IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO.

9, SEPTEMBER 2006 2771

Moving Window-Based Double Haar Wavelet

Transform for Image Processing
Xin Wang, Member, IEEE

Abstract—Image denoising is a lively research field. The classical

nonlinear filters used for image denoising, such as median filter,
are based on a local analysis of the pixels within a moving window.
Recently, the research of image denoising has been focused on the
wavelet domain. Compared to the classical nonlinear filters, it is
based on a global multiscale analysis of images. Apparently, the
wavelet transform can be embedded in a moving window. Thus, a
moving window-based local multiscale analysis is obtained. In this
paper, based on the Haar wavelet, a class of nonorthogonal multi-
channel filter bank with its corresponding wavelet shrinkage called
Lee shrinkage is derived. As a special case of this filter bank, the
double Haar wavelet transform is introduced. Examples show that Fig. 1. Structure of discrete M -channel filter bank.
it is suitable for a moving window-based local multiscale analysis
used for image denoising, edge detection, and edge enhancement.
Index Terms—Edge detection, edge enhancement, Haar wavelet, used for denoising should be selected carefully so as to convert
image denoising, Lee shrinkage, moving window. an optimal estimate of the wavelet coefficients to an optimal
estimate of the original signal.
In this paper, based on the Haar wavelet, a class of multi-
I. INTRODUCTION channel nonorthogonal filter bank is designed. Also, its corre-
ILTERING image without smearing edges is an important sponding wavelet shrinkage called Lee shrinkage is derived for
F research field of image processing. Based on a moving
filter window, some classical nonlinear filters were presented.
yielding an optimal estimate of the original signal. As a special
case of this filter bank, a double Haar wavelet transform is in-
Median filter [1], Lee filter [2], order statistics filter [3], non- troduced. By embedding the double Haar wavelet transform in a
linear mean filter [4], multistage median filter [5], and stack moving window, a local multiscale analysis of signal is derived.
filter [6] are some examples. They attempt to remove the effects It is very suitable for image denoising, edge detection, and edge
of noise while retaining the edge of images. enhancement. Examples show that the proposed algorithms are
In recent years, much attention has been devoted to multires- effective.
olution representations for signal denoising. In their pioneer The rest of this paper is organized as follows. In Section II,
work, Mallat et al. [7] introduced a complete signal reconstruc- the Haar wavelet-based multichannel filter bank (HMF) is de-
tion by wavelet coefficient maxima. Donoho et al. [8] proposed signed. In Section III, based on the HMF, the wavelet shrinkage
a soft thresholding to obtain a minimax mean-squared error esti- is developed. In Section IV, the double Haar wavelet transform
mate of wavelet coefficients. Xu et al. [9] developed a spatially (DHWT) is introduced. In Section V, the DHWT is extended
selective noise filter technique. Nowak [10] derived a wavelet to two-dimensional (2-D) for image denoising. In Section VI,
Wiener filter for Rician noise removal. Moulin et al. [11] pre- the Prewitt operator-based optimal edge detector is proposed. In
sented a Maximum a posteriori (MAP) estimate of wavelet co- Section VII, the median filter-based edge enhancer is presented.
efficients. Hansen et al. [12] derived a MDL criterion-based The conclusions are made In Section VIII.
wavelet shrinkage rule. Chang et al. [13] found a wavelet thresh-
II. HAAR WAVELET-BASED -CHANNEL FILTER BANK
olding by minimizing Bayes risk.
It is known that there is limitation for the time-frequency lo- To give the definition of Haar wavelet-based -channel filter
calization of a single wavelet function [14]. Based on multi- bank, we should introduce the concept of discrete -channel
wavelet filter bank, a method for image denoising was devel- filter bank first.
oped [15]. The discrete -channel filter bank is described in Fig. 1, in
Essentially, denoising in the wavelet domain is to obtain an which the input signal is split by a low-pass filter
optimal estimate of the original signal. Thus, the wavelet basis and high-pass filters , into the reference
signal and the detail signals , ,
Manuscript received May 7, 2004; revised October 25, 2005. This work was
each of which is decimated by a factor of . For reconstruc-
supported by the National Natural Science Foundation of China under Grant tion, interpolation by a factor of is performed, followed by
60172022. The associate editor coordinating the review of this manuscript and reconstruction filters , .
approving it for publication was Dr. Truong Q. Nguyen. The -channel orthogonal filter bank was constructed by
The author is with the School of Information Science and Engineering, Shan-
dong University, Jinan 250061, China (e-mail: xwang@sdu.edu.cn). Vadiyanathan [16] and Nguyen [17] for a number of commu-
Digital Object Identifier 10.1109/TIP.2006.877316 nication applications such as subband coders for speech sig-
1057-7149/$20.00 © 2006 IEEE
2772 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

nals. However, it is interesting to apply a multichannel filter III. HMF-BASED LLMMSE ESTIMATE
bank to the denoising of signals. The key of this problem is to
For the HMF, an operation on the wavelet coefficients will be
find such a multichannel filter bank whose high-pass filters and
derived in this section so as to obtain an optimal estimate of the
low-pass filter are good at edge detecting and noise smoothing,
original signal.
respectively.
Suppose that is the original signal and the input noisy
It is known that the Haar wavelet has the most compact spa-
signal can be described as
tial support of all wavelets and is also an optimal edge matching
filter [18]. Thus, based on the Haar wavelet and a simple shift
operation [19], a -channel filter bank can be designed. Its (6)
high-pass and low-pass filters are defined as
where is white Gaussian noise with zero mean and vari-
(1) ance . Based on the filter bank shown in Fig. 1, (6) can be
rewritten as a vector notation
and
(7)
(2) where ,
and ,
respectively. . Suppose that
To obtain the reconstruction filters , , the samples are i.i.d. random variables
we need to express the -channel analysis filters as a matrix and the variance of the original signal is . A linear estimate
notation. Let of the wavelet coefficients can be defined as

(8)
.. ..
. .
where ,
, and

.. .. .. .. .. (3)
. . . . .

.. ..
. .
The inverse matrix of (see Appendix ) can be written as

Let
.. .. .. .. .. (4)
. . . . .
.. .. .. .. ..
. . . . .
Then, the reconstruction filters, , , are
given by [20]

.. .. .. ..
. . . .
.. .. .. .. .. .. ..
. . . . . . .
where and are the matrices obtained by deleting the
(5) last row of and the last column of , respectively. The
where denotes the transpose of matrix. reconstruction process shown in Fig. 1 can be expressed as a
Note that the span of the low-pass filter is just . After vector notation
decimating with a factor of , the noise in the decimated refer-
ence signal will preserve its independent property. This property
is very useful for some denoising operators that ask the noise to (9)
be white in the different scales of wavelet transform domain.
Now, the design of the Haar wavelet-based -channel where .
filter bank (HMF) is finished. When , the HMF is Our goal is to minimize the mean-squared error
nonorthogonal. by selecting some proper values of
WANG: MOVING WINDOW-BASED DOUBLE HAAR WAVELET TRANSFORM 2773

. For this purpose, we can derive the which can yield a local linear minimum mean-square error
equation [21] (LLMMSE) estimate [21] of the original signal.

(10) IV. DOUBLE HAAR WAVELET TRANSFORM

The HMF with is called the DHWT. In this paper, we
where and
only discuss the applications of the DHWT in image processing.
.
About the DHWT, we have
Since the values of the original signal at different positions
have been supposed to be i.i.d. random variables with variance
, we have

.. .. ..
. . . Suppose that , and are the samples of
the input signal at a given time. The estimate of the center pixel
is especially important for some applications of image
processing. According to the definition of DHWT we have the
.. .. .. reconstructed
. . .

(16)
Then, (10) can be solved for to obtain

Then, from (9), (13), and (15), the estimate of is ob-

(11) tained by

It is note that the term can be replaced with the local

variance defined by (17)

where
(12)

or expressed as a weighted sum of the wavelet coefficients given (18)

by
and

(19)

(13) The double Haar wavelets is as well as the Haar wavelet in

edge detection, but better in denoising for its improved low-pass
The (11) becomes filter. Particularly, the short support makes it very suitable for a
moving window-based multiscale analysis that will be discussed
in the following sections.
(14)
V. DHWT FOR IMAGE DENOISING
Since , the case of
Similar to the two-channel orthogonal wavelet transform,
says that a flat part of signal comes [2]. Thus, by letting
the DHWT can be extended to 2-D for image denoising. Let
, , the wavelet coefficients of noise are
be an image of pixels. The steps of the
cleaned out entirely. It follows
2-D discrete double Haar wavelet transform is defined by the
following steps.
1) In the horizontal direction, the original image is
filtered by the filters , , and , respec-
(15) tively. Three images and
are produced.
where is the threshold. Based on (8), (13), and (15), 2) In the vertical direction, the three images
the wavelet shrinkage, or called Lee shrinkage (LS), is derived, and are filtered by
2774 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

TABLE I
DENOISING RESULTS OF THE LENA IMAGE

the filters , and , respectively. This

gives nine images , .
3) Down-sampling the images , , with
an interval of three, we obtain nine subimages ,
.
4) Steps 1)–3) can be repeated on the subimage so
as to get the other subimages in the next scale.
In the above steps, by letting the wavelet coefficients be
shrunken with the algorithm given in (15), the 2-D DHWT-
based LS is developed. Similarly, the process for reconstructing
the original image from the subimages can be designed.
Soft thresholding (ST) [8] is one of the well known noise
smoothers in wavelet domain. Here, we first investigate the
effectiveness of the Spline biorthogonal (bior3.1) wavelet
[22]-based ST and the DHWT-based LS in image denoising.
The reason for selecting the bior3.1 wavelet is that it is a linear
phase filter and almost has the same filter support length as the
double Haar wavelets.
Example 1: In this example, a 512 512 Lena image is used.
First, three noisy images are set by adding the white Gaussian
noise with zero mean and variance 100, 400, and 900 to the
original Lena image, respectively. Then, applying the bior3.1-
based ST, and the DHWT-based LS to the noisy images, we
obtain the filtered images. By computing, the PSNR [23] values 2
Fig. 2. (a) Part of the original Lena image with 512 512 pixels. (b) The orig-
inal image corrupted by Gaussian white noise(PSNR = 22:14 dB). (c) The
are shown in the center two columns of Table I. Part of some of filtered image with the soft thresholding(PSNR = 25:31 dB). (d) The fil-
the images are shown in Fig. 2: (a) the original Lena image, (b) tered image with the LS (PSNR = 27:90 dB). (e)The filtered image with the
2
MWD-based LS of 3 3 window (PSNR = 28:88 dB). (f)The filtered image
the Lena image corrupted by Gaussian white noise with variance 2
with the MWD-based LS of 9 9 window (PSNR = 30:04 dB).
400, (c) the noisy image filtered with the ST, and (d) the noisy
image filtered with the LS, respectively. In this example, the
denoising is done only on the first scale. simplicity, the moving window-based denoising in wavelet do-
Unfortunately, the DHWT-based LS may produce “mosaic,” main is called MWD. Since a pixel of an image bears no relation
which can be observed from Fig. 2(d). The reason for this is that to the pixels far from it in distance, a small size of the moving
too much details of image are lost. To improve it, we may change window will be good enough for the MWD.
the value of threshold, or replace the double Haar wavelet with a To explore the different between the traditional denoising in
smoother wavelet. However, considering that a moving window- wavelet domain and the MWD, an example is given below.
based mean filter does not produce “mosaic,” we may embed the Example 2: Fig. 3(a) shows nine neighbor pixels of an image.
DHWT in a moving window so as to delete the “mosaic.” Based on the 2-D DHWT of the image, those pixels may use
For this purpose, a rectangular window should be the same wavelet coefficients to obtain their estimates. In the
defined first. Suppose that, at a given time, the center of this worst case of denoising, all of the wavelet coefficients are set to
window is at the pixel position of an image. By applying zero. Thus, by reconstruction, the estimates of the nine pixels are
the 2-D DHWT-based LS to the pixels within the window, we equal to the average of their grays, which just forms a “mosaic.”
obtain the estimate of pixel with the number of de- To describe the process of MWD, Fig. 3(b) shows that the
compositions. Then, we move the center of the window to the center of a 3 3 moving window is at the pixel of the
neighbor pixel so as to compute the next one estimate by a new image at a given time. By applying the 2-D DHWT-based LS
cycle of the denoising operation. This procedure continues until to the nine pixels within the , the estimate of the pixel is
all of the estimates of the pixels in the image are obtained. For obtained. Then, the center of the window is shifted by one pixel
WANG: MOVING WINDOW-BASED DOUBLE HAAR WAVELET TRANSFORM 2775

Fig. 3. (a) Part of an image; (b) 3 2 3 moving window centered at the pixel e; (c) by shifting one pixel, the 3 2 3 moving window is centered at the pixel h.

The output of the optimum Prewitt operator-based edge detector

(OPED) can be defined as

(20)
Fig. 4. Masks used by the Prewitt operator.
One of the advantages of the OPED is that the distances
between detected edges are enlarged. For example, suppose
to the neighbor pixel shown in Fig. 3(c). By applying the same that there are two edges with steps and ( ),
denoising operation to the pixels within the at the new posi- respectively. From (18)–(20), the estimates of the two edges
tion, the estimate of the pixel is subsequently obtained. In the are and . It means
worst case, the estimates of the pixels and are the average of that the distance between them increases from to
the grays of the nine pixels in their respective windows, which . This property cannot be achieved
just forms a 3 3 moving window-based mean filtering. with a linear noise smoother used for preprocessing.
The last two columns of Table I show the PSNR values ob- Based on the Prewitt masks shown in Fig. 4, the OPED can
tained by Applying the DHWT-based MWD to the three noisy be extended to 2-D easily. Suppose that, at a given time, the
Lena images, with the window length of three and nine, respec- pixels of an image in a 3 3 rectangular moving window are
tively. The corresponding filtered images, which were corrupted , . Let
by the Gaussian white noise with variance 400, are shown in
Fig. 2(e) and (f), respectively. Those images demonstrate that
the “mosaic” phenomenon is suppressed effectively.
Compared to the traditional denoising in wavelet transform (21)
domain, the MWD uses different wavelet coefficients for dif- (22)
ferent pixels to obtain the estimates. Thus, the estimate error of
the wavelet coefficients for a pixel does not influence the esti-
mates of the other pixels in the image.
(23)
VI. DHWT-BASED EDGE DETECTION
Thus, the , and represent a one-dimen-
Edges are important features for analyzing images. The clas- sional (1-D) signal in the moving window of the image. Using
sical edge detectors are based on some standard masks such as the 1-D edge detector defined by (20), we obtain an estimate
the Sobel operator [24], and the Prewitt operator [25]. Simple of edges in the horizontal direction of the image. Simi-
and effective, those two properties make them commonly used larly, we can obtain the estimate of edges in the vertical
today. In general, the Prewitt operator is better in suppress noise direction of the image. Finally, the output of the edge detector
for some three point mean filters were employed in its masks is defined as
shown in Fig. 4. Unfortunately, it is still sensitive to the present
of noise. Thus, a low-pass filter, such as the Gaussian filter, is
often used for preprocessing. (24)
Since the DHWT-based LS provides an noise smoother, we
may use it to improve the property of the Prewitt operator in The following example shows the difference between the Pre-
suppressing noise. The key for this is to relate the wavelet co- witt operator and the OPED in edge detection.
efficients of the DHWT to the Prewitt operator. Note that, in Example 3: Fig. 5(a) shows a Balloon image corrupted by
the DHWT, the equation is white Gaussian noise with mean zero and variance 100. Fig. 5(b)
just the -transform of the Prewitt operator in one dimension. and (c) are the edge images detected by the Prewitt operator
2776 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

with a median filter [1]. Thus, a median filter-based edge

enhancer (MFEE) is derived.
For the double Haar wavelets-based filter bank, suppose that
, , and are the samples of the input signal
at a given time. The output of the MFEE is written as

(25)

Apparently, by this replacement, the signal cannot be recon-

structed exactly at the output of the MFEE. Since the median
filter almost has the same output as the low-pass filter in the
smooth parts of signals, the reconstruction error is small except
for edges.
In some applications, we hope that the degree of edge en-
hancement can be adjusted. For this purpose, we replace the
output of the median filter with a new value defined by

(26)

where is a parameter to be adjusted. Being two special

cases, will be the output of the low-pass filter (
) or the median filter ( ).
Example 4: Fig. 7(a) shows an original “block” signal.
Fig. 5. (a) Noisy Balloon image. (b) Edge images detected by the Prewitt op-
erator. (c) Edge images detected by the OPED. (d) Edge detected by the Prewitt
Fig. 7(b) and (c) are the reconstructed block signals filtered
operator with a Gaussian filter as preprocessing. by the MFEE with 0.2 and 0.6 , respectively. It shows
how the degree of edge enhancement varies with the change of
parameter .
The approach of edge enhancement can be extended to 2-D
by the following procedure. First, a 3 3 cross moving window
shown in Fig. 6(b) is defined. Thus, the pixels of an image in this
filter window are sorted into two groups along the horizontal and
vertical directions, respectively.
Suppose that at a given time, the pixels in the two groups of
the moving window are , , , and
Fig. 6. (a) Typical high-pass mask used for edge enhancement; (b) 3 2 3 cross , , , respectively. Applying the
filter window.
approach of the 1-D edge enhancer to the two pixel groups, we
obtain the outputs
and the OPED, respectively. It is shown that the OPED is better
than the Prewitt operator in noise suppressing. Fig. 5(d) shows (27)
the edge image obtained by using the Prewitt operator with a (28)
7 7 Gaussian filter [26] as preprocessing for smoothing noise.
Compared with Fig. 6(c), the detected edges are significantly
Then, the output of the 2-D MFEE is defined as
blurred.

VII. DHWT-BASED EDGE ENHANCEMENT (29)

Edge enhancement is a useful tool for image processing,
especially for sharpening the image blurred by unknown degra- It is known that the unsharp masking-based edge enhancer
dation models. A simple image edge enhancer is called unsharp is sensitive to noise. Since the median filter is employed, the
masking [27] and accomplished by a high-pass filter whose 2-D FMEE will have a good property in suppressing noise while
mask is shown in Fig. 6(a). enhancing edges.
Certainly, the edge enhancement can be realized in wavelet Example 5: Fig. 8(a) shows an original fruits image. Fig. 8(b)
domain. From Fig. 1, we can see that there are two ways to en- is the blurred fruits image corrupted by Gaussian white noise
hance the edge of signals. One way is by enhancing the compo- with mean zero and variance 100. Fig. 8(c) and (d) is the en-
nents of edges in the details, but this work is often too complex hanced images filtered by the unsharp masking and the 2-D
to be done. The other way is by enhancing the edges in the ref- MFEE with , respectively. It is shown that the later is
erence. It can be finished easily by replacing the low-pass filter better in suppressing noise.
WANG: MOVING WINDOW-BASED DOUBLE HAAR WAVELET TRANSFORM 2777

Fig. 7. (a) Original block signal. (b) The edge enhanced block signal with  = 0:2. (c) The edge enhanced block signal with  = 0:6.

bank with its corresponding wavelet shrinkage is derived. Ex-

amples show that based on a moving window, they are flexible to
be used for image denoising, edge detection, and edge enhance-
ment. How to make better multichannel filter bank for image
processing is an interesting problem.

APPENDIX
SOLUTION OF THE INVERSE MATRIX
The matrix is defined in Section II for the Haar wavelet-
based -channel filter bank. In this appendix, we will derive
its inverse matrix as follows. Let

.. ..
. .

and

.. .. .. .. .. (A1)
Fig. 8. (a) Original fruits image. (b) The blurred image corrupted by Gaussian . . . . .
noise. (c) The images enhanced by unsharp masking. (d) The images enhanced
by the MFEE.

We have
VIII. CONCLUSION
In this paper, the concept of moving window-based local mul- (A2)
tiscale analysis of images is introduced. Consequently, based on
the Haar wavelet, a class of nonorthogonal multichannel filter (A3)
2778 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

Suppose that the inverse matrix has the form ACKNOWLEDGMENT

The author would like to thank Prof. T. Q. Nguyen and
the anonymous reviewers for their comments and suggestions
.. .. ..
. . . (A4) which greatly improved the quality of this paper.

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WANG: MOVING WINDOW-BASED DOUBLE HAAR WAVELET TRANSFORM 2779

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