is used as reference.

b) Histogram comparison: To better understand how thesedifferences in the PSNR values between the three methods canbe explained, the histograms of the denoised images were com-pared to the histogram of the ground truth. Fig. 12(b) shows thatthe Optimized Blockwise NL-means filter is the only methodable to retrieve a histogram similar to the ground truth. The

NL-means-based restoration schemes clearly distinguish thethree main peaks representing the white matter, the gray matterand the cerebrospinal fluid. The sharpness of the peaks showshow the Optimized Blockwise NL-means filter increasesthe contrast between denoised biological structures (see alsoFig. 13). The distances between these histograms are estimatedwith the Bhattacharyya coefficient defined as

(19)

where and are the two histograms be to compared and isthe bin index. A close to 1 means and are very similar.Each histogram of denoised images is compared to the “groundtruth” one (see Table III). The distance between the histogramof the noisy image and the histogram of the “ground truth” isused as a reference. The BC distance shows that the restoredhistogram obtained with the Optimized Blockwise NL-meansfilter is the closest to the “ground truth,” as visually assessed inFig. 12(b). Finally, Table III suggests that the NL-means-basedapproach could improve the registration of images, since themutual information (MI) computed between the restored image


and the “ground truth” is higher in comparison with AD filterand TV minimization. The MI is a similarity measure commonlyused in image registration.

c) Visual assessment: Figs. 13 and 14 show the restoredimages and the removed noise obtained with the three com-pared methods. As shown in the previous analysis, we observethat the homogeneity of white matter is higher when the imageis denoised with the Optimized Blockwise NL-means filter.Moreover, focusing on the structure of the removed noise, itclearly appears that NL-means-based restoration schemes betterpreserves the high-frequency components of the image corre-sponding to anatomical structures while removing efficientlythe high frequencies due to noise. According to the “methodnoise” introduced in [57], the NL-means is a better denoisingmethod since the removed noise is the most similar to a whiteGaussian noise. Finally, the difference between the “groundtruth” and the denoised image is presented in order to showwhich structures are removed during the denoising process. InFig. 13, this difference shows that: 1) the AD filter tends tospoil the edges especially on the skull; 2) the TV minimizationslightly better preserves the edges but does not remove allthe noise; and 3) the Optimized Blockwise NL-means filtervisually better preserves the edges while efficiently removingthe noise (especially for white matter).

VI. VALIDATION ON A PHANTOM DATA SET

WITH ADDED RICIAN NOISE

In this section, the same experiments are performed on aphantom data set corrupted by Rician noise in order to study theimpact of the Gaussian assumption. Table IV shows the com-putation time and the denoising performance of the differentcompared NL-means filters. These results show that the opti-mized NL-means versions outperform the classical ones also forRician noise. Fig. 15 presents the comparison with AD filter andTV minimization in terms of PSNR values and histogram anal-ysis. As for the AD filter and TV minimization, the NL-means-based denoising is able to correctly restore an image corruptedby Rician noise using a Gaussian approximation. When the his-tograms are compared, low values of intensity ( 20) are incor-rectly restored for all the filters; the Gaussian approximation isnot appropriate in that case. Nevertheless, it seems the under-lying assumption is well suited to high values ( 60).

As for Gaussian noise, the NL-means-based restorationclearly emphasizes the three main peaks corresponding to thewhite matter, the gray matter and the cerebrospinal fluid. Fig. 16shows the visual results obtained when the methods are com-pared on phantom data with Rician noise. Compared to Fig. 13,the denoising of background is worse in the Rician case, but thecerebral structures are correctly restored with the NL-meansfilter especially the white matter (see Fig. 16). Finally, Fig. 17shows the PSNR results of the parameter screening for the ADfilter and the TV minimization at 9% of Rician noise. All theseresults on Rician noise show that the PNSR values slightlydecrease due to more pronounced noise compared to Gaussiancase for a same level (see Section IV-A2 for an explanation),but the general performance of the filters is preserved.

Fig. 16. Comparison with AD, TV, and NL-means denoising on synthetic T1-wimages. Top: zooms on T1-w BrainWeb images. Left: the “ground truth.” Right:the noisy images with 9% of Rician noise. Middle: the results of restoration ob-tained with the different methods and the images of the removed noise (i.e., thedifference (centered on 128) between the noisy image and the denoised image.Bottom: the “method noise” which is the difference (centered on 128) betweenthe denoised image and the ground truth. Left: AD denoising. Middle: TV mini-mization process. Right: Optimized Blockwise NL-means filter. The NL-meansbased restoration better preserves the anatomical structure in the image whileefficiently removing the noise, it can be seen in the image of removed noise.

VII. EXPERIMENTS ON CLINICAL DATA

A. High Field MRI

The restoration results presented in Fig. 18 show good preser-vation of the cerebellum contours. Fully automatic segmenta-tion and quantitative analysis of such structures are still a chal-lenge, and improved restoration schemes could greatly improvethese processings.

B. MS Pathological Context

Fig. 19 shows that the optimized blockwise NL-means filterpreserves the lesions while removing the noise. The impacton further processing is not the scope of this paper and is notstudied here. Nevertheless, visually the lesions appears morecontrasted and as seen on the difference image the proposedNL-means approach does not include any structure of lesion


Fig. 17. Result for AD filter and TV minimization on phantom images with 9% of Rician noise. For AD filter K varies from 0.05 to 1 with step of 0.05 and thenumber of iterations varies from 1 to 10. For TV minimization � varies from 0.01 to 1 with step of 0.01 and the number of iterations varies from 1 to 10.

Fig. 18. NL-means filter on a real T1-w MRI. Fully-automatic restoration obtained with the Optimized Blockwise NL-means filter on a 3 T T1-w MRI data of256 voxels in less than 3 min on a Intel Pentium D CPU 3.40 GHz with 2G of RAM. From left to right: original image, denoised image, and difference imagewith gray values centered on 128. Whole image is shown on top, and a detail is displayed on bottom.

in the estimated noise image. This was confirmed by an expe-rienced neurologist.

VIII. DISCUSSION AND CONCLUSION

This paper presents an optimized blockwise version of theNL-means filter, applied to 3-D medical data. Validation wasperformed on the BrainWeb dataset [2] and showed that theproposed Optimized Blockwise NL-means filter outperformsthe classical implementation of the NL-means filter and somestate-of-the-art techniques, such as the AD approach [3] and theTV minimization process [4] on both Gaussian and Rician noise.These first results show that the image-redundancy assumption

required for NL-means based restoration holds for 3-D MRI.Compared to the classical NL-means filter, our implementation(with voxel preselection, multithreading, and blockwise imple-mentation) considerably decreases the required computationaltime (up to a factor of 60 on a Xeon at 3 GHz) and increases thePSNR value of the denoised image. Nevertheless, the problemof the computational burden can still be investigated with otherfaster implementations such as the “plain multiscale” schemealso suggested in [1]. Further works should be pursued for com-paring NL-means based restoration with recent promising de-noising methods, such as TV in wavelet domain [43] or adaptiveestimation method [28], [50]. Moreover, the efficiency of the


Fig. 19. NL-means filter on a real T2-w MRI with MS. Fully-automatic restoration obtained with the Optimized Blockwise NL-means filter on a 1.5 T T2-w MRIdata with MS lesions of 512 � 512 � 28 voxels in less than 2 min on a Intel Pentium D CPU 3.40 GHz with 2G of RAM. From left to right: original image,denoised image, and difference image with gray values centered on 128. Whole image is shown on top, and a detail is exposed on bottom.

technique limiting the staircasing effect proposed in [58] needsto be studied for MRI.

We show on sample pathological cases (patients with MS le-sions) that the filter preserves the major visual signature of thegiven pathology. However, the impact on specific pathologiesneeds to be further investigated.

Finally, the impact of the NL-means-based denoising on theperformances of postprocessing algorithms, like segmentationand registration schemes also should be studied. Nonetheless,the first results presented on the MI suggest that the proposedOptimized Blockwise NL-means filter could improve theimage registration process. Indeed, the MI computed betweenthe restored image and the “ground truth” is higher with theOptimized Blockwise NL-means filter than with the ADapproach and the TV minimization process.

REFERENCES

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

[2] D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani,C. J. Holmes, and A. C. Evans, “Design and construction of a realisticdigital brain phantom,” IEEE Trans. Med. Imag., vol. 17, no. 3, pp.463–468, Jun. 1998.

[3] P. Perona and J. Malik, “Scale-space and edge detection usinganisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., vol.12, no. 7, pp. 629–639, Jul. 1990.

[4] L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation basednoise removal algorithms,” Physica D, vol. 60, pp. 259–268, 1992.

[5] J. Mazziotta, A. Toga, A. Evans, P. Fox, J. Lancaster, K. Zilles, R.Woods, T. Paus, G. Simpson, B. Pike, C. Holmes, L. Collins, P.Thompson, D. MacDonald, M. Iacoboni, T. Schormann, K. Amunts,N. Palomero-Gallagher, S. Geyer, L. Parsons, K. Narr, N. Kabani,G. Le Goualher, D. Boomsma, T. Cannon, R. Kawashima, and B.Mazoyer, “A probabilistic atlas and reference system for the humanbrain: International Consortium for Brain Mapping (ICBM),” Philos.Trans. R. Soc. Lond. B Biol. Sci., vol. 356, no. 1412, pp. 1293–1322,Aug. 2001.

[6] A. P. Zijdenbos, R. Forghani, and A. C. Evans, “Automatic “pipeline”analysis of 3-D MRI data for clinical trials: Application to multiplesclerosis,” IEEE Trans. Med. Imag., vol. 21, no. 10, pp. 1280–1291,Oct. 2002.

[7] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution,and the Bayesian restoration of images,” IEEE Trans. Pattern Anal.Mach. Intell., vol. 6, pp. 721–741, 1984.

[8] D. Mumford and J. Shah, “Optimal approximations by piecewisesmooth functions and variational problems,” Commun. Pure Appl.Math., vol. 42, pp. 577–685, 1989.

[9] D. Tschumperlé, “Curvature-preserving regularization of multi-valuedimages using PDE’s,” in ECCV, Graz, Austria, 2006, pp. 428–433.

[10] M. J. Black and G. Sapiro, “Edges as outliers: Anisotropic smoothingusing local image statistics,” in Proc. Scale-Space Theories ComputerVision, Corfu, Greece, Sep. 1999, pp. 259–270.

[11] P. Saint-Marc, J.-S. Chen, and G. Medioni, “Adaptive smoothing: Ageneral tool for early vision,” IEEE Trans. Pattern Anal. Mach. Intell.,vol. 13, no. 6, pp. 514–529, Jun. 1991.

[12] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by waveletshrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, 1994.

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

[14] J. Portilla and E. P. Simoncelli, “Image restoration using Gaussian scalemixtures in the wavelet domain,” in Int. Conf. Image Process., 2003,pp. 965–968.

[15] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color im-ages,” in ICCV ’98: Proc. 6th Int. Conf. Computer Vision, Washington,DC, 1998, p. 839.

[16] S. M. Smith and J. M. Brady, “SUSAN—A new approach to low levelimage processing,” Int. J. Comput. Vis., vol. 23, no. 1, pp. 45–78, May1997.

[17] M. Elad, “On the origin of the bilateral filter and ways to improve it,”IEEE Trans. Image Process., vol. 11, no. 10, pp. 1141–1151, Oct. 2002.

[18] J. van de Weijer and R. van den Boomgaard, “Local mode filtering,”in IEEE Computer Soc. Conf. Computer Vis. Pattern Recognit., Kauai,HI, Dec. 2001, pp. 428–433.

[19] T. F. Chan and H. M. Zhou, “Total variation improved wavelet thresh-olding in image compression,” in Int. Conf. Image Process. , 2000, vol.2, pp. 391–394.

[20] S. Durand and J. Froment, “Reconstruction of wavelet coefficientsusing total variation minimization,” SIAM J. Sci. Comput., vol. 24, no.5, pp. 1754–1767, 2002.


[21] S. Lintner and F. Malgouyres, “Solving a variational image restorationmodel which involves L constraints,” Inverse Problems, vol. 20, no.3, pp. 815–831, 2004.

[22] P. Mrazek, J. Weickert, and A. Bruhn, “On robust estimation andsmoothing with spatial and tonal kernels,” in Geometric Propertiesfrom Incomplete Data. Dordrecht: Springer, 2004, pp. 335–352.

[23] D. Barash, “A fundamental relationship between bilateral filtering,adaptive smoothing, and the nonlinear diffusion equation,” IEEETrans. Pattern Anal. Mach. Intell., vol. 24, no. 6, pp. 844–847, Jun.2002.

[24] G. Sapiro, “From active contours to anisotropic diffusion: Connec-tions between basic PDE’s in image processing,” in ICIP’96: Int. Conf.Image Process., 1996, vol. 1, pp. 477–480.

[25] J. Polzehl and V. Spokoiny, “Adaptive weights smoothing with appli-cation to image restoration,” J. Roy. Stat. Soc. B, vol. 62, pp. 335–354,2000.

[26] V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window sizeimage de-noising based on intersection of confidence intervals (ICI)rule,” J. Math. Imag. Vis., vol. 16, no. 3, pp. 223–235, May 2002.

[27] C. Kervrann, “An adaptive window approach for image smoothingand structures preserving,” in Proc. 8th Eur. Conf. Computer Vision,Prague, Czech Republic, May 2004, pp. 132–144.

[28] C. Kervrann and J. Boulanger, “Unsupervised patch-based imageregularization and representation,” in Proc. Eur. Conf. Comp. Vis.(ECCV’06), Graz, Austria, May 2006.

[29] C. Kervrann and J. Boulanger, “Optimal spatial adaptation for patch-based image denoising,” IEEE Trans. Image Process., vol. 15, no. 10,pp. 2866–2878, Oct. 2006.

[30] S. C. Zhu, Y. Wu, and D. Mumford, “Filters, random fields and max-imum entropy (frame): Towards a unified theory for texture modeling,”Int. J. Comput. Vis., vol. 27, no. 2, pp. 107–126, 1998.

[31] W. T. Freeman, E. C. Pasztor, and O. T. Carmichael, “Learning low-level vision,” Int. J. Comput. Vis., vol. 40, no. 1, pp. 25–47, Oct. 2000.

[32] S. Roth and M. J. Black, “Fields of experts: A framework for learningimage priors,” in 2005 IEEE Comput. Soc. Conf. Computer Vision Pat-tern Recognit., San Diego, CA, Jun. 2005, pp. 860–867.

[33] S. P. Awate and R. T. Whitaker, “Higher-order image statistics for un-supervised, information-theoretic, adaptive, image filtering,” in IEEEComputer Soc. Conf. Comput. Vision Pattern Recognit., San Diego,CA, Jun. 2005, pp. 44–51.

[34] G. Gerig, R. Kikinis, O. Kübler, and F. Jolesz, “Nonlinear anisotropicfiltering of MRI data,” IEEE Trans. Med. Imag., vol. 11, no. 2, pp.221–232, Jun. 1992.

[35] J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficientand reliable schemes for nonlinear diffusion filtering,” IEEE Trans.Image Process., vol. 7, no. 3, pp. 398–410, Jul. 1998.

[36] S. L. Keeling, “Total variation based convex filters for medicalimaging,” Appl. Math. Comput., vol. 139, no. 1, pp. 101–119, 2003.

[37] W. C. K. Wong, A. C. S. Chung, and S. C. H. Yu, Trilateral filtering forbiomedical images Arlington, VA, Apr. 2004, pp. 820–823.

[38] R. Nowak, “Wavelet-based Rician noise removal for magnetic res-onance imaging,” IEEE Trans. Image Process., vol. 8, no. 10, pp.1408–1419, Oct. 1999.

[39] J. C. Wood and K. M. Johnson, “Wavelet packet denoising of magneticresonance images: Importance of Rician noise at low SNR,” Magn.Reson. Med., vol. 41, no. 3, pp. 631–635, Mar. 1999.

[40] S. Zaroubi and G. Goelman, “Complex denoising of MR data viawavelet analysis: Application for functional MRI,” Magn. Reson.Imag., vol. 18, no. 1, pp. 59–68, Jan. 2000.

[41] M. E. Alexander, R. Baumgartner, A. R. Summers, C. Windischberger,M. Klarhoefer, E. Moser, and R. L. Somorjai, “A wavelet-based methodfor improving signal-to-noise ratio and contrast in MR images,” Magn.Reson. Imag., vol. 18, no. 2, pp. 169–180, Feb. 2000.

[42] P. Bao and L. Zhang, “Noise reduction for magnetic resonance im-ages via adaptive multiscale products thresholding,” IEEE Trans. Med.Imag., vol. 22, no. 9, pp. 1089–1099, Sep. 2003.

[43] A. Ogier, P. Hellier, and C. Barillot, “Restoration of 3-D medical im-ages with total variation scheme on wavelet domains (TVW),” in Proc.SPIE Med. Imag. 2006: Image Process., San Diego, Feb. 2006.

[44] Y. Wang and H. Zhou, “Total variation wavelet-based medical imagedenoising,” Int. J. Biomed. Imag., vol. 2006, 2006.

[45] M. J. McDonnell, “Box-filtering techniques,” Comput. Vis., Graph.,Image Process., vol. 17, no. 1, pp. 65–70, Sep. 1981.

[46] L. P. Yaroslavsky, Digital Picture Processing—An Introduction. NewYork: Springer-Verlag, 1985.

[47] J. S. Lee, “Digital image smoothing and the sigma filter,” Comput. Vis.,Graph. Image Process., vol. 24, pp. 255–269, 1983.

[48] R. Coifman and D. Donoho, “Translation invariant de-noising,” in Lec-ture Notes in Statistics: Wavelets and Statistics. New York: Springer,, Feb. 1995, Lecture Notes Statistics, pp. 125–150.

[49] T. Gasser, L. Sroka, and C. J. Steinmetz, “Residual variance andresidual pattern in nonlinear regression,” Biometrika, vol. 73, no. 3,pp. 625–633, 1986.

[50] J. Boulanger, C. Kervrann, and P. Bouthemy, “Adaptive spatio-tem-poral restoration for 4D fluoresence microscopic imaging,” presentedat the Int. Conf. Medical Image Comput. Computer Assisted Interven-tion (MICCAI’ 05), Palm Springs, CA, Oct. 2005.

[51] A. Buades, B. Coll, and J.-M. Morel, “Nonlocal image and movie de-noising,” Int. J. Comput. Vis., vol. 76, no. 2, pp. 123–140, 2008.

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

[53] H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRIdata,” Magn. Reson. Med., vol. 34, pp. 910–914, 1995.

[54] A. Macovski, “Noise in MRI,” Magn. Reson. Med., vol. 36, pp.494–497, 1996.

[55] S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative reg-ularization method for total variation-based image restoration,” Multi-scale Model. Simul., vol. 4, no. 2, pp. 460–489, 2005.

[56] E. Tadmor, S. Nezzar, and L. Vese, “A multiscale image representa-tion using hierarchical (BV; L ) decompositions,” Multiscale Model.Simul., vol. 2, no. 4, pp. 554–579, 2004.

[57] A. Buades, B. Coll, and J.-M. Morel, A Non Local Algorithm for ImageDenoising. San Diego, CA, IEEE Computer Society, Jun. 2005, pp.60–65.

[58] A. Buades, B. Coll, and J. M. Morel, “The staircasing effect in neigh-borhood filters and its solution,” IEEE Trans. Image Process., vol. 15,no. 6, pp. 1499–1505, Jun. 2006.

ieee transactions on medical imaging, vol. 27, no. 4, …read.pudn.com/downloads675/doc/2731726/an...

Documents