IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009 2221
Nonlocal Means-Based Speckle Filtering for
Ultrasound Images
Pierrick Coupé, Pierre Hellier, Charles Kervrann, and Christian Barillot
Abstract—In image processing, restoration is expected to im-
prove the qualitative inspection of the image and the performance
of quantitative image analysis techniques. In this paper, an adap-
tation of the nonlocal (NL)-means filter is proposed for speckle
reduction in ultrasound (US) images. Originally developed for
additive white Gaussian noise, we propose to use a Bayesian
framework to derive a NL-means filter adapted to a relevant
ultrasound noise model. Quantitative results on synthetic data
show the performances of the proposed method compared to
well-established and state-of-the-art methods. Results on real
images demonstrate that the proposed method is able to preserve
accurately edges and structural details of the image.
I. INTRODUCTION
I N ultrasound imaging, denoising is challenging since thespeckle artifacts cannot be easily modeled and are known
to be tissue-dependent. In the imaging process, the energy of
the high frequency waves are partially reflected and transmitted
at the boundaries between tissues having different acoustic im-
pedances. The images are also log-compressed to make easier
visual inspection of anatomy with real-time imaging capability.
Nevertheless, the diagnosis quality is often low and reducing
speckle while preserving anatomic information is necessary to
delineate reliably and accurately the regions of interest. Clearly,
the signal-dependent nature of the speckle must be taken into ac-
count to design an efficient speckle reduction filter. Recently, it
has been demonstrated that image patches are relevant features
for denoising images in adverse situations [1]–[3]. The related
methodology can be adapted to derive a robust filter for US im-
ages. Accordingly, in this paper we introduce a novel restoration
scheme for ultrasound (US) images,strongly inspired from the
NonLocal (NL-) means approach [1] introduced by Buades et al.
[1] to denoise 2-D natural images corrupted by an additive white
Gaussian noise. In this paper, we propose an adaptation of the
NL-means method to a dedicated US noise model [4] using a
Bayesian motivation for the NL-means filter [3]. In what fol-
lows, invoking the central limit theorem, we will assume that
Manuscript received September 26, 2008; revised April 28, 2009. First pub-
lished May 27, 2009; current version published September 10, 2009. The as-
sociate editor coordinating the review of this manuscript and approving it for
publication was Dr. John Kerekes.
P. Coupé, P. Hellier, and C. Barillot are with the University of Rennes I-
CNRS UMR 6074, IRISA, Campus de Beaulieu, F-35042 Rennes, France, and
also with the INRIA, VisAGeS U746 Unit/Project, IRISA, Campus de Beaulieu,
F-35042 Rennes, France, and also with the INSERM, VisAGeS U746 Unit/
Project, IRISA, Campus de Beaulieu, F-35042 Rennes, France.
C. Kervrann is with the INRIA, VISTA Project, IRISA, Campus de Beaulieu,
F-35042 Rennes, France, and also with the INRA, UR341 Mathématiques et
Informatique Appliquées, Domaine de Vilvert 78352 Jouy en Josas, France.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2009.2024064
the observed signal at a pixel is a Gaussian random variable with
mean zero and a variance determined by the scattering proper-
ties of the tissue at the current pixel.
The remainder of the paper is organized as follows. In
Section II, we give an overview of speckle filters and re-
lated methods. Section III described the proposed Bayesian
NL-means filter adapted to speckle noise. Quantitative results
on artificial images with various noise models are presented in
Section IV. Finally, qualitative result on real 2-D and 3-D US
images are proposed in Section V.
II. SPECKLE REDUCTION: RELATED WORK
The speckle in US images is often considered as undesirable
and several noise removal filters have been proposed. Unlike the
additive white Gaussian noise model adopted in most denoising
methods, US imaging requires specific filters due to the signal-
dependent nature of the speckle intensity. In this section, we
present a classification of standard adaptive filters and methods
for speckle reduction.
A. Adaptive Filters
The adaptive filters are widely used in US image restoration
because they are easy to implement and control. The commonly
used adaptive filters—the Lee’s filter [5], Frost’s filter [6], and
Kuan’s filter [7]—assume that speckle noise is essentially a
multiplicative noise. Many improvements of these classical fil-
ters have been proposed since. At the beginning of the 1990s,
Lopes et al. [8] suggested to improve the Lee’s and Frost’s
filters by classifying the pixels in order to apply specific pro-
cessing to the different classes. Based on this idea, the so-called
Adaptive Speckle Reduction filter (ASR) exploits local image
statistics to determine specific areas to be processed further. In
[9], the kernel of the adaptive filter is fitted to homogeneous re-
gions according to local image statistics. Analyzing local homo-
geneous regions was also investigated in [10], [11] to spatially
adapt the filter parameters. Note that the Median filter has been
also examined for speckle reduction in [4]. Very recently, a sto-
chastic approach to ultrasound despeckling (SBF) has been de-
veloped in [12] and [13]. This local averaging method removes
the local extrema assumed to be outliers in a robust statistical
estimation framework. Finally, the Rayleigh-Maximum-Likeli-
hood (R-ML) filter has been derived with similar methodolog-
ical tools in [14].
B. Partial Differential Equations (PDE) -Based Approaches
Adapted formulations of the Anisotropic Diffusion filter
(AD) [15] and the Total Variation minimization scheme (TV)
[16] have been developed for US imaging. In [17] and [18], the
1057-7149/$26.00 © 2009 IEEE
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2222 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009
Speckle Reducing Anisotropic Diffusion (SRAD) was intro-
duced and involves a noise-dependent instantaneous coefficient
of variation. In [19] the Nonlinear Coherent Diffusion (NCD)
filter is based on the assumption that the multiplicative speckle
in US signals is transformed into an additive Gaussian noise
in Log-compressed images. Recently, the Oriented SRAD
(OSRAD) filter has been proposed in [20]; this filter takes into
account the local directional variance of the image intensity,
i.e., the local image geometry. Finally, the TV minimization
scheme has been adapted to ultrasound imaging in [21] and
[22]. Unlike the previous adaptive speckle filters, all the consid-
ered PDE-based approaches are iterative and produce smooth
images while preserving edges. Nevertheless, meaningful
structural details are unfortunately removed during iterations.
C. Multiscale Methods
Several conventional wavelet thresholding methods [23]–[25]
have also been investigated for speckle reduction [26]–[28] with
the assumption that the logarithm compression of US images
transforms the speckle into an additive Gaussian noise. In order
to relax this restrictive assumption, Pizurica et al. [29] proposed
a wavelet-based Generalized Likelihood ratio formulation and
imposed no prior on noise and signal statistics. In [30]–[33],
the Bayesian framework was also explored to perform wavelet
thresholding adapted to the non-Gaussian statistics of the signal.
Note that other multiscale strategies have been also studied in
[34]–[36] to improve the performance of the AD filter; in [37],
the Kuan’s filter is applied to interscale layers of a Laplacian
pyramid.
D. Hybrid Approaches
The aforementioned approaches can be also combined in
order to take advantage of the different paradigms. In [38],
the image is preprocessed by an adaptive filter in order to
decompose the image into two components. A Donoho’s soft
thresholding method is then performed on each component.
Finally, the two processed components are combined to reduce
speckle. PDE-based approaches and a wavelet transform have
been also combined as proposed in [39].
III. METHOD
The previously mentioned approaches for speckle reduction
are based on the so-called locally adaptive recovery paradigm
[40]. Nevertheless, more recently, a new patch-based nonlocal
recovery paradigm has been proposed by Buades et al. [1]. This
new paradigm proposes to replace the local comparison of pixels
by the nonlocal comparison of patches. Unlike the aforemen-
tioned methods, the so-called NL-means filter does not make any
assumptions about the location of the most relevant pixels used
to denoise the current pixel. The weight assigned to a pixel in the
restoration of the current pixel does not depend on the distance
between them (neither in terms of spatial distance nor in terms
of intensity distance). The local model of the signal is revised
and the authors consider only information redundancy in the
image. Instead of comparing the intensity of the pixels, which
may be highly corrupted by noise, the NL-means filter analyzes
the patterns around the pixels. Basically, image patches are com-
pared for selecting the relevant features useful for noise reduc-
tion. This strategy leads to competitive results when compared
to most of the state-of-the-art methods [3], [41]–[46]. Neverthe-
less, the main drawback of this filter is its computational burden.
In order to overcome this problem, we have recently proposed
a fast and optimized implementation of the NL-means filter for
3-D magnetic resonance (MR) images [46].
In this section, we rather revise the traditional formulation of
the NL-means filter, suited to the additive white Gaussian noise
model, and adapt this filter to spatial speckle patterns. Accord-
ingly, a dedicated noise model used for US images is first con-
sidered. A Bayesian formulation of the NL-means filter [3] is
then used to derive a new speckle filter.
A. Nonlocal Means Filter
Let us consider a gray-scale noisy image
defined over a bounded domain , (which is usually
a rectangle of size ) and is the noisy observed
intensity at pixel . In the following, denotes the
image grid dimension ( or respectively for
2-D and 3-D images). We also use the notations given below.
Original pixelwise NL-means approach
square search volume centered at pixel
of size , ;
square local neighborhood of of size
, ;
vector
gathering the intensity values of ;
• true intensity value at pixel ;
restored value of pixel ;
weight used for restoring given
and based on the similarity of
patches and .
Blockwise NL-means approach
square block centered at of size
, ;
• unobserved vector of true values of block
;
vector gathering the intensity values of
block ;
restored block of pixel ;
weight used for restoring given
and based on the similarity of
blocks and .
Finally, the blocks are centered on pixels with
and represents the
distance between block centers.
1) Pixelwise Approach: In the original NL-means filter [1],
the restored intensity of pixel , is the weighted
average of all the pixel intensities in the image
(1)
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COUPÉ et al.: NONLOCAL MEANS-BASED SPECKLE FILTERING FOR ULTRASOUND IMAGES 2223
Fig. 1. Pixelwise NL-means filter ( and ). The restored value at
pixel (in red) is the weighted average of all intensity values of pixels in
the search volume . The weights are based on the similarity of the intensity
neighborhoods (patches) and .
where is the weight assigned to value for
restoring the pixel . More precisely, the weight evaluates the
similarity between the intensities of the local neighborhoods
(patches) and centered on pixels and , such that
and (see Fig. 1).
The size of the local neighborhood and is .
The traditional definition of the NL-means filter considers that
the intensity of each pixel can be linked to pixel intensities of
the whole image. For practical and computational reasons, the
number of pixels taken into account in the weighted average is
restricted to a neighborhood, that is a “search volume” of
size , centered at the current pixel .
For each pixel in , the Gaussian-weighted Euclidean
distance is computed between the two image patches
and as explained in [1]. This distance is the tra-
ditional -norm convolved with a Gaussian kernel of standard
deviation . The standard deviation of the Gaussian kernel is
used to assign spatial weights to the patch elements. The cen-
tral pixels in the patch contribute more to the distance than the
pixels located at the periphery. The weights are then
computed as follows:
(2)
where is a normalization constant ensuring that
, and acts as a filtering parameter
controlling the decay of the exponential function.
2) Blockwise Approach: As presented in [46], a blockwise
implementation of the proposed NL-means-based speckle filter
is able to decrease the computational burden. The blockwise ap-
proach consists of: i) dividing the volume into blocks with over-
lapping supports; ii) performing a NL-means-like restoration of
these blocks; iii) restoring the pixel intensities from the restored
blocks. Here, we describe briefly these three steps and refer the
reader to [46] for additional detailed explanations.
i) A partition of the volume into overlapping blocks
containing elements is per-
Fig. 2. Blockwise NL-means Filter ( , and ). For each
block centered at pixel , a NL-means-like restoration is performed from
blocks . The restored value of the block is the weighted average of all
the blocks in the search volume. For a pixel included in several blocks,
several estimates are obtained and fused. The restored value of pixel is the
average of the different estimations stored in vector .
formed, i.e., . These blocks are cen-
tered on pixels which constitute a subset of .
The pixels are equally distributed at positions
where
represents the distance between the centers of .
ii) A block is restored as follows:
(3)
where is
an image patch gathering the intensities of the
block , is a normalization constant ensuring
and
(4)
iii) For a pixel included in several blocks , several es-
timates of the same pixel from different
are computed and stored in a vector of size (see
Fig. 2). We denote the th element of vector .
The final restored intensity of pixel is the mean of the
restored values .
The main advantage of this approach is to significantly reduce
the complexity of the algorithm. Indeed, for a volume of
size , the global complexity is
. For instance, if we set , the complexity
is divided by a factor of 4 in 2-D and 8 in 3-D.
B. NL Means-Based Speckle Filter
1) Bayesian Formulation: In [3], a Bayesian formulation of
the NL-means filter was proposed. Equivalent to the conditional
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2224 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009
mean estimator, it has been shown that an empirical estimator
of a block can be defined as (see the Appendix)
(5)
where denotes the probability density func-
tion (pdf) of given the noise free and unknown patches
. Since is unknown, an estimator is classically
computed by substituting for . Hence, we get
(6)
where denotes the pdf of condi-
tionally to . In the case of an additive white Gaussian
noise, the likelihood will be proportional
to , and the corresponding Bayesian
estimator is then similar to the initial NL-means method
[see (3)].
In what follows, this general Bayesian formulation is used to
derive an adapted filter to a dedicated ultrasound noise model.
2) Noise Models for Log-Compressed US Images: The rele-
vant noise pdfs useful for US image denoising cannot be easily
exhibited. Basically, we should consider the complex image
formation process, i.e.,: i) local correlation due to periodic ar-
rangements of scatterers [17] and finite beamwidth; ii) envelope
detection and logarithm amplification of radio-frequency sig-
nals performed on the display image [19]; iii) additive Gaussian
noise of sensors [19]; iv) additive Gaussian noise introduced
by the acquisition card. All these factors tend to prove that
the Rayleigh model used for RF signals is not suitable for
analyzing US Log-compressed images. Usually, in the wavelet
denoising domain [19], [26], [27], multiplicative speckle noise
is supposed to be transformed into an additive Gaussian noise
by the logarithmic compression. However, recent studies re-
lated to US images demonstrate also that the distribution of
noise is satisfyingly approximated by a Gamma distribution
[47] or a Fisher–Tippett distribution [48].
Consequently, we have decided to choose the following gen-
eral speckle model:
(7)
where is the original image, is the observed image,
is a zero-mean Gaussian noise. This model
is more flexible and less restrictive than the usual RF model
and is able to capture reliably image statistics since the factor
depends on ultrasound devices and additional processing related
to image formation.
Contrary to additive white Gaussian noise model, the noise
component in (7) is image-dependent. In [4], based on the ex-
perimental estimation of the mean versus the standard devia-
tion in Log-compressed images, Loupas et al. have shown that
model fits better to data than the multiplicative model
or the Rayleigh model. Since, this model has been used suc-
cessfully in many studies [38], [49]–[51]. Clearly, this model is
relevant since it is confirmed that the speckle is higher in regions
of high intensities versus regions of low intensities [47], [49].
3) A New Statistical Distance for Patch Comparison: The
Pearson Distance: Based on the Bayesian formulation [see (6)],
we introduce a new distance to compare image patches if we
consider the noise model (7). For each pixel, we assume
(8)
which yields
(9)
Given a block , the likelihood can be factorized as (condi-
tional independence hypothesis)
(10)
Accordingly, the so-called Pearson distance defined as
(11)
is substituted to the usual -norm (see (4)). In the reminder of
the paper, is considered in the proposed filter.
A pixel selection scheme similar to [46] based on tests on
the mean will be used to select the most relevant patches. This
selection is controlled by the thresholding as previously de-
scribed in [46]. As a result, the denoising results are improved
and the algorithm is faster. In addition, a parallel implementa-
tion is used in all experiments to speed up the algorithm.
IV. SYNTHETIC IMAGES EXPERIMENTS
In this section, we propose to compare different filters with
experiments on synthetic data, with different noise models,
image data and quality criteria. Two simulated data are de-
scribed in this section.
• In Section IV-A, a 2-D phantom and a noise model avail-
able in MATLAB are considered for the experiments, and the
signal-to-noise Ratio (SNR) is used to compare objectively
several methods.
• In Section IV-B, the realistic speckle simulation proposed
in Field II is performed and the ultrasound despeckling
assessment index ( ) proposed in [12] and [13] is chosen
for objective comparisons.
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COUPÉ et al.: NONLOCAL MEANS-BASED SPECKLE FILTERING FOR ULTRASOUND IMAGES 2225
TABLE I
OPTIMAL SET OF PARAMETERS USED FOR THE MATLAB “PHANTOM” EXPERIMENT. FOR THE NL-MEANS BASED FILTERS, WE SET AND
Fig. 3. Results obtained with different filters applied to the “Phantom” image
corrupted with signal-dependent noise ( ). The quantitative evaluation,
measured using the signal-to-noise ratio, is presented in Table II.
A. 2-D Phantom Corrupted By the Theoretical (MATLAB)
Noise Model
1) Speckle Model and Quality Criterion: In this experi-
ment, the synthetic image “Phantom” (see Fig. 3), available
in MATLAB, was corrupted with different levels of noise. The
“Phantom” image is a 32 bit image of 256 256 pixels. First,
an offset of 0.5 was added to the image (to avoid naught areas)
before performing a multiplication of intensities by a factor 20.
Then, the MATLAB speckle simulation based on the following
image model
(12)
TABLE II
SNR VALUES OBTAINED WITH SEVERAL FILTERS APPLIED TO THE
2-D MATLAB “PHANTOM” IMAGE
was applied to the “Phantom” image. Three levels of noise
were tested by setting . To assess denoising
methods, the SNR values [52] were computed between the
“ground truth” and the denoised images
(13)
where is the true value of pixel and the restored
intensity of pixel .
2) Compared Methods: In this experiment, we compared
several speckle filters and the parameters were adjusted to get
the best SNR values:
• Lee’s filter (2-D MATLAB implementation) [5];
• Kuan’s filter (2-D MATLAB implementation) [7];
• SBF filter (2-D MATLAB implementation provided by the
authors) [12];
• Speckle Reducing Anisotropic Diffusion (SRAD)
(MATLAB implementation of Virginia University1); [17]
• blockwise implementation of the classical NL-means filter
[1];
• blockwise implementation of the proposed method de-
noted as Optimized Bayesian NL-means with block
selection (OBNLM).
In this experiment, we have chosen to compare our method
with well-known adaptive filters, such as the Lee’s and Kuan’s
filters, and with two competitive state-of-the-art methods:
SRAD and SBF filters. We limited the comparison to this set of
methods but the results produced by other recent filters could
be also analyzed further (e.g., see [14]). Moreover, the usual
NL-means filter has been applied to the same images in order
to quantitatively evaluate the performance of our speckle-based
NL-means filter. For each method and for each noise level, the
optimal filter parameters were searched within large ranges.
These parameters are given in Table I.
1http://viva.ee.virginia.edu/downloads.html
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2226 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009
TABLE III
OPTIMAL SET OF PARAMETERS USED FOR VALIDATION ON THE 2-D SYNTHETIC DATA SIMULATED
WITH FIELD II. FOR THE NL-MEANS-BASED FILTERS AND
3) Results: Table II gives the SNR values obtained for each
method. The denoised images corresponding to are
presented in Fig. 3. For all levels of noise, the OBNLM filter
obtained the best SNR value. For this experiment, the use of
the Pearson distance combined with block selection enabled
to improve the denoising performances of the NL-means filter
for images corrupted by signal-dependent noise. Visually, the
NL-means based filter satisfyingly removed the speckle while
preserving meaningful edges.
B. Field II Simulation
1) Speckle Model and Quality Assessment: In order to eval-
uate the denoising filters with a more relevant and challenging
simulation of speckle noise, the validation framework proposed
in [12], [13] was used. This framework is based on Field II sim-
ulation [53]. The “Cyst” phantom is composed of 3 constant
classes presented in Fig. 4. The result of the Field II simu-
lation is converted into an 8 bit image of size 390 500 pixels.
Since the geometry of the image is known, but not the true value
of the image (i.e., without speckle), the authors introduced the
ultrasound despeckling assessment index ( ) defined as
(14)
to evaluate the performance of denoising filters for this simu-
lation. We denote as the mean and as the variance of
class after denoising. To avoid the sensitivity to image res-
olution, is normalized by . The new index
is high if the applied filter is able to produce a new image with
well separated classes and small variances for each class. Ac-
cording to [12] and [13], an image is satisfyingly denoised if
is high enough.
2) Method Comparison: For this experiment, we compared
the SBF filter, SRAD filter, NL-means filter and the proposed
OBNLM filter. The filter parameters are given in Table III. Com-
pared to the previous experiment, the patch size by using the
NL-means-based filters is increased. Indeed, the patch size re-
flects the scale of the “noise” compared to the image resolution.
In the previous experiment, the resolution of the added noise was
about one pixel. In this experiment, the objects to be removed
are composed of several pixels; thus, the patch size is increased
to evaluate the restoration performance of each object.
3) Results: The denoised images and the quantitative results
are given in Fig. 4. In this evaluation framework, the OBNLM
filter produced the highest index. Similar values than those
presented in [12] and [13] were found for the SRAD and SBF
filters. Compared to the original NL-means filter, the proposed
Fig. 4. Denoised images obtained with the compared filters for the Field II
experiment and the corresponding indexes.
adaptations for US images improved the index of the denoised
image.
V. EXPERIMENTS ON REAL DATA
In this section, a visual comparison of 2-D intraoperative ul-
trasound brain images (Section V-A) and a visual inspection of
3-D ultrasound image of a liver (Section V-B) are proposed.
A. Intraoperative Ultrasound Brain Images
In this paragraph, we propose a visual comparison of the SBF
and SRAD filters and the OBNLM method on real intraoperative
brain images. The parameters for the SBF and SRAD filters are
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COUPÉ et al.: NONLOCAL MEANS-BASED SPECKLE FILTERING FOR ULTRASOUND IMAGES 2227
Fig. 5. Results obtained with the SRAD and SBF filters and the proposed filters
on real intraoperative brain images. The OBNLM filter efficiently removes the
speckle while enhancing the edges and preserving the image structures.
the parameters given respectively in [17] and [12]. The param-
eters of the OBNLM filter were set as follows : , ,
, and .
Fig. 5 shows the denoising results. Visually, the OBNLM
filter efficiently removes the speckle component while en-
hancing the edges and preserving the image structures. The
visual results produced on real image by our method are com-
petitive compared to SRAD and SBF filters.
B. Experiments on a 3-D Liver Image
In this section, result of the proposed filter on 3-D US image
is proposed. These data are freely available on Cambridge
University website.2 The B-scans were acquired with a Lynx
ultrasound unit (BK Medical System) and tracked by the
magnetic tracking system miniBIRD (Ascension Technology).
The reconstruction of the volume was performed with the
method described in [54]. The reconstructed volume size
was 308 278 218 voxels with an isotropic resolution of
0.5 0.5 0.5 mm .
The denoising results obtained with the OBNLM method on
the liver volume are shown in Fig. 6. As for the previous exper-
iment on 2-D US brain images, the visual results on this 3-D
dataset show edge preservation and efficient noise removal pro-
duced by our filter.
Fig. 7 shows zooming views of hepatic vessels. The edge
preservation of the OBNLM filter is visible on the removed
noise image that does not contain structures. Moreover, the dif-
ference in dark areas (hepatic vessels) and gray areas (hepatic
2http://mi.eng.cam.ac.uk/milab.html
Fig. 6. Top: 3-D volume of the liver. Bottom: the denoising result obtained
with the OBNLM filter with , , , and .
Fig. 7. From left to right: Original “noisy” volume, the denoising result ob-
tained with the OBNLM filter with and and the removed noise
component. The edge preservation of the OBNLM is visually appreciated by
inspecting the removed noise component which contains few structures. More-
over, the level of noise estimated for the dark areas (hepatic vessels) and the
gray areas (hepatic tissues) demonstrates the relevance of the signal-dependent
modeling.
tissues) shows the smoothing adaptation according to the signal
intensity. The noise in brighter areas is drastically reduced.
VI. CONCLUSION
In this paper, we proposed a nonlocal (NL)-means-based filter
for ultrasound images by introducing the Pearson distance as
a relevant criterion for patch comparison. Experiments were
carried out on synthetic images with different simulations of
speckle. During the experiments, quantitative measures were
used to compare several denoising filters. Experiments showed
that the Optimized Bayesian Nonlocal Means (OBNLM) filter
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2228 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009
proposes competitive performances compared to other state-of-
the-art methods. Moreover, as assessed by quantitative results,
our adaptation of classical NL-means filter to speckle noise pro-
poses a filter more suitable for US imaging. Experiments on real
ultrasound data were conducted and showed that the OBNLM
method is very efficient at smoothing homogeneous areas while
preserving edges. Further work will be pursued on the auto-
matic tuning of the OBNLM filter and on the influence study
on postprocessing tasks such as image registration or image
segmentation.
APPENDIX
Details of the Bayesian NL-Means Filter: The following
Bayesian definition of the NL-means filter is presented in [3].
By considering the quadratic loss function, the optimal Bayesian
estimator of a noise-free patch can be written as
(15)
where denotes the probability density function
(pdf) of conditionally to , is the observed in-
tensity and is the true intensity of block . According to
the Bayes’ and marginalization rules, the so-called conditional
mean estimator can be rewritten as (see [3])
(16)
where denotes the distribution of condi-
tionally to . Since and cannot be
estimated from only one realization of the image, these pdfs
are estimated from observations (blocks ) taken in a search
window of a block . According to [3] and [55], the fol-
lowing approximations can be written:
(17)
(18)
If we assume the prior distribution uniform. This
leads to the empirical Bayesian estimator uses in (5)
(19)
We refer the reader to [3] for detailed explanations.
ACKNOWLEDGMENT
The authors would like to thank P. Tay for providing the
MATLAB code of the SBF filter. They would also like to thank
the reviewers for their fruitful comments.
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Pierrick Coupé, photograph and biography not available at the time of
publication.
Pierre Hellier, photograph and biography not available at the time of
publication.
Charles Kervrann, photograph and biography not available at the time of
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Christian Barillot, photograph and biography not available at the time of
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