Image Denoising by Pointwise Thresholding of the Undecimated Wavelet Coefficients: A Global Sure Optimum

Abstract

We devise a new undecimated wavelet thresholding for denoising images corrupted by additive Gaussian white noise. The first key point of our approach is the use of a linearly parameterized pointwise thresholding function. The second key point consists in optimizing the parameters globally by minimizing Stein's unbiased MSE estimate (SURE) directly in the image-domain, and not separately in the wavelet subbands. Amazingly, our method gives similar results to the best state-of-the-art algorithms, despite using only a simple pointwise thresholding function; we demonstrate it in simulations over a wide range of noise levels for a representative set of standard grayscale images

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IMAGE DENOISING BY POINTWISE THRESHOLDING OF THE UNDECIMATED
WAVELET COEFFICIENTS: A GLOBAL SURE OPTIMUM
Florian Luisier and Thierry Blu
Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
ABSTRACT
We devise a new undecimated wavelet thresholding for de-
noising images corrupted by additive Gaussian white noise.
The first key point of our approach is the use of a linearly pa-
rameterized pointwise thresholding function. The second key
point consists in optimizing the parameters globally by mini-
mizing Stein’s unbiased MSE estimate (SURE) directly in the
image-domain, and not separately in the wavelet subbands.
Amazingly, our method gives similar results to the best
state-of-the-art algorithms, despite using only a simple point-
wise thresholding function; we demonstrate it in simulations
over a wide range of noise levels for a representative set of
standard grayscale images.
Index Terms— Image denoising, undecimated wavelet
thresholding, SURE minimization
1. INTRODUCTION
The multi-resolution analysis performed by the wavelet trans-
form has been proved to be particularly efficient in image
denoising. Since the early use of the classical orthonormal
wavelettransformforremovingadditiveGaussianwhitenoise
through thresholding [1], a lot of work has been done [2, 3, 4,
5], leading to two main observations:
• betterperformancescanbeachievedwithshift-invariant
transformations such as the undecimated wavelet trans-
form [6];
• further improvements can be obtained with more so-
phisticated thresholding functions which incorporates
inter- and intra-scale dependencies.
In this paper, we will only concentrate on the first of these
two observations, without forgetting the pioneer work of Do-
noho, to workoutanewundecimatedwavelet thresholding al-
gorithm based on the minimization of Stein’s unbiased MSE
estimate (SURE) [7]. Up to our knowledge, the SURE has
only been properly used in the framework of orthonormal
This work was supported by the Center for Biomedical Imaging (CIBM)
of the Geneva - Lausanne Universities and the EPFL, the foundations
Leenaards and Louis-Jeantet, as well as by the Swiss National Science Foun-
dation under grant 200020-109415.
wavelet transform. Indeed, its MSE preservation property
allows a wavelet-domain subband-independent SURE mini-
mization. Unfortunately, thisclassicalapproachissub-optimal
withtheundecimatedwaveletrepresentation, becausetheMSE
is not preserved anymore in the reconstruction, despite the
fact that the undecimated wavelet transform (UWT) is a tight
frame. In this paper, we show how to overcome this diffi-
culty and efficiently apply the SURE to undecimated wavelet
thresholding. The results show that, despite not using any
form of inter- or intra-scale dependencies, our simple thresh-
olding strategy rivals the best state-of-the-art algorithms in
the undecimated wavelet transform setting1.
The paper is organized as follows: first, we recall the ma-
trix formulation of the undecimated wavelet transform and
describe the main details of our denoising strategy; then, we
expose our new SURE-based undecimated wavelet threshold-
ing; finally, we demonstrate its efficiency.
2. CONTEXT
2.1. The undecimated wavelet transform
In this paper, we propose to use the classical undecimated
wavelet transform whose dyadic analysis/synthesis filterbank
for 1D signal is presented in Figure 1. The redundancy fac-
tor of this overcomplete wavelet expansion is J + 1 where J
standsforthe numberofdecompositionlevels. For2D signals
like images, we simply extend this representation in a sepa-
rable way, leading to an overall redundancy factor of 3J + 1.
Thanks to this redundancy, the UWT is a completely shift-
invariant transformation.
The linearity of the UWT allows the following matrix for-
mulation:
w = Dx
where: D = [D1...Dj...DJ+1]Tis a matrix of size (J +
1)N×N, where each Djis a square matrix of size N×N and
x ∈ RNis an input vector. The columns of the sub-matrices
Djare circularly shifted versions of the impulse response of
the corresponding wavelet filter (or scaling filter for DJ+1).
Since the wavelet filters G and the scaling filter H are nor-
malized, diag(DT
(1)
jDj) = 1 ∈ RN, ∀j ∈ [1;J + 1].
1The best UWT for these methods turn out to be the Haar UWT for the
vast majority of test images.

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Perfect reconstruction RD = I is then achieved as fol-
lows:
x = Rw
(2)
where: R = [1


(J +1)N ×(J +1)N which compensates for the redundancy
of the transformation.
2D1...1
0
...
...
...
2jDj...
0
1
2JDJ


1
2JDJ+1] = DTΛ.
Λ =

1
2I
0
...
0
...
...
...
1
0
2JI0
1
2JI

is a diagonal matrix of size
(A)
x
G(z−1)
H(z−1)
G(z−2)
H(z−2)
w1
w2
w3
(B)
w1
w2
w3
1
2G(z2)
1
2H(z2)
?
1
2H(z)
1
2G(z)
?
x
Fig. 1. The classical undecimated wavelet filterbank for 1D
signal. (A) Analysis. (B) Synthesis.
2.2. Our denoising strategy
The aim of the denoising algorithm is to find the best estimate
ˆ x ∈ RNof an unknown signal x ∈ RNfrom a noisy obser-
vation of it y = x + b ∈ RN. Every element of the noise b
is assumed to follow an independent normal law specified by
a zero-mean and a known variance σ2.
In this context, our denoising strategy consists of the fol-
lowing steps:
1. Apply the UWT to the noisy signal y to get the trans-
formed noisy coefficients w = Dy = (wj)j∈[1;J+1],
where wj∈ RN;
2. Apply a pointwise thresholding function Γ to the trans-
formed noisy coefficients to get:
ˆ w = Γ(w),
a vector of (J + 1) × N components ˆ wj,n= γj(wj,n),
where γj, j = 1···(J + 1), are nonlinear functions;
3. Come back to the original domain by applying the in-
verse UWT to the thresholded coefficients.
The whole denoising process can thus be fully described by a
function of the noisy input coefficients:
ˆ x = F(y) = RΓ(Dy)
(3)
The first key point of our approach is to build F as a linear
combinationofsimple —possibly nonlinear—functionsFj,k,
that is:
J+1
?
where: Γj,k(w) = [0T···0TΓk(wj)T0T···]T, i.e., the
simple denoising function Fj,kmodifies only subband j.
Thanks to this linear parametrization, the search for the
optimal—intheminimummeansquareerror(MMSE)sense—
parameters akboils down to the resolution of a linear system
of equations (see section 3.3), because the MSE, defined as
1
N?ˆ x − x?2=
is a quadratic function in F(y) and thus, in the aj,k’s.
Of course, since we do not have access to the noise-free
signal x, we can not compute the actual MSE. To overcome
this difficulty, we will use Stein’s unbiased MSE estimate
(SURE) [7] (see section 3.1).
The second key point of our approach is that the parame-
ters optimization is performed in the image domain, contrary
to what is usually done in other UWT-based denoising algo-
rithms. This image domain optimization is unavoidable when
usingtheUWT,becausetheMSEisnotpreservedinthetrans-
formed domain due to the non-orthonormality of the transfor-
mation.
F(y) =
j=1
K
?
k=1
aj,kRΓj,k(Dy)
?
???
Fj,k(y)
,
(4)
MSE =
1
N?F(y) − x?2,
(5)
3. SURE-BASED ALGORITHM FOR
UNDECIMATED WAVELET THRESHOLDING
3.1. MSE estimation
It is possible to have an unbiased estimate of the MSE us-
ing only the noisy observation y, not the original noise-free
signal x. There is even no need to put any statistical assump-
tion on this unknown underlying signal. This unbiased MSE
estimate, known as SURE, takes the following expression:
1
N
+1
N?y?2− σ2
In general, it is quite painful to express the divergence
term which appears in equation (6). Fortunately, in the frame-
work of the undecimated wavelet transform presented in sec-
tion 2, this computation becomes simple, leading to the fol-
lowing form of the MSE estimate:
?
=
?
?F(y)?2− 2F(y)Ty + 2σ2div{F(y)}
?
(6)

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Theorem 1 After a pointwise undecimated wavelet process-
ing, the MSE can be unbiasedly estimated by the following
random variable:
1
N
+1
N?y?2− σ2
i.e.:
E{?} =
where E{·} stands for the expectation operator and
Γ?(w) = [Γ?
Since the variance of the MSE estimate given by Theo-
rem 1 mainly depends on the number of samples N, ? is really
close to the actual MSE in image denoising applications.
?
=
?
?F(y)?2− 2F(y)Ty + 2σ2diag(Λ)TΓ?(Dy)
?
(7)
1
NE{?F(y) − x?2}
1(w1)T···Γ?
J+1(wj+1)T]T.
3.2. Parametric pointwise denoising function
ToapplyTheorem1, weneedadifferentiablepointwisethresh-
olding function; for an easier optimization, it is really ap-
propriate to use a linear parametrization. Both of these con-
straints are satisfied by the following function
γ(w) =
K
?
k=1
akwe−(k−1)
?
w2
3K2σ2
???
θk(w)
(8)
which is simply a weighted sum of first derivatives of Gaus-
sians. For example, for K = 2 terms, we obtain:
γ(w) = a1w + a2we−
w2
12σ2,
(9)
which has been already shown in [8] to be more efficient than
the classical soft-thresholding.
Furthermore, weproposetolinkthenumberofparameters
K to the level of decomposition j as follows:
Kj= min(J − j + 2,4)
(10)
to speed up the algorithm by assigning less parameters to the
coarsest scales.
3.3. Parameters optimization
Our aim is to find the remaining linear parameters aj,kwhich
will minimize the MSE estimate ?. Starting from equation (7),
these are solved for in the following way:
∀(k,j) ∈ [1;K] × [1;J + 1], we have:
∂?
∂aj,k
where: a, c and M are vector/matrix straightforwardly built
using, respectively, the coefficients aj,k, the SURE expres-
sions
Fj,k(y)Ty − σ2diag(Λ)TΓ?
products Fj,k(y)TFj?,k?(y).
Because of this linear feature, our algorithm is fast and
simple to implement.
= 0 ⇐⇒ a = M−1c
(11)
?
j,k(Dy)
?
and the scalar
4. EXPERIMENTS
4.1. Optimization method assessment
Before comparing our algorithm with the best state-of-the-
art denoising schemes, we have performed several experi-
ments on simulated additive Gaussian white noise to validate
our image-domain optimization method. We have noticed an
overall gain of up to 1 dB compared to an independent undec-
imated wavelet in-band MSEs minimization. This large gap is
caused by the high correlation of the UWT wavelet subbands.
4.2. Comparison with state-of-the-art denoising schemes
We propose to compare our pointwise undecimated wavelet
thresholding with two other wavelet denoising procedures:
the BiShrink [5] and the BLS-GSM [3] which gives, up to our
knowledge, the best results known in the literature. Contrary
to our approach, both of these methods integrate the parent
—coefficient at the next coarser scale— and a local neighbor-
hood to better estimate the current noisy coefficient. We have
run these algorithms under the same conditions as ours using
therespectiveMatlabcodesoftheauthors, kindlyprovidedon
their respective webpages. Their parameters have been cho-
sen according to what was suggested in their papers [5, 3].
We have applied five decomposition stages of an undec-
imated wavelet transform for each of the various denoising
procedures. Surprisingly, the best results were obtained for
all methods with the Haar filters for over a vast majority of
test images. To perform a reliable PSNR2comparison, we
have then averaged the results over eight noise realizations in
an input PSNR range of 8.13 − 34.15 dB.
In Table 1, we have reported the output PSNRs of the
various algorithms and, in Figure 2, we show a visual result.
As it can be observed, our method is really competitive: we
achieve the same average quality as the BLS-GSM and an av-
erage gain of 0.4 dB against the BiShrink. These observations
have been confirmed over most standard test images, to the
notable exception of Barbara, where we obtain similar re-
sults as the BiShrink, but experience an average loss of 0.5
dB compared to the BLS-GSM. This may suggest that with
textured images, the integration of local neighborhood infor-
mation may become important.
From a practical point of view, the computation time of
our method is approximately 5s for 256 × 256 images and
about 35s for 512 × 512 images, whereas the BLS-GSM re-
quired respectively 12s and 60s under the same conditions.
5. CONCLUSION
WehavepresentedanewpointwiseSURE-basedundecimated
wavelet thresholding algorithm. The parameters involved in
2defined as PSNR = 10log102552
MSE

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Table 1. Comparison of some of the most efficient denoising
methods (UWT Haar).
σ σ σ
510
Input PSNR
34.1528.13
Method House 256 × 256
BiShrink
38.4034.94
BLS-GSM
38.3535.29
Our method38.78 35.49
Method
BiShrink
37.9834.73
BLS-GSM
38.26
35.19
Our method38.29
35.08
Method
BiShrink
36.65 33.12
BLS-GSM
36.8233.51
Our method37.1833.57
Method
BiShrink
38.2034.79
BLS-GSM
38.89 35.51
Our method39.0535.60
203050 100
8.1322.11 18.59 14.15
31.90
32.39
32.44
Lena 512 × 512
31.71
32.10
32.00
Boat 512 × 512
29.97
30.31
30.26
Al 512 × 512
31.85
32.42
32.46
30.04
30.53
30.59
27.62
28.01
28.15
24.14
24.59
24.79
29.93
30.25
30.21
27.66
27.93
28.03
24.56
25.05
25.28
28.17
28.44
28.42
25.96
26.17
26.21
23.17
23.44
23.59
30.20
30.60
30.72
28.00
28.30
28.55
24.72
25.26
25.59
Note: output PSNRs have been averaged over eight noise realizations.
thethresholdingfunctionarelinearandtherefore, canbesolved
for elegantly by minimizing Stein’s unbiased MSE estimate
(SURE). Due to the non-orthonormality of the undecimated
wavelettransform, wehaveproposedarigorousimage-domain
optimizationwhichfinallymakesourpointwisealgorithmcom-
petitive with the best state-of-the-art denoising procedures.
6. REFERENCES
[1] David L. Donoho and Iain M. Johnstone, “Adapting to
Unknown Smoothness via Wavelet Shrinkage,” Journal
of the American Statistical Association, vol. 90, no. 432,
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[2] S. Grace Chang, Bin Yu, and Martin Vetterli, “Adaptive
Wavelet Thresholding for Image Denoising and Com-
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9, no. 9, pp. 1532–1546, September 2000.
[3] Javier Portilla, Vasily Strela, Martin J. Wainwright, and
Eero P. Simoncelli, “Image Denoising using Scale Mix-
tures of Gaussians in the Wavelet Domain,” IEEE Trans-
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[4] Aleksandra Piˇ zurica and Wilfried Philips, “Estimating
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in Multiresolution Single- and Multiband Image Denois-
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[5] Levent Sendur and Ivan W. Selesnick, “Bivariate Shrink-
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2002.
(A)
(B)(C)
(D)(E)
Fig. 2. (A) Part of the noise-free Al image. (B) Noisy image:
PSNR = 14.15 dB. (C) BiShrink: PSNR = 28.05 dB. (D)
BLS-GSM: PSNR = 28.34 dB. (E) Our method: PSNR =
28.62 dB.
[6] S. Mallat, “A wavelet Tour of Signal Processing,” Aca-
demic Press, 1998.
[7] C. Stein, “Estimation of the Mean of a Multivariate Nor-
mal Distribution,” The Annals of Statistics, vol. 9, pp.
1135–1151, 1981.
[8] Florian Luisier, Thierry Blu, and Michael Unser,
New SURE Approach to Image Denoising: Inter-scale
Orthonormal Wavelet Thresholding,” To appear in IEEE
Transactions on Image Processing, 2007.
“A