An efficient parameter selection criterion for image denoising
ABSTRACT The performance of most image denoising systems depends on some parameters which should be set carefully based on noise distribution and its variance. As in some applications noise characteristics are unknown, in this research, a criterion which its minimization leads to the best parameter set up is introduced. The proposed criterion is evaluated for the wavelet shrinkage image denoising algorithm using the cross validation procedure. The criterion is tested for some different values of thresholds, and the output leading to the minimum criterion value is selected as the final denoised output. The resulting outputs of our method and the previous threshold selection scheme for the wavelet shrinkage, i.e. the median absolute difference (MAD), are compared. The objective and subjective test results show the improved efficiency of the proposed denoising algorithm
- John Wiley, New York, ISBN 0-471-85233-3. 01/1987;
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ABSTRACT: With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels; however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. We describe a new principle for spatially-adaptive estimation: selective wavelet reconstruction. We show that variableknot spline fits and piecewise-polynomial fits, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients. RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as we...11/1998;
Article: On image denoising methods[show abstract] [hide abstract]
ABSTRACT: The search for e-cient image denoising methods still is a valid challenge, at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions, but fail in general and create artifacts or remove image flne structures. The main focus of this paper is, flrst, to deflne a general mathematical and experimental methodology to compare and classify classical image denoising algorithms, second, to propose an algorithm (Non Local Means) addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the \method noise", deflned as the difierence between a digital image and its denoised version. The NL-means algorithm is also proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods are compared in four ways ; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of L2 distances of the denoised version to the original image. The most powerful evaluation method seems, however, to be the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method.
An Efficient Parameter Selection Criterion for Image Denoising
Hamed Pirsiavash1, Shohreh Kasaei2, Farrokh Marvasti1
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
1Multimedia Research Lab., Sharif University of Technology, Tehran, Iran
2Department of Computer Engineering, Sharif University of Technology, Tehran, Iran
Abstract - The performance of most image denoising systems
depends on some parameters which should be set carefully based
on noise distribution and its variance. As in some applications
noise characteristics are unknown, in this research, a criterion
which its minimization leads to the best parameter set up is
introduced. The proposed criterion is evaluated for the wavelet
shrinkage image denoising algorithm using the cross validation
procedure. The criterion is tested for some different values of
thresholds, and the output leading to the minimum criterion value
is selected as the final denoised output. The resulting outputs of our
method and the previous threshold selection scheme for the wavelet
shrinkage, i.e. the median absolute difference (MAD), are
compared. The objective and subjective test results show the
improved efficiency of the proposed denoising algorithm.
Keywords – image denoising, wavelet shrinkage, noise estimation,
Nowadays, by improving image acquisition systems,
many types of cameras are available. Some of these cameras
use very simple hardware in order to have low cost and to be
embedded in other devices like mobile phones. Hence, the
output images of these devices are noisy and poor. In
addition, in most image processing systems, the taken image
should be fed to some processing stages like compression
and recognition. The parasitic noise in the input image could
suffer the other processes and make them inefficient.
To overcome these shortcomings, many image denoising
algorithms have been developed during recent years. For
instance, Gaussian smoothing, neighborhood filtering, and
wavelet shrinkage can be mentioned .
In general, all denoising methods have some parameters
and thresholds which should be adjusted to gain the best
performance. Generally, these parameters depend on the
noise distribution and its variance. Most algorithms suppose
the noise to have a white Gaussian distribution with a known
variance. However, in practical situations, we have no
information about the noise variance. Hence, another
problem rises which is the parameter and threshold selection
algorithm. During recent years, some researchers considered
this problem and made some solutions [2, 3, and 4]. The
generalized cross validation method is proposed by Jansen
et al. for multiple wavelet threshold selection [2, 3]. They
defined a criterion which its minimum roughly minimizes
the mean square error (MSE), but their method works in
some special conditions and as proved in , it works only
for wavelet shrinkage with orthogonal transforms. In
addition, as they mentioned in their paper, its output has low
MSE, but it is not guaranteed to yield a good visual quality.
In this paper, assuming the additive noise to have an
arbitrary distribution, a novel criterion for image denoising
is introduced. The minimization of this criterion leads to
near optimum parameter set for denoising purposes. In order
to evaluate the performance of this criterion, it is applied for
optimum parameter selection in a popular image denoising
algorithm, the wavelet thresholding.
The layout of this paper is as follows: Section 2
introduces the proposed criterion and its efficiency in
parameter selection. In section 3, wavelet shrinkage
algorithm is described briefly. The experimental results and
the performance comparison are presented in Section 4.
Finally, Section 5 concludes the paper.
2. PROPOSED CRITERION FOR PARAMETER
In image denoising algorithms with additive noise, the
input image is assumed to be the summation of original
image and an additive random noise. An important
knowledge which is used in the proposed criterion is the
independency of these two signals (the original image and
the additive noise). Here, the aim of denoising algorithms is
to remove the parasitic noise. In fact, the difference between
the input and the output of the denoising stage is the
estimated noise which has been removed (see Figure 1).
Therefore, the distribution of the estimated noise should
approach that of the additive noise.
The estimated noise for image denoising with two
distinct parameter sets is shown in Figure 2. In this figure,
the estimated noise is exaggerated to be shown clearly. It
could be seen that there is a large similarity between the
estimated noise and the original image, but this similarity in
Figure 2(f) is less than that in Figure 2(e). It means that for
an optimum image denoising algorithm, the correlation
between the estimated noise and the output image which is
an expectation of the original image should be minimized.
This result is in agreement with the assumption of
independency between the original image and the additive
noise. Now, the correlation for each parameter set can be
computed. Consequently, by minimizing that, the best
parameter set for the denoising algorithm can be found.
Fig. 1. Noise estimation
Fig. 2. Form left to right and top to bottom: (a) Original image(Lena), (b) noisy image (AWGN, sigma=15), (c) blurred denoised image, (d) denoised
image using proper parameters, (e) estimated noise of (c), (f) estimated noise of (d). (The estimated noise is exaggerated to be seen clearly).
According to our knowledge, most denoising
algorithms assume the original image to have more energy
in lower frequency components compared to the noise. In
addition, the additive noise usually has near flat spectrum
(i.e. white noise). Hence, a huge energy of noise can be
removed by removing the higher frequency components.
But, we know that using these methods, the edge points
which have higher frequency components will be blurred.
As we know, because the human visualize system is more
sensitive to the edges, the blurring effect will be perceived
obviously. In this research, in order to adapt to the human
visualize system, the criterion is altered. As can be seen in
Figure 2, the edge points can be found in the estimated
noise. Therefore, the correlation between the estimated
noise and the edge map of the output image is used.
Because the output image for some parameter sets has
high amount of noise, the edge map should be extracted
using a robust edge detection method; thus, here, Canny
edge detector is used . Then, the following value should
(C Correlation Estimated noise output image edge
With weak denoising parameters, the estimated noise
approaches to a zero field and makes the correlation to
have a low value. Consequently, the minimum correlation
will be found for the weak denoising parameters. In order
to suppress this defect, the estimated noise energy is used
in the proposed criterion. The final criterion can be written
Input image energy
The minimum of this criterion is found for the
optimum parameters set. Some optimization methods like
the genetic algorithms can be used to achieve the
minimum, yet in this paper, showing the performance of
the proposed criterion is the final goal; consequently,
finding the minimum is performed using a simple method,
i.e. cross validation. This criterion is computed for some
parameter sets which are predefined multiplications of the
parameter set of the MAD method, described in the next
section, and the minimum value is chosen.
3. WAVELET SHRINKAGE
As the defined criterion should be evaluated and
compared with the other parameter selection methods, in
this research, a usual image denoising algorithm, i.e. the
wavelet shrinkage is implemented. In this section, a brief
description of this method
implementation results and details are discussed in Section
Wavelet shrinkage is an efficient signal denoising
algorithm introduced by Donoho et al. in [1, 6, and 7].
That method is based on the idea that the original image
has large wavelet coefficients and the noise is distributed
over all coefficients. Thus, by thresholding the small
coefficients, the image will not be damaged although a
large amount of noise energy will be removed. The hard
thresholding is applied using:
where T is a predetermined threshold value. This basic
idea causes some oscillations near the edges. As a result,
they proposed soft thresholding method in which small
wavelet coefficients are cancelled and the others are
changed in order not to destroy the continuity in wavelet
where Sign(x) denotes the signum function. Using this
method, oscillations are suppressed .
In wavelet thresholding methods, the selection of
thresholds for each resolution level is very important
because according to the other denoising algorithms,
wrong selection can make the output image blurred or
Correlation Estimated noise output image edge
estimated noise energy
is presented. The
noisy. Some threshold selection methods are introduced
for Gaussian noise distributions with known variances.
Three commonly used methods are the universal, SURE,
and MiniMax. The mathematical details can be found in
[6, 7]. For instance, universal method is as follows:
where N is the number of data points and
noise variance defined below. In most denoising
algorithms, the median of absolute difference (MAD) is
used for noise variance estimation .
6745 . 0
This estimation yields to good results for Gaussian
distributed noise. A typical wavelet shrinkage algorithm is
shown in Figure 3.
6745 . 0
) )x( Medianx Median
Fig. 3. Block diagram of a simple wavelet shrinkage denoising system.
4. EXPERIMENTAL RESULTS
In this section, the implementation results and the
performance of the proposed algorithm when compared to
other available approaches are presented. An efficient
criterion is computed for several parameter sets in a
denoising algorithm and the parameters leading to the
minimum of the criterion are chosen as the best
parameters for the input image and the related noise
statistics. The proposed algorithm is implemented using
Matlab package for wavelet shrinkage image denoising
As briefly discussed in Section 3, wavelet shrinkage is
a powerful image denoising algorithm, and thus many
researchers have proposed different modified versions of
that algorithm. In this research, wavelet shrinkage is
implemented in two resolution levels. Here, Daubechies
wavelet with 6 tabs is used. The initial threshold for each
subspace is chosen independently based on the MAD
variance estimation and MiniMax threshold selection
methods. Next, the criterion is computed for 11 different
multiplications of this initial threshold set. The
multiplications are chosen uniformly in the logarithmic
scale in the range of
is taken as the best solution. The MSE of this method and
the MAD method for some standard images are listed in
Table 1. The last column contains the best MSE among 11
tested threshold sets. As can be seen, the obtained
denoising system (with selected variance) approaches the
minimum MSE. The resulting PSNR and the calculated
criterion for a sample image are plotted in Figure 4. As
seen in this figure, choosing the minimum value of the
proposed criterion matches the maximum PSNR that leads
to the best parameter selection for denoising purposes. A
sample output result is shown in Figure 5.
As seen in Table 1, for most tested images the MSE
obtained from the proposed algorithm is less than that of
the MAD method. Moreover, for some cases the obtained
MSE by our algorithm is close to the minimum available
As the MSE is not the best measurement for
performance analysis in image processing systems, the
outputs should be examined in a subjective test as well.
The results of the subjective test among 20 boys and girls
are presented in Table 2. Some particular cases in which
the MSEs of our method are high are examined in this
test. For instance, in the 21st row of Table 1, the resulting
MSE is higher than that of the MAD method. The outputs
of the 21st case are shown in Figure 6. It is obvious that
the output of our method is subjectively better than the
MAD output, which proves that both objective and
subjective tests should be run. In fact, because the
criterion uses the edge map, our algorithm leads to less
defects in the edge areas and thus results in higher
subjective performance; although it may have a higher
MSE. Another result obtained from these implementations
is that for lower input noise variances, our method
performs much better than the MAD method. Because for
images with a low level of noise, after denoising the edge
map can be extracted more efficiently, and thus the
proposed algorithm can better calculate the minimum that
matches the maximum of the PSNR. As another result, for
images with small size (about 256x256), our method
performs better than the MAD and MiniMax methods,
because our method is less directly dependent to the
statistics of the images. This fact motivated us to examine
this method in spatially adaptive wavelet shrinkage
algorithms introduced in .
In this paper, an efficient criterion for performance
analysis of denoising systems is introduced. It is shown
that using a cross validation procedure, we can adjust the
system parameters to achieve a better performance. This
criterion is examined for wavelet shrinkage as a common
8 . 02 . 1
. Finally, the minimum
denoising algorithm. According to the results, the obtained
subjective tests show the superiority of the proposed
algorithm when compared to the MAD approach. Another
important advantage of this method is its independency on
the noise distribution and its variance. As mentioned
above, the algorithm performs even better for images with
lower noise variances.
Table 1. MSE of our method in comparison with MAD and minimum
1 Barbara512 5
2 Barbara512 10
3 Barbara512 15
4 Barbara512 20
5 Barbara512 25
6 Barbara256 5
7 Barbara256 10
8 Barbara256 15
9 Barbara256 20
10 Lena512 5
11 Lena512 10
12 Lena512 15
13 Lena512 20
14 Lena512 25
15 Lena256 5
16 Lena256 15
17 Boat 5
18 Boat 10
19 Boat 15
20 Boat 15
21 Boat 20
22 Peppers 5
23 Peppers 10
Table 2. Subjective test results among 20 boys and girls (score 5 is
assigned to the original image).
1 Barbara512 25
2 Barbara256 20
3 Lena256 15
4 Boat 15
The authors would like to acknowledge the support of
Iran Telecommunication Research Center (ITRC) for this
 A. Buades, B. Coll, and J. M. Morel, “On image denoising
methods,” CMLA Preprint (http://www.cmla.ens-cachan.fr/Cmla/),
to appear in SIAM Multiscale Modeling and Simulation, 2004.
 M. Jansen, M. Malfait and A. bultheel, "Generalized cross
validation for wavelet thresholding," Signal Processing, Vol. 56,
pp. 33-34, Jan 1997.
 M. Jansen and A. Bultheel, "Multiple wavelet threshold estimation
by generalized cross validation for images with correlated noise,"
IEEE Tarns. on Image Processing, Vol. 8, No. 7, pp. 947-953, Jul
 Tai-Chiu Hsung and D. Pak-Kong Lun, "Generalized cross
validation for multiwavelet shrinkage," IEEE Signal Processing
Letters, Vol.11, No. 6, pp. 549-552, Jun 2004.
 Canny and John, "A Computational Approach to Edge Detection,"
IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.
PAMI-8, No. 6, pp. 679-698, 1986.
 D. Donoho and I. Johnstone, "Ideal spatial adaptation via wavelet
shrinkage," Biomedika, vol. 81, pp. 425–455, 1994.
 D. Donoho, "Denoising by soft-thresholding," IEEE Trans. on
Information Theory, Vol. 41, pp.613-627, 1995.
 P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier
Detection, New York: Wiley, 1987.
 S. Grace Chang, Bin yu and Martin Vetterli, "Spatially adaptive
wavelet thresholding with context modeling for image denoising,"
IEEE Trans. on Image Processing, vol. 9, No. 9, pp. 1522-1531,
Fig. 4. Results for Lena256 (with Gaussian noise, standard deviation=15) (a) PSNR vs. threshold ratio, (b) proposed criterion vs. threshold ratio.
(Threshold is the multiplication of the threshold ratio and the MAD threshold, i.e. setting threshold ratio equal to one leads to the MAD method).
Fig. 5. From left to right and top to bottom: (a) Original image (Barbara256), (b) noisy image (Gaussian, standard deviation=15), (c) MAD output
(MSE=295), (d) output of our method (MSE=131).
Fig. 6. From left to right and top to bottom: (a) Original image (fishing boat), (b) noisy image (AWGN, standard deviation=20), (c) MAD output
(MSE=157), (d) output of our method (MSE=192).