IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
VOL. 30, NO. 12,
DECEMBER 2008
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Nonlinear Scale Space with
Spatially Varying Stopping Time
Guy Gilboa
Abstract—A general scale-space algorithm is presented for denoising signals and images with spatially varying dominant scales. The
process is formulated as a partial differential equation with spatially varying time. The proposed adaptivity is semilocal and is in
conjunction with the classical gradient-based diffusion coefficient, designed to preserve edges. The new algorithm aims at maximizing
a local SNR measure of the denoised image. It is based on a generalization of a global stopping time criterion presented recently by the
author and his colleagues. Most notably, the method also works well for partially textured images and outperforms any selection of a
global stopping time. Given an estimate of the noise variance, the procedure is automatic and can be applied well to most natural
images.
Index Terms—Image denoising, stopping time, scale space, nonlinear diffusion, SNR, spatially varying parameters.
Ç
1
INTRODUCTION
S
CALE-SPACE techniques, based on partial differential
equations (PDEs), have been used successfully in the
past two decades for a wide variety of tasks in image
processing and computer vision, such as image enhancement, denoising, sharpening, segmentation, optical flow,
and more. For some background on the theory and
applications, see [2], [43], [10], [29].
The more classical linear and nonlinear scale spaces [46],
[21], [31], [44] were realized as diffusion-type PDEs. Many
generalizations were made, such as various geometric
evolutions [35], [34], [38], [1], complex-valued processes
[13], variational scale spaces [14], [39], [40], scale spaces
based on morphological operators [18], [24], and even
understanding robust statistics techniques [6], bilateral
filters [5], and wavelet shrinkage [9], [39] as scale spaces.
The relations between functional regularization and diffusion scale spaces were examined in [36]. A scale space based
on robust statistical approach that can also accommodate
heavy-tailed noise well is presented in [16]. See also the
recently proposed inverse-scale-space methods, with excellent denoising capabilities, which presents a new
formalism combining iterated variational techniques and
PDEs [28], [8].
To make the presentation straightforward, the model is
based on the classical Perona-Malik (P-M)-type nonlinear
scale space [31]:
ut ¼ divðgðjrujÞruÞ;
ujt¼0 ¼ f;
ð1Þ
where f is the input noisy image and the regularization is
achieved by evolving u for a finite time t ¼ T . In order to
. The author is with 3DB Systems Ltd., P.O.B. 249, 2 Carmel St., Yokneam
20692, Israel. E-mail: [email protected].
Manuscript received 6 Oct. 2006; revised 7 May 2007; accepted 3 Dec. 2007;
published online 15 Jan. 2008.
Recommended for acceptance by J. Weickert.
For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number TPAMI-0706-1006.
Digital Object Identifier no. 10.1109/TPAMI.2008.23.
0162-8828/08/$25.00 ß 2008 IEEE
preserve edges, the diffusion coefficient g is a decreasing
function of the gradient magnitude of the filtered image
jruj. Neumann boundary conditions are assumed (thus, the
mean gray-level value is preserved). We shall stay with this
definition of nonlinear diffusion in this paper. Naturally,
the idea can be extended to other types of scale spaces.
When (1) is used for image denoising, a certain scale for
u is chosen to represent the original clean image. The choice
of this scale is a significant issue that greatly affects the
performance of the process. This scale depends on both the
level of the noise and the characteristics of the processed
image. As time represents scale in nonlinear diffusion, this
is often called the stopping time problem. We will now
summarize the various main methods that have been
suggested to automatically select the stopping time.
1.1 The Stopping Time Problem
Several studies have addressed the stopping time problem
in recent years, trying to suggest robust and systematic
mechanisms. Weickert [44] suggested stopping when an
estimate of the variance of the clean image is reached.
Mrázek and Navara [27] argued that one should expect to
find the best stopping time when the correlation between
the restored signal and its residual is minimal. This simple
and elegant condition does not need an estimate of the noise
variance, but may oversmooth textures, which are often
only loosely correlated with the main structural part (as
shown in [14]). A very early method by Morozov [26] in
regularization theory, referred to as the discrepancy
principle [11], is to select a solution such that the variance
of the residual equals the variance of the noise. In [30], a
comprehensive method based on cross-validation techniques was proposed. Physical considerations for solving the
stopping time problem for the viscoplastic fluid model were
presented in [12].
A reliable method aimed at maximizing the SNR was
recently suggested in [14]. We base our local algorithm in
the present paper on this method, which is explained in
detail in Section 3.
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1.2 Local Modeling Approaches
Our proposed algorithm essentially selects different scales
in different image locations and can therefore be regarded
as belonging to local modeling approaches. Local modeling
is used in various tasks such as image partitioning and
segmentation [22], [37] and local feature selection [23]. The
nonlinear scale spaces mentioned above can also be
regarded as belonging to this category (since the diffusion
coefficient is spatially varying). Our algorithm differs from
most of these methods in that it focuses on the noise and not
on the image. We are not concerned with scale invariance as
in [23]. In terms of scale, the noise always has a dominant
component at the finest scale (of a single pixel). Our
window of locality should only be large enough to get
reasonable estimations regarding the noise behavior. The
goal is always the same, for all types of images, and that is
to locally increase the SNR.
The method controls only the extent of filtering and not
its qualitative features. For nonlinear diffusion, for example,
edges would be preserved very well and no oscillations
would be created. For highly textured parts, however,
where nonlinear diffusion does not perform well, the
process can improve the performance only by locally
reducing the degree of filtering (keeping both the noise
and the textures). Visually, one sees a considerable
improvement compared to the classical evolutions since
the noise in textured regions is much less noticeable than
that in smooth regions.
There are some studies on local models of the variational
approach. Spatially varying constraints for variational
denoising were proposed in [15]. In the present study,
however, a completely different approach is taken which is
based not on a priori constraints but on estimating the local
SNR. Image smoothing by a data-driven adaptive window
mechanism is suggested in [19]. Black and Sapiro (B-S) [7]
suggested a spatially varying threshold parameter for the
Perona-Malik process which is based on arguments from
robust statistics. We shall examine this approach more
closely in Section 6 and compare it to our method. To the
extent of the author’s knowledge, there are no previous
studies concerning spatially varying stopping time.
1.3 Motivation for Spatially Varying Stopping Time
It is obvious that there is no single global stopping time that
is optimal for all possible images. Consequently, one can
have the following thought experiment: Let us take a single
image and divide it into two parts, evolving each part
separately and joining them back together at the end. As
diffusion filtering is essentially a local operator, apart from
the joint boundary between the two parts, the result should
be quite similar to denoising the image as a whole. In most
cases, however, the two parts do not have exactly the same
image features (for example, textured grass versus smooth
sky in natural images or different anatomical parts in
medical images). Therefore, it is very likely that the optimal
stopping time for each part would not be exactly the same.
By repeating this argument for smaller size windows, we
reach the conclusion that a spatially varying stopping time,
if properly chosen, is likely to perform better than a single
global stopping time.
VOL. 30, NO. 12,
DECEMBER 2008
The problem is how to do it in a methodical and
systematic manner, without introducing additional significant parameters, and having the algorithm general enough
to be successfully applied to most types of images. We try to
present such a method in this paper.
1.4 Summary of the Proposed Method
Our method consists of two stages. We first divide the
image into subregions and find, for each region, the best
stopping time according to the method in [14]. We perform
this procedure twice with different subregions and smooth
the averaged stopping time to avoid artifacts. This serves as
an input for a nonlinear diffusion equation where time is
reparameterized in a spatially varying manner (see (16) in
Section 4).
2
PROBLEM STATEMENT
AND
DEFINITIONS
Below we set the basic definitions and notations of this
paper. It is assumed that the input signal f is composed of
the original signal s and additive uncorrelated noise n of
variance 2 . The aim is to find the u that best approximates
s. We denote v as the residual part of f. Thus, we have
f ¼ s þ n ¼ u þ v:
ð2Þ
Let be the image domain with an area of jj. The
empirical notions of mean, variance, and covariance are
used, with the following definitions of the mean
R
R
1
1
Þ2 d, and
q :¼ jj
qd, the variance RV ðqÞ :¼ jj ðq q
1
the covariance covðq; rÞ :¼ jj ðq qÞðr rÞd. We recall
the identity V ðq þ rÞ V ðqÞ þ V ðrÞ þ 2covðq; rÞ.
The SNR of the recovered signal u is defined as
SNRðuÞ :¼ 10 log
V ðsÞ
V ðsÞ
¼ 10 log
;
V ðu sÞ
V ðn vÞ
ð3Þ
:
where log ¼ log10 . The initial SNR of the input signal,
denoted by SNR0 , where no processing is carried out
ðu ¼ f; v ¼ 0Þ, is according to (3) and (2):
SNR0 :¼ SNRðfÞ ¼ 10 log
V ðsÞ
V ðsÞ
¼ 10 log 2 :
V ðnÞ
ð4Þ
Let us define the optimal SNR of a certain process
applied to an input image f as
SNRopt :¼ max SNRðuðtÞÞ:
t
ð5Þ
In the main parts of this paper, we use a simple scalar
diffusion coefficient in (1) and its spatially varying variations presented later:
g ¼ ð1 þ jruj2 Þ1=2 :
ð6Þ
Since images with gray-scale range 0:255 are processed,
near edges, we have jruj2 1 and the process preserves
edges well (but does not enhance them since it is derived
from a convex energy functional). A common more general
formula introduces a scale parameter K, in the form
gK ¼ ð1 þ ðjruj=KÞ2 Þ1=2 . Thus, the equation above is the
case where K ¼ 1. We shall stay with the simpler form of (6)
in the rest of this paper.
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2177
In Section 6, we show that the algorithm also performs
well for the classical smoothing-enhancing process of
Perona-Malik. The implementation is based on the discretization scheme in [43] for scalar diffusion coefficients.
An explicit time marching scheme is used.
3
ESTIMATING THE OPTIMAL GLOBAL SNR
(GSZ METHOD)
In this section, we describe the method that was recently
suggested by Gilboa-Sochen-Zeevi (GSZ) in [14] for SNR
motivated single scale parameter selection. The method
was primarily formulated in the context of choosing the
best fidelity parameter in the variational context:
max fSNRðuðf; ÞÞg such that u is the minimizer of
JðuÞ þ kf uk2L2 , where RJðuÞ is a convex regularizer like
the total variation JðuÞ ¼ jrujd. In this case, V ðvÞ has a
1:1 correspondence with and one can alternate between
the scale parameters and V ðvÞ. The analog method for
choosing the stopping time was described rather briefly.
Therefore, we give it in some more details here.
3.1 Necessary Condition for Maximal SNR
For any smooth SNRðuðtÞÞ with respect to t, one can write
¼ 0. From (3),
the necessary condition for extrema @ SNRðuðtÞÞ
@t
we see that, since the log function is monotonically
increasing and V ðsÞ is constant in time, maximizing the
SNR is equivalent to minimizing V ðn vÞ. Expanding
V ðn vÞ to V ðnÞ þ V ðvÞ 2 covðn; vÞ yields
1 @V ðvÞ @ cov ðn; vÞ
¼
:
2 @t
@t
ð7Þ
A typical behavior of the SNR as a function of time is to
first increase monotonically and then to decrease monotonically; see Fig. 1 (top). In these cases, condition (7) is
sufficient for choosing the optimal stopping time in the SNR
or mean-square-error (MSE) sense. One may also consider
the boundary case where the SNR decreases from the
beginning (the maximum is at t ¼ 0). In this case,
@ SNRðuÞ
jt¼0 < 0. We can thus formulate the following
@t
theoretical condition for stopping the evolution (which
applies for both cases):
1
T ¼ min t : @t V ðvðtÞÞ @t cov ðn; vðtÞÞ :
2
This condition is only theoretical since we do not know the
term covðn; vÞ as the noise n is unknown. For a practical
stopping condition, we first have to estimate this term.
3.2 Estimating covðn; vÞ
The model assumes prior estimation of the noise statistics
(for zero-mean white Gaussian noise, this means that we
have an estimate of the variance of the noise 2 ). Our
experiments indicate that the variance is the sole significant
parameter to provide and that white noise can be assumed
to have Gaussian statistics. This feature is validated
analytically in the case of linear diffusion (Section 7).
The evaluation is accomplished by generating noise with
statistics similar to n, denoted by n~. One then evolves an
image f ¼ n~ that contains only pure noise and measures
Fig. 1. Processing the Lena image. Top: SNR plot as a function of t.
Middle: @ covð~
n; vðtÞÞ=@V ðvðtÞÞ ¼ CðtÞ=ð@t V ðvðtÞÞÞ (“Estimated”) and
@ covðn; vðtÞÞ=@V ðvðtÞÞ (“Real”) as a function of t. A dot-dashed straight
line is plotted at the 12 value, indicating the theoretical and practical
stopping criteria (7) and (11), respectively. Bottom: Precomputed term
CðtÞ ¼ @ covð~
n; vÞ=@t as a function of t ðlog scaleÞ.
~
@ covðn;vÞ
.
@t
This is done a single time for each noise variance
and can be regarded as a lookup table (see Fig. 1 (bottom)).
The approximation is
@ cov ðn; vðtÞÞ
jf¼sþn
@t
~ vðtÞÞ
@ cov ðn;
jf¼n~ :
@t
ð8Þ
Let us define
CðtÞ :¼
~ vðtÞÞ
@ cov ðn;
jf¼~n :
@t
CðtÞ can be viewed as an estimation of
:¼ E @ cov ðn; vÞj
CðtÞ
f¼n ;
@t
ð9Þ
ð10Þ
where E½s stands for the expected value of a random
variable s. The validity of these approximations is straightforward in the linear case (see Section 7). In the nonlinear
case, a formal analysis is much harder and is based on
strong experimental indications. We currently investigate
the possibility to analyze several specific cases in the
nonlinear setting.
We can now have a practical method for stopping the
evolution:
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TGSZ
1
:¼ min t : @t V ðvðtÞÞ CðtÞ :
2
ð11Þ
This gives a good approximation of the optimal stopping
time in the SNR sense. See Fig. 1 for an example of the SNR
obtained by the method versus the optimal one (Fig. 1 (top))
versus the ground truth
and the approximation of @ covðn;vÞ
@V ðvÞ
(Fig. 1 (middle)). The example depicts the processing of the
Lena image.
We give the precise definition of two additional stopping
time methods that are used in the experiments below. The
stopping condition of Mrazek-Navara (MN) [27] aims at
finding the minimal correlation between u and v:
T ¼ argmint fcorrðuðtÞ; vðtÞÞg, where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
corrðu; vÞ :¼ covðu; vÞ= V ðvÞV ðuÞ:
However, in practice, one does not check for all possible t
and the process is stopped as soon as the correlation begins
to increase:
TMN :¼ minft : @t corrðuðtÞ; vðtÞÞ 0g:
ð12Þ
Thus, we do not need to require that V ðvðtÞÞ increase
monotonically with time, although it most often does. The
method suggested in [44] is somewhat similar in nature and
more rigorous theoretically (since V ðuðtÞÞ is a Lyapunov
functional, see [43]), but usually does not perform as well
(tends to stop the process very early, as explained in [27]).
4
PROPOSED METHOD
First, we define the notions of time and stopping time in a
spatially varying sense.
Spatially varying stopping time. Let us examine the
following modification of (1):
ut ¼ HðT ðx; yÞ tÞdivðgðjrujÞruÞ;
ujt¼0 ¼ f;
ð14Þ
where HðÞ is the Heaviside function:
1; q 0
HðqÞ ¼
0; otherwise:
For T const, this simply incorporates the stopping
time T within the evolution equation such that uðtÞ ¼
uðT Þ for any t T . In case we want to evolve a part of
the image 1 for a time of T1 and the other part 2 for
a time of T2 (where 1 [ 2 ¼ ), we can define T ðx; yÞ
as T ðx; yÞ ¼ fT1 ; 8ðx; yÞ 2 1 ; T2 ; 8ðx; yÞ 2 2 g. The evolution of (14) can be viewed as nonlinear diffusion with a
spatially varying stopping time T ðx; yÞ. A problem may
arise in creating artificial edges (or artifacts) in this way
along the different region boundaries.
A similar behavior to spatially varying stopping time can
be achieved by using spatially varying reparameterization
of time. This avoids the artificially imposed boundary
conditions within the image domain and reduces artifacts.
DECEMBER 2008
The flow of information between image regions is kept
throughout the whole evolution. Moreover, the time
transitions between regions can be made very smooth and
do not rely on the time step size of the actual implementation (as in the stopping time case).
Time reparameterization. Let us examine another modification of (1):
uðx;y;tÞ ¼ divðgðÞruÞ;
uj0 ¼ f;
ð15Þ
where ðx; y; tÞ is a spatially varying reparameterization of
time defined as ðx; y; tÞ :¼ ðx; yÞt and t is a scalar time. We
use a linear reparameterization, which is the simplest and
relates directly to the evolution with spatially varying
stopping time (the close resemblance between these two
equations will be used in our algorithm). The following
simple change of variables,
uðx;y;tÞ ¼
@u
@t
ut
¼
;
@t @ðx; y; tÞ ðx; yÞ
yields a more tractable equation:
2
The Morozov discrepancy principle (“ method”), a
popular denoising strategy in the variational framework
(for example, [33]), can be adopted to nonlinear diffusion by
selecting the following stopping time:
ð13Þ
T2 :¼ min t : V ðvðtÞÞ 2 :
VOL. 30, NO. 12,
ut ¼ ðx; yÞdivðgðÞruÞ;
ujt¼0 ¼ f:
ð16Þ
Setting
ðx; yÞ ¼
T ðx; yÞ
;
Tmax
ð17Þ
where Tmax ¼ maxðx;yÞ ðT ðx; yÞÞ, and evolving (16) for Tmax
time (using as the solution uðTmax Þ) gives similar results to
the solution of (14). The similarity comes from the fact that
each region is effectively evolved the same amount of time
in the two schemes. The difference comes from the different
interactions within the neighborhood. More coherent results
are achieved when the interaction takes place throughout
the evolution and there are no “frozen” parts.
Remark. Similarly, we could have set ðx; yÞ ¼ T ðx; yÞ and
evolve (16) a unit time. Since we want to keep the standard
numerical implementations of explicit schemes and use
the same CFL, we should set the bound ðx; yÞ 1 and,
therefore, (17) is preferred.
4.1 Algorithm
Now that the necessary tools are defined, the algorithm can
be described (Table 1):
1.
2.
3.
4.
5.
Inputs: image, estimated noise variance 2 . Retrieve
CðtÞ from the lookup table (see Section 3.2). Set
k ¼ 1.
Divide the image into subregions ki .
Evolve (14). Find spatially varying stopping time
T k ðx; yÞ by stopping in any region ki according to
the SNR condition (11), where V ðvÞ is evaluated
locally in ki , or when @V@tðvÞ jk ". Terminate the
i
evolution when all the regions are stopped.
Set k ¼ 2, repeat Steps 2 and 3 for a different
partition ki , and average the stopping times:
Tðx; yÞ ¼ ðT 1 ðx; yÞ þ T 2 ðx; yÞÞ=2.
Construct ðx; yÞ as a smoothed version of Tðx; yÞ
(see (18)). Evolve (16) for Tmax time.
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TABLE 1
Algorithm Illustration
Fig. 2. Two partitions of the image into subregions: 1i (full line) and 2i
(dashed line). Each square is of size ‘ ‘ pixels.
1
"
@t V ðvðt1 ÞÞji Cðt1 Þ
2
2
and the region will be deactivated according to the first
stopping condition.
u
t
To construct ðx; yÞ, we use
Tðx; yÞ
ðx; yÞ ¼ g ;
T max
ð18Þ
where Tðx; yÞ is the average of T k ðx; yÞ of the two evolutions,
g is a Gaussian of standard deviation , and denotes
convolution. It is suggestedpto
ffiffiffiffiffiffichoose as a function of the
j j
subregion size. We set ¼ 4 i . For example, in our choice,
ji j ¼ 12 12 ¼ 144 and we have ¼ 3.
4.2 Termination of the Algorithm
Here, we show that the algorithm terminates in a finite time.
The part that should be examined more carefully is stage 3 of
the algorithm, where (14) is evolved and the stopping is done
locally according to GSZ. It is noted that this proposition is
also true for the global GSZ method (in the original paper [14],
this topic is not regarded). Below we prove that the maximal
time for the process to complete (denoted by t1 ) is defined by
the precomputed function CðtÞ (9).
Proposition 1. For any " > 0, 9t t1 , for which the
evolution in stage 3 of the algorithm terminates, where
t1 :¼ minft : CðtÞ 2"g.
Proof. In the evolution of (1), u converges in the L2 sense to f
and v to f f (see, for example, [2]). Consequently,
¼ V ðfÞ. Therefore,
covðf; vÞ converges to covðf; f fÞ
there exists a time t
for which @t covðf; vðt
ÞÞ 2" . This is
also valid for the precomputed function CðtÞ with pure
~ Hence, t1 as defined above exists and is finite.
noise f ¼ n.
It is easy to check that, after the time t1 , there cannot be any
active region i : If @t V ðvðt1 ÞÞji < ", the algorithms stops
according to the second stopping condition (Step 3). On the
other hand, if @t V ðvðt1 ÞÞji ", then
4.3 Image Partitioning
The partition into subregions ki should follow the guideline
that the local measurements would be as uncorrelated as
possible between evolutions. In our proposed algorithm, we
advocate the use of only two evolutions ðk ¼ 2Þ and the
partitioning is rather straightforward: For a square subregion
of size ji j ¼ ‘ ‘ pixels, the second partition is displaced by
‘
2 in each direction (see Fig. 2). With regard to the boundaries,
one can have either smaller subregions near the boundaries or
ones larger than ‘ ‘. Both options are acceptable; we
observed slightly better behavior by taking smaller subregions (the averaging and smoothing at the later stage
considerably reduces the overall effect of these choices).
5
EXPERIMENTAL RESULTS
The proposed spatially varying time algorithm was
compared with three global stopping time methods mentioned earlier, MN (12), the 2 method (13), and the global
SNR method (GSZ) (11), as well as with the theoretical
global optimal result (“Opt”) attaining the highest SNR (or
smallest MSE). Note that this is the optimal global result. It is
selected by checking the SNR at every time step. Naturally,
it is not a practical stopping criterion since, in order to
obtain it, we need the clean image at hand. Moreover, we
will see that, when we have the freedom to select different
stopping times in different regions, as proposed here, we
are able to considerably outperform this result.
The SNR results of denoising six classical test images
(Fig. 3) with additive white Gaussian noise of standard
deviations ¼ 10 and ¼ 20 are summarized in Table 2.
The proposed method (“Prop”) achieves the best results in
terms of SNR in all of our experiments. The row in each
SNR
opt ” shows the difference of
table denoted by “SNR
the average SNR of the six images from the average optimal
(global) SNR.
Fig. 3. Test images.
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DECEMBER 2008
TABLE 2
SNR Results of Test Images (in dB)
In Fig. 4, a part of the Barbara image is shown (white
Gaussian noise ¼ 20) containing both textural and smooth
parts. The proposed method is shown in comparison with
Fig. 5. Results of denoising the Sailboat image (an enlarged part is
shown). Top row (from the left): Clean image s, input image f ð ¼ 10Þ.
Middle row (from the left): Filtering result u using the proposed method
and filtering result u using the optimal global stopping time. Bottom row:
Corresponding contrasted residuals: 2v ¼ 2ðf uÞ.
Fig. 4. Results of denoising the Barbara image (an enlarged part is
shown). Top row (from the left): Clean image s, input image f ð ¼ 20Þ.
Middle row (from the left): Filtering result u using the proposed method
and filtering result u using the optimal global stopping time. Bottom row:
Corresponding contrasted residuals: 2v ¼ 2ðf uÞ.
the SNR-optimal global result. One can clearly see how the
proposed method adapts to the different features of the
image, denoising the smooth parts well while keeping the
textures intact. Examining the residual part (Figs. 4c, left), it
is apparent that the amount of denoising changes considerably in different regions. In Fig. 5, a part of the Sailboat
image, with a lower level of noise ð ¼ 10Þ, is shown.
Similar characteristics appear where, in the proposed
method, the sky is smoother and the textures of the sea
are not eroded compared with the global stopping
approach.
In Fig. 6, the Cameraman image is processed. We show
here that the process also performs very well for uniform
noise. In this case, a uniform noise in the range ½35; 35 is
added. The SNR results of the various methods are given
within the subfigure titles. The proposed local method
clearly outperforms all the global methods both visually
and in terms of SNR. In the enlargement (Fig. 7), one can see
that the details of the camera are better preserved compared
to the global stopping methods, whereas the noise in the
sky region is almost completely removed. In Fig. 8, we
summarize the SNR results graphically: SNRðuÞ is plotted
as a function of time. On that curve, the stopping times and
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2181
Fig. 7. Cameraman—enlargement. Top (from the left): The clean, noisy,
and proposed methods. Bottom: GSZ and MN. Whereas GSZ preserves
the camera details well and MN denoises the smooth background well,
the proposed method does both.
6
VARIATIONS
ON THE
MODEL
In this section, we elaborate more on various aspects of the
algorithm. We show that it can also work for a PeronaMalik type diffusivity [31] and explain how to compute CðtÞ
in this case. We examine the spatially varying diffusion
method of Black and Sapiro (B-S) [7], explain the different
motivation, and show that our method is clearly better for
denoising (B-S is actually directed for image simplification).
Last, we perform various extensive tests and experiments
that show the robustness of the method to various
parameters (which are fixed in our algorithm). An essential
input of the algorithm is the estimated noise variance. We
compute an SNR curve that depicts the sensitivity to errors
in the noise variance estimation. We observe that overestimation of the variance is preferred to underestimation.
Moreover, it is shown experimentally that, for white noise,
the variance is the only significant statistic for our process
(therefore, one can compute CðtÞ using Gaussian noise).
6.1 Perona-Malik Process
Our method can also work successfully for enhancing
processes, such as ones based on the Perona-Malik (P-M)
diffusivities:
gP M1 ¼ ð1 þ ðjruj=Þ2 Þ1 or
ð19Þ
Fig. 6. Cameraman image, uniform noise in the range ½35; 35. Top
(from the left): Clean and noisy images. Second row (from the left): u of
the proposed method and u of GSZ [14]. Third row: Discrepancy
principle [26] and M-N [27]. The fourth and fifth rows depict the
corresponding residuals 2v ¼ 2ðf uÞ. (Optimal global SNR:
SNRopt ¼ 16:18).
the corresponding SNR of the three experimented global
methods are marked. Also, the optimal stopping time is
denoted by “
.” Two additional SNR values are overlaid for
comparison in dashed lines: that of the noisy image SNR0
and the SNR of the proposed method.
Fig. 8. Cameraman image. SNR as a function of time (solid) and the
SNR of the proposed method (dot-dashed line).
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2
gP M2 ¼ eðjruj=Þ :
DECEMBER 2008
ð20Þ
For our experiments, we use (19). It is well known (see, e.g.,
[2]) that nonlinear diffusion with such diffusion coefficients
can be viewed as the steepest descent to minimize a
nonconvex functional. Such evolutions entail both properties of denoising and enhancement [42].
The P-M process depends on a parameter , which can
be understood as a soft threshold parameter of the gradients
to be preserved and even enhanced (as they are expected to
belong to genuine edges of the image). In this case, the CðtÞ
of (9) should be computed with the same parameter as
that used for denoising the image. The additional complexity is usually not very significant (less than 10 percent, as
will be discussed later). In [7], nonlinear P-M diffusion with
a spatially varying parameter ðx; yÞ is suggested, following
the ideas in [6] for the global parameter selection. We
examine this method and compare it to ours in the
following.
6.2 Black and Sapiro’s Spatially Varying Process
In [6], Black et al. followed ideas from robust statistics [32]
and suggested using the following “robust scale” (in their
terminology) for the P-M parameter:
:¼ c MADðruÞ ¼ c medianu ðjru medianu ðjrujÞjÞ;
ð21Þ
where MAD stands for the median absolute deviation and c
is calculated such that, for white Gaussian noise with unit
variance, we get ¼ 1 (the theoretical number is c ¼ 1:4826,
but this may depend on the specific discretization of the
gradient). In [7] (B-S), a spatially varying process was
suggested where the MAD value is calculated locally in a
window for each pixel to obtain a spatially varying
parameter ðx; yÞ.
In essence the local method of B-S is for simplification or
what is now referred to as structure-texture decomposition
(see [25], [41], [3], [4]) and is less adequate for denoising. For
instance, it is shown that, in terms of SNR, their simple
global parameter performs better (see Fig. 9). The most
notable difference is that the method of B-S does not treat
textures as outliers and therefore smoothes them along with
the noise. Still, it is shown that GSZ works well for both the
global and the local B-S methods (see SNR plots, Fig. 9). In [4],
the use of a correlation criterion in the variational context is
suggested, similar to that of M-N [27], as a method to find a
good decomposition parameter (using the assumption that
in many cases, the structures and textures of the image are
not correlated). It is therefore suggested to view the local
B-S with M-N stopping time as a simple reasonable flow to
achieve structure-texture decomposition. For comparisons
between the proposed method and that of B-S, see Figs. 10
and 11.
6.3
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Robustness of the Proposed Method
6.3.1 Robustness to Parameter Choice
The size of each subregion ji j has to be selected, which
determines how local the method is. As in any local
approach, an uncertainty principle governs the basic tradeoffs: As we get more accurate in terms of resolution (smaller
window size), we lose accuracy in the estimates, in our case,
Fig. 9. Local versus global, Black-Sapiro method. A graph of the SNR as
a function of time of two versions of the Black-Sapiro method are plotted
(Barbara image, see Fig. 10). The solid line represents the local method
(spatially varying ); the dashed line represents the global method
(single ). The global method most often gives better results for
denoising. On the plots, the stopping selections of GSZ (circle) and M-N
(square) are shown.
the local noise level and the covariance condition. In Fig. 12,
we show an experiment where the SNR result of the
proposed method is checked for different sizes of ji j and
different noise levels. In general, the algorithm is robust and
works well for a large range of sizes. The experiments in
Table 2 are done with subregions of size 12 12 pixels for
both levels of noise.
Other variations that may be applied to the algorithm are
the number of runs k and the width of the convolving
Gaussian, which are resolved in our proposed method. Our
experiments indicate that the most noticeable improvement
is achieved by computing two stopping times T k ðx; yÞ
versus a single one. If one repeats this process four times
(using different nonoverlapping partitions i ), there is
some improvement, of about 0:1dB, which does not seem to
be worth the additional computational effort. Averaging
over 8 or 16 runs makes almost no further improvement.
We propose making the standard deviation of the
smoothing kernel dependent on the size of the subregion
such that blocky artifacts are not visible and the resolution
of the window size is preserved. Here, again, the process is
robust and we did not observe significant changes by
changing .
6.4 Errors in the Noise Variance Estimation
A significant input for this process is the variance of the
noise. This is used for the computation of CðtÞ. We have
therefore performed extensive experiments to examine the
reliability of the method in cases where the noise estimations are not reliable. Our experiments are based on the
Kodak image photos [20] (23 natural images). We have
added white Gaussian noise with variance 2 ¼ 400. CðtÞ
was taken for 11 different values of variance ranging from
0:52 to 1:52 in steps of 0:12 . In Fig. 13, a summary of the
results is given, where an average of the SNR result is
plotted for each noise variance estimation. We performed
this experiment with both the nonenhancing diffusion
coefficient (6, left) and with the P-M coefficient (19).
Moreover, we tested corruption by uniform white noise.
We also tested corruption by uniform white noise where the
GILBOA: NONLINEAR SCALE SPACE WITH SPATIALLY VARYING STOPPING TIME
2183
Fig. 11. Comparison of the proposed method (top right) to local B-S with
GSZ and M-N stopping criteria. On the bottom right, the local MADbased threshold is depicted. The B-S algorithm regards texture and
noise similarly and smooths both (only edges are considered outliers by
this approach).
We observe that overestimation of the variance is preferred to underestimation. For instance, for the nonenhancing
diffusivity (6), we observe a reduction of about 0:5dB in
the SNR for a 40 percent overestimation in the noise
variance (from a peak of 14.18 for the correct estimation to
13.76 for a variance estimation of 1:42 ). This SNR is about
the same as the original GSZ with a correct variance
estimation (13.73). Visually, the denoising is still at
acceptable quality.
Large errors in the estimation of the noise variance may
happen, especially for “clean” or very high SNR images,
where the noise estimator may confuse random texture with
noise. We show in Fig. 14 that our algorithm is relatively
robust. A low level of noise with variance 2orig ¼ 4 was
added to a raft image (taken from the Kodak collection [20]).
We assume an estimation of 10 times the original noise
variance, that is, 2est ¼ 40. CðtÞ is taken according to est .
Fig. 10. Comparison of the proposed method to variations of B-S with
GSZ and M-N stopping criteria. Our proposed method is clearly better
suited for denoising.
computation of CðtÞ assumes Gaussian white noise. We see
that there is very little difference in the results (Figs. 13,
bottom plots). Our intuition for that is that we are
optimizing second-moment statistics (SNR or MSE) and,
therefore, higher order moments are less relevant for the
nonlinear case.
Fig. 12. Performance for different sizes of subregion ji j. The SNR
results of the proposed scheme applied to the Lena image are plotted as
a function of the size of ji j for different noise levels. The subregions
tested (from left to right) are squares of 8 8, 12 12, 16 16, 20 20, 24 24, 28 28, and 32 32 pixels. Each line (from top to bottom)
depicts trials with the following standard deviation of white Gaussian
noise: ¼ 5, 10, 15, 20, 25, and 30, respectively.
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
Fig. 13. Average SNR of the Kodak image collection for different
estimations of the noise variance. First and second rows: White
Gaussian noise ð ¼ 20Þ. First row: Experiment with nonenhancing
diffusivity (6). Second row: Perona-Malik process (19). Third and fourth
rows: Uniform noise (range ½20; 20, g as in (6)). A solid line means that
the CðtÞ is computed using uniform noise; a dashed line means that CðtÞ
is computed using Gaussian noise of the same estimated variance.
Fourth row: The difference in average SNR between these two
computations. One observes that the exact statistics of white noise
are not significant for the proposed method.
We compare the proposed method to M-N and the
2 method (13). Our algorithm is shown to be more robust
to the estimation error than the 2 method. Moreover, it is
clear that, although M-N does not need the estimation of the
noise, very low levels of noise can significantly mislead the
algorithm, resulting in noticeable oversmoothing (in other
words, the implied noise estimation of the M-N decorrelation criterion works poorly in such cases).
6.5 Complexity
The complexity when CðtÞ is precomputed involves two
evolutions for the stopping times (stages 3 and 4 of the
algorithm) and one evolution of stage 5. Therefore, it takes
about three times of an evolution with a global time. If one
needs to compute CðtÞ, this may be slightly longer. In the
VOL. 30, NO. 12,
DECEMBER 2008
Fig. 14. Large error in the noise estimation (high SNR image). The
original noise variance is four; it is estimated as 2 ¼ 40. We show how
all algorithms fail, although the proposed algorithm is relatively better.
The M-N algorithm does not rely on a noise estimator but, for low levels
of noise, tends to oversmooth the textures (image taken from the Kodak
collection [20]).
P-M process, one has a parameter , which is often image
dependent. It may not be reasonable to have a lookup table
for many values of both and the noise variance 2 .
However, for training CðtÞ, one needs only a patch of about
80 80 pixels. For images of size 512 512, this process is
negligible (about 2.5 percent additional time). For spatially
varying parameters, such as the local B-S, where we have
ðx; yÞ, the computation of CðtÞ should use the same
parameters as the denoising process and, therefore, the
patch should be the same size as the image (which requires
an additional full evolution).
GILBOA: NONLINEAR SCALE SPACE WITH SPATIALLY VARYING STOPPING TIME
7
LINEAR ANALYSIS
OF THE
ut ¼ u;
E½< g s; n >
Z Z
¼E
gð~
x; y~; tÞsðx x~; y y~Þd~
xd~
y nðx; yÞdxdy
Z
Z
gð~
x; y~; tÞE
sðx x~; y y~Þ nðx; yÞdxdy d~
xd~
y
¼
GSZ METHOD
In this section, we develop the GSZ algorithm [14] in the
linear case. It is shown that the method selects the expected
value of the optimal global time (where optimality is in the
MSE sense). Also, it may give the reader some more
intuition on the process and specifically on the role of CðtÞ
(9). In Section 8, we discuss and draw some conclusions
relating the analytic results of the linear case to the
experimental results of the nonlinear processes shown in
Section 6.
It is shown below that the standard assumption of
uncorrelated additive noise is the main requirement of the
GSZ algorithm. In the linear case, we only require that s and
n and their translations are not correlated. However, for the
broader nonlinear case, we may need full statistical
independence between both signals (and their translations).
The property of statistical independence is valid for
additive white noise. Taking gðjrujÞ 1 in (1) gives the
standard linear diffusion process (linear scale space [17],
[46]; see [45] for the axiomatics and a broader historical
review):
ujt¼0 ¼ f:
ð22Þ
For simplicity (without loss of generality), let us assume an
infinite domain ¼ IR2 to have the following explicit
solution:
uðx; y; tÞ ¼ gðx; y; tÞ fðx; yÞ;
where gðx; y; tÞ is a two-dimensional Gaussian of standard
pffiffiffiffiffi
deviation ðtÞ ¼ 2t. Here, we use the MSE criterion
(instead of maximizing the SNR). Therefore, we use slightly
different notations; the derivation, however, is the same. We
replace the variance by the L2 norm and the covariance by
the L2 inner product. (As the mean values of f and u are the
same, the definitions coincide, up to a scaling factor.)
We would like to choose the optimal t which minimizes
the MSE:
e2 :¼ ks uk2 ¼ kn vk2 :
2185
Replacing v :¼ f u by fðx; yÞ gðx; y; tÞ fðx; yÞ and f by
s þ n, we get
E½< n; v > ¼ E½< n; n g n >:
F fE½< n; n g n >g ¼ Nð!Þð1 Gð!ÞÞ;
where F is the Fourier transform, Nð!Þ is the power
spectral density of the noise n, and Gð!; tÞ :¼ F fgðx; y; tÞg.
Thus,
@
@
CðtÞ :¼ E
< n; v > ¼ E½< n; n g n >
@t
@t
@
ð24Þ
¼ F 1 fNð!Þð1 Gð!; tÞÞg
@t
¼ F 1 fNð!ÞGt ð!; tÞg;
where F 1 is the inverse Fourier transform and
@
Gt ð!; tÞ ¼ @t
Gð!; tÞ. Hence, in the linear case, CðtÞ
has a
closed form expression, (10) is its generalization in the
nonlinear case, and (9) is its numerical estimation.
8
DISCUSSION
The proposed algorithm relies on the GSZ method [14] for
choosing the stopping time. Its properties are therefore
closely related to this method. The linear analysis above
shows the validity of GSZ in the linear case and reveals
some properties of the method. Experimentally, it appears
that many of the properties hold (at least approximately) for
nonlinear diffusion processes (relating to the minimization
of both convex and nonconvex energies). We can summarize the main findings as follows:
.
2
@
@
< n; v > þ
kvk2 ¼ 0:
@t
2@t
As we have v, the right term of the left-hand side is
known. Let us examine the expected value of the left
@
@
term, that is, E½@t
< n; v > ¼ @t
E½< n; v >. We first show
that E½< g s; n > ¼ 0. We use the fact that any spatial
translation of n or s yields uncorrelated signals, that is,
Z
E
sðx x~; y y~Þnðx; yÞdxdy ¼ 0; 8~
x; y~ 2 :
As g is deterministic, we can change the integration order
and use the above relation to have
ð23Þ
This can be written in the Fourier domain as
2
A necessary condition is @e
@t ¼ 0. Expanding kn vk and
taking the derivative with respect to t yields the following
condition:
¼ 0:
.
.
The expected value of the optimal solution (in the
MSE sense) is found by the GSZ method in the linear
case. In the nonlinear case, a good approximation is
obtained. This paper presents the adaptive (or
spatially varying) version of the filter, which performs significantly better. An estimation of the noise
variance is required as an input, although the
algorithm is quite robust and handles overestimation well (up to about 40 percent more than the
actual noise variance).
The computation of CðtÞ is justified by the analysis
of the statistically correct way to estimate the
unknown term covðn; vÞ in the linear case. In the
general nonlinear setting, CðtÞ can be understood as
an approximation for the (time derivative of) the
expected value of the filtering response of the noise.
An important consequence of the linear analysis is
that one does not need the full statistics of the noise,
only its autocorrelation function (or power spectral
2186
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
density). Our experimental results (see Fig. 13)
indicate that a similar behavior seems to hold for
nonlinear diffusion processes. Therefore, for computing CðtÞ in the case of white noise, one can
assume that it is Gaussian.
9
CONCLUSION
Nonlinear diffusion is primarily adequate for denoising
piecewise smooth images with large structures. In denoising natural images, the performance degrades mainly due
to textures and fine-scale details that are part of the image.
We propose a general automatic algorithm where the
amount of filtering changes smoothly in a semilocal
manner, trying to achieve the maximal local SNR based
on the covariance condition presented in [14]. Our experiments indicate that the method outperforms any choice of
global stopping time that could be suggested, both visually
and with respect to the SNR results. The complexity is
approximately three times that of a classical evolution with
a global time. We are currently searching for ways to reduce
the runtime, perhaps by finding suitable estimates of ðx; yÞ
that can be computed “on the fly” in a single evolution.
A simple partition algorithm is used where the local
regions are predetermined. To reduce the creation of
artifacts, this procedure is repeated twice in a blocky
stopping time formulation. The final result is based on a
spatially varying time, calculated as a smoothed function of
the averaged stopping times. More elaborated schemes that
use segmentation algorithms to select image-driven regions
may also be suggested.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
ACKNOWLEDGMENTS
[23]
This work was supported by grants from the US National
Science Foundation under Contracts DMS-0714087, ITR
ACI-0321917, and DMS-0312222, and the US National
Institutes of Health under Contract P20 MH65166. The
main part of the paper was done during the stay of the
author in the Department of Mathematics at the University
of California, Los Angeles.
[24]
[25]
[26]
[27]
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2187
Guy Gilboa received the BSc degree in
electrical engineering from Ben-Gurion University, Israel, in 1997 and the PhD degree in
electrical engineering from the Technion-Israel
Institute of Technology in 2004. Prior to his PhD
studies, he worked for three years at the Intel
Development Center, Haifa, Israel, in the design
of processors. He was a researcher in the
Department of Mathematics at the University of
California, Los Angeles (hosted by Professor
Stanley Osher) on advanced variational and PDE-based methods
applied to image processing. He has recently joined 3DV Systems
Ltd., Yokneam, Israel.
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