1532 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 9, SEPTEMBER 2000
Adaptive Wavelet Thresholding for Image Denoising
and Compression
S. Grace Chang, Student Member, IEEE, Bin Yu, Senior Member, IEEE, and Martin Vetterli, Fellow, IEEE
Abstract—The first part of this paper proposes an adaptive,
data-driven threshold for image denoising via wavelet soft-thresh-
olding. The threshold is derived in a Bayesian framework, and the
prior used on the wavelet coefficients is the generalized Gaussian
distribution (GGD) widely used in image processing applications.
The proposed threshold is simple and closed-form, and it is adap-
tive to each subband because it depends on data-driven estimates
of the parameters. Experimental results show that the proposed
method, called BayesShrink, is typically within 5% of the MSE
of the best soft-thresholding benchmark with the image assumed
known. It also outperforms Donoho and Johnstone’s SureShrink
most of the time.
The second part of the paper attempts to further validate
recent claims that lossy compression can be used for denoising.
The BayesShrink threshold can aid in the parameter selection
of a coder designed with the intention of denoising, and thus
achieving simultaneous denoising and compression. Specifically,
the zero-zone in the quantization step of compression is analogous
to the threshold value in the thresholding function. The remaining
coder design parameters are chosen based on a criterion derived
from Rissanen’s minimum description length (MDL) principle.
Experiments show that this compression method does indeed re-
move noise significantly, especially for large noise power. However,
it introduces quantization noise and should be used only if bitrate
were an additional concern to denoising.
Index Terms—Adaptive method, image compression, image de-
noising, image restoration, wavelet thresholding.
I. INTRODUCTION
A
N IMAGE is often corrupted by noise in its acquisition or
transmission. The goal of denoising is to remove the noise
while retaining as much as possible the important signal fea-
tures. Traditionally, this is achieved by linear processing such
as Wiener filtering. A vast literature has emerged recently on
Manuscript received January 22, 1998; revised April 7, 2000. This work
was supported in part by the NSF Graduate Fellowship and the Univer-
sity of California Dissertation Fellowship to S. G. Chang; ARO Grant
DAAH04-94-G-0232 and NSF Grant DMS-9322817 to B. Yu; and NSF Grant
MIP-93-213002 and Swiss NSF Grant 20-52347.97 to M. Vetterli. Part of this
work was presented at the IEEE International Conference on Image Processing,
Santa Barbara, CA, October 1997. The associate editor coordinating the
review of this manuscript and approving it for publication was Prof. Patrick L.
Combettes.
S. G. Chang was with the Department of Electrical Engineering and Computer
Sciences, University of California, Berkeley, CA 94720 USA. She is now with
Hewlett-Packard Company, Grenoble, France (e-mail: grchang@yahoo.com).
B. Yu is with the Department of Statistics, University of California, Berkeley,
CA 94720 USA (e-mail: binyu@stat.berkeley.edu)
M. Vetterli is with the Laboratory of Audiovisual Communications, Swiss
Federal Institute of Technology (EPFL), Lausanne, Switzerland and also with
the Department of Electrical Engineering and Computer Sciences, University of
California, Berkeley, CA 94720 USA.
Publisher Item Identifier S 1057-7149(00)06914-1.
signal denoising using nonlinear techniques, in the setting of
additive white Gaussian noise. The seminal work on signal de-
noising via wavelet thresholding or shrinkage of Donoho and
Johnstone ([13]–[16]) have shown that various wavelet thresh-
olding schemes for denoising have near-optimal properties in
the minimax sense and perform well in simulation studies of
one-dimensional curve estimation. It has been shown to have
better rates of convergence than linear methods for approxi-
mating functions in Besov spaces ([13], [14]). Thresholding is
a nonlinear technique, yet it is very simple because it operates
on one wavelet coefficient at a time. Alternative approaches to
nonlinear wavelet-baseddenoising can be found in, for example,
[1], [4], [8]–[10], [12], [18], [19], [24], [27]–[29], [32], [33],
[35], and references therein.
On a seemingly unrelated front, lossy compression has been
proposed for denoising in several works [6], [5], [21], [25],
[28]. Concerns regarding the compression rate were explicitly
addressed. This is important because any practical coder must
assume a limited resource (such as bits) at its disposal for repre-
senting the data. Other works [4], [12]–[16] also addressed the
connection between compression and denoising, especially with
nonlinear algorithms such as wavelet thresholding in a mathe-
matical framework. However, these latter works were not con-
cerned with quantization and bitrates: compression results from
a reduced number of nonzero wavelet coefficients, and not from
an explicit design of a coder.
The intuition behind using lossy compression for denoising
may be explained as follows. A signal typically has structural
correlations that a good coder can exploit to yield a concise rep-
resentation. White noise, however, does not have structural re-
dundancies and thus is not easily compressable. Hence, a good
compression method can provide a suitable model for distin-
guishing between signal and noise. The discussion will be re-
stricted to wavelet-based coders, though these insights can be
extended to other transform-domain coders as well. A concrete
connection between lossy compression and denoising can easily
be seen when one examines the similarity between thresholding
and quantization, the latter of which is a necessary step in a prac-
tical lossy coder. That is, the quantization of wavelet coefficients
with a zero-zone is an approximation to the thresholding func-
tion (see Fig. 1). Thus, provided that the quantization outside
of the zero-zone does not introduce significant distortion, it fol-
lows that wavelet-based lossy compression achieves denoising.
With this connection in mind, this paper is about wavelet thresh-
olding for image denoising and also for lossy compression. The
threshold choice aids the lossy coder to choose its zero-zone,
and the resulting coder achieves simultaneous denoising and
compression if such property is desired.
1057–7149/00$10.00 © 2000 IEEE
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