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- Permanent Link:
- http://digital.auraria.edu/AA00003496/00001
## Material Information- Title:
- Image denoising and compression based on wavelets
- Creator:
- Ly, Dieu M
- Publication Date:
- 2010
- Language:
- English
- Physical Description:
- xii, 103 leaves : illustrations ; 28 cm
## Thesis/Dissertation Information- Degree:
- Master's ( Master of Science)
- Degree Grantor:
- University of Colorado Denver
- Degree Divisions:
- Department of Electrical Engineering, CU Denver
- Degree Disciplines:
- Electrical engineering
## Subjects- Subjects / Keywords:
- Electronic noise ( lcsh )
Image processing -- Digital techniques ( lcsh ) Wavelets (Mathematics) ( lcsh ) Electronic noise ( fast ) Image processing -- Digital techniques ( fast ) Wavelets (Mathematics) ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 100-103).
- General Note:
- Department of Electrical Engineering
- Statement of Responsibility:
- by Dieu M. Ly.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- |Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 656249389 ( OCLC )
ocn656249389 - Classification:
- LD1193.E54 2010m L9 ( lcc )
## Auraria Membership |

Full Text |

IMAGE DENOISING AND COMPRESSION BASED ON WAVELETS
Dieu M. Ly B.S., University of Colorado Denver, 2007 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2010 This thesis for the Master of Science degree by Dieu M. Ly has been approved by 4Va/A Date Ly, Dieu M. (M.S., Electrical Engineering) Image Denoising and Compression Based on Wavelets. Thesis directed by Professor Jan T. Bialasiewicz ABSTRACT In modem communication, the received signal or image through transmission is always corrupted with noise. Noise removal or denoising involves in manipulating the image data for better visual image qualities. For the past two decades, wavelets are well known for noise removal from the transmitted visual information in the form of digital image and still become the popular subject in other signal processing tasks such as image compression and enhancement. First, this thesis reviews deeply the literature survey of wavelet transform. Secondly, it introduces the basic concept of Gaussian noise, which is widely used in signal and image processing. The thesis then investigates the denoising techniques based on wavelet approach, which are VisuShrink, SureShrink, and BayesShrink. The processes of the study are: corrupt the image with Gaussian noise and then acquire the Discrete Wavelet Transform (DWT); perform hard or soft thresholding on the wavelet coefficients based on the noisy image; and then apply the Inverse Discrete Wavelet Transform (IDWT) to reconstruct the image. Lastly, the thesis focuses on an application of image compression, the Wavelet Transform Modulus Maxima (WTMM). When implemented, WTMM uses a trous algorithm to generate the approximation, horizontal and vertical detail coefficients of the test image. Instead of down-sampling the filtered image, the a trous lowpass and highpass filters are obtained by just inserting zero between each filter coefficient during the decomposition. This creates holes (A trous in French). The Conjugate Gradient Accelerated Algorithm is then used to reconstruct an image from its WTMM. The data collected from each denoising technique and compression provides for comparative study. The performance of each application is compared in terms of Root Mean-Square-Error (RMSE), Peak Signal-to-Noise Ratio (PSNR) values, compression rate, and visual image qualities. This abstract accurately presents the content of the candidates thesis. I recommend its publication. Signed Jan T. Bialasiewicz DEDICATION I dedicate this thesis to my parents, Phet, Saphine and my wife, Sanh for their constant supports, which give me an opportunity to pursue higher educations. My deepest gratitude goes to them for believing that I could make it to graduate school. Thanks to my wife, who has been patient and provides me the encouragement to complete this thesis. ACKNOWLEDGEMENT My thanks to Jan Bialasiewicz, thesis advisor, for his conctribution and guidance for my research. Through his interest in the subjects, I took classes such as digital signal processing, digital image processing which helped me understand wavelet applications within image processing. I also thank all the members of my committee for their participation. TABLE OF CONTENTS Figures....................................................................ix Tables....................................................................xii Chapter 1. Introduction............................................................1 1.1 Objective...............................................................1 1.2 Thesis Outlines.........................................................2 2. Wavelet Transform.......................................................4 2.1 Continuous Wavelet Transform (CWT)......................................4 2.2 Multiresolution Analysis (MRA)..........................................6 2.3 Discrete Wavelet Transform (DWT).......................................11 2.4 Mallats Decomposition and Reconstruction Algorithm....................16 2.5 Filter Banks...........................................................19 2.6 Summary................................................................23 3. Noise Model and Error Measurements.....................................24 3.1 Gaussian Noise.........................................................24 3.2 Error Measurements.....................................................25 3.3 Summary................................................................27 4. Wavelet Denoising Methods..............................................28 4.1 Thesholding............................................................29 vii 4.1.1 Hard Thresholding..................................................30 4.1.2 Soft Thresholding..................................................30 4.1.3 Universal Thresholding.............................................31 4.2 VisuShrink...........................................................32 4.3 SureShrink...........................................................34 4.4 BayesShrink.........................................................37 4.5 Summary.............................................................39 5. Wavelet Transform Modulus Maxima (WTMM)...............................41 5.1 The A Trous Algorithm................................................42 5.2 Reconstruction from Wavelet Transform Modulus Maxima (WTMM)..........49 5.3 Summary..............................................................53 6. Results and Conclusions...............................................54 6.1 Results..............................................................54 6.2 Conclusions..........................................................64 Appendix A. VisuShrink Matlab Program............................................65 B. SureShrink Matlab Program............................................70 C. BayesShrink Matlab Program...........................................78 D. Wavelet Transform Modulus Maxima Matlab Programs.....................86 References..............................................................100 viii LIST OF FIGURES Figure 2.1 The Morlet Wavelet Function...........................................5 2.2 The Relationship between Scaling and Wavelet Function Spaces..........7 2.3 Haar Scaling and Wavelet Function, (a) Haar scaling function, (b) Haar wavelet function............................................9 2.4 Daubechies Wavelet for 4 Vanishing Moments, (a) Daubechies scaling function, (b) Daubechies wavelet function....................11 2.5 The Analysis Filter Bank for Two-Dimensional Wavelet Transform.......15 2.6 One (left) and Two (right) Levels Two-Dimensional DWT................15 2.7 The Synthesis Filter Bank for Two-Dimensional Wavelet Transform......16 2.8 Three-Scale Forward Wavelet Transform Analysis Filter Bank...........19 2.9 Three-Scale Inverse Wavelet Transform Synthesis Filter Bank..........19 2.10 Two-Channel Filter Banks for One-Dimensional Subband Coding..........20 3.1 Graphical Plot of Gaussian Function..................................25 3.2 Coin Image with Gaussian Noise, (a) original image, (b) Gaussian noise image (with default values of the mean and noise variance of 0 and 0.01 respectively)...................................................25 4.1 Block Diagram of Wavelet Denoising...................................28 4.2 Thresholding Curves, (a) original thresholding, (b) hard thresholding, (c) soft thresholding...............................................31 4.3 Denoising Lena Image by VisuShrink with Hard and Soft Thresholding, using a = 20, (a) original Lena, (b) noisy Lena, (c) denoised Lena with hard thresholding (PSNR = 27.29 dB, RMSE = 11.02), (d) denoised Lena with soft thresholding (PSNR = 27.97 dB, RMSE = 14.39)........33 IX 4.4 Denoising Mandrill Image by SureShrink with Hard and Soft Thresholding, using a = 25, (a) original image, (b) noisy image, (c) denoised image with hard thresholding (PSNR = 22.74 dB, RMSE = 18.60), (d) denoised image with soft thresholding (PSNR = 23.56 dB, RMSE = 16.93).......................................36 4.5 Denoising Goldhill Image by BayesShrink with Hard and Soft Thresholding, using a = 10, (a) original image, (b) noisy image, (c) denoised image with hard thresholding (PSNR = 30.48 dB, RMSE = 7.63), (d) denoised image with soft thresholding (PSNR = 30.60 dB, RMSE = 7.53)........................................39 5.1 Levels 0,.. .,2 of an A Trous Decomposition...........................44 5.2 Quadratic Spline Wavelet (left) and Scaling Function (right)..........46 5.3 Decomposition Filter Bank for the Algorithm A Trous...................48 5.4 Reconstruction Filter Bank for the Algorithm A Trous..................48 5.5 3- Levels of Decomposition of WTMM Representation of Lena Image.......51 5.6 Wavelet Transform Modulus Maxima Representation of Image Lena for 3- Levels Decomposition, (a) wavelet transform modulus, (b) angle of wavelet transform modulus, (c) modulus maxima..................52 5.7 Image Lena using WTMM with a = 0.5, (a) original image, (b) reconstructed image, PSNR = 47.45 dB, compression rate = 59.03%..53 6.1 Denoised Lena Images by VisuShrink, SureShrink and BayesShrink, using a = 10, (a) hard VisuShrink (b) soft VisuShrink, (c) hard SureShrink, (d) soft SureShrink, (e) hard BayesShrink, (f) soft BayesShrink......56 6.2 Denoised Goldhill Images by VisuShrink, SureShrink and BayesShrink, using a = 20, (a) hard VisuShrink, (b) soft VisuShrink, (d) hard SureShrink, (c) soft SureShrink, (e) hard BayesShrink, (f) Soft BayesShrink......58 6.3 Denoised Mandrill Images by VisuShrink, SureShrink and BayesShrink x using a = 25, (a) hard VisuShrink, (b) soft VisuShrink, (c) hard SureShrink, (d) soft SureShrink, (e) hard BayesShrink, (f) soft BayesShrink........60 6.4 Reconstructed Head Image from WTMM with a = 2, (a) original image, (b) reconstructed image................................................62 6.5 Reconstructed Peppers Image from WTMM with c = 0.9, (a) original image, (b) reconstructed image............................62 6.5 Reconstructed Lena Image from WTMM with o = 10, (a) original image, (b) reconstructed image............................63 XI LIST OF TABLES Table 4.1 Comparisons of PSNR outputs of Lena image by VisuShrink with hard and soft thresholding at different a values.........................34 4.2 Comparisons of PSNR outputs of Mandrill image by SureShrink with hard and soft thresholding at different a values...............36 4.3 Comparisons of PSNR outputs of Goldhill image by BayesShrink using hard and soft thresholding at different o values....................38 5.1 Spline filter coefficients used in the algorithm a trous............47 6.1 Comparisons of PSNR and RMSE values of Lena image, using a = 10....55 6.2 Comparisons of PSNR and RMSE values of Goldhill image, using a = 20.........................................................57 6.3 Comparisons of PSNR and RMSE values of Mandrill image, using a = 25.........................................................59 6.4 Comparisons of PSNR, RMSE values, and CRate (Compression Rate) of Head, Peppers, and Lena images at different a values.............63 xii 1. Introduction Since the age of modem communication, variety of digital image has been generated and introduced to everyday life. They include natural image, geographical information, astronomy, digital commercial television, and magnetic resonance images, etc. When images are transmitted, they become a major information source. They are corrupted by different kinds of noise. Denoising them requires sophisticated algorithms in the image processing and computer vision. 1.1 Objective There are many approaches and techniques for denoising and compressing images. The objective of this thesis is to denoise and compress images using wavelets. The thesis assumes that images are corrupted by Gaussian noise. The denoising methods are based on the existing techniques developed by Dohono and Johnstone [10, 11] and Yang, Yu, and Vetterli [19, 20, 21, 22]. For compression, we use a technique suggested by Mallat and Zhong [29]. Suggested by Donaho and Johnston, VisuShrink is the technique applying both hard and soft thresholds utilizing the wavelet shrinkage with the universal threshold, a single threshold applied globally to all coefficients of all levels. They also proposed another method called SureShrink, which is based on the threshold derived from the minimization of the Steins Unbiased Risk Estimate (SURE) for one-dimensional signal. The choice of thresholding in this technique depends on the shrinkage functions and the multiresolution levels. For two-dimensions, the threshold can be achieved on the multiresolution level or subband. Yang, Yu, and Vetterli proposed BayesShrink, which is based on the mathematical Bayesian risk. BayesShink is a well known method for obtaining higher image quality. It is subband threshold dependent to be nearly optimal favoring soft 1 threshold over hard threshold. The Bayesian thresholding is done at each subband resolution in the wavelet decomposition. Lastly, the wavelet transform modulus maxima (WTMM), one of multiscale signal presentations, is used for image compression. Mallat and Zhong showed that wavelet transform is proportional to gradient smooth function when the detail coefficients are the derivative of the approximation coefficients. Thus multiscale edge detection is implemented by looking into the modulus of the wavelet detail coefficient to detect the local maxima. For this reason, WTMM is implemented by using an a trous algorithm instead of discrete wavelet transform. 1.2 Thesis Outlines This thesis is outlined as follows: Chapter 2 reviews in depth the literature survey of wavelet transforms. It introduces the continuous wavelet transform (CWT), multiresolution analysis (MRA), and the discrete wavelet transform (DWT) respectively. Section 2.4 of this chapter presents Mallats decomposition and reconstruction algorithm. Section 2.5 explains how two channel filter banks can be used to implement the forward discrete wavelet transform and the inverse discrete wavelet transform (IDWT). Chapter 3 introduces basic concepts of Gaussian noise and error measurements. The quantitative measurements of images are in terms of mean square error (MSE), root mean square error (RMSE), signal-to-noise ratio (SNR), and peak signal-to-noise ratio (PSNR). Chapter 4 presents the applications of the wavelet denoising. First, the hard, soft, and universal thresholds are introduced. Then, the applications of wavelet denoising VisuShrink, SureShrink, and BayesShrink are presented and implemented. Chapter 5 describes an application of wavelet transform modulus maxima (WTMM). Because WTMM is implemented by using a trous algorithm, the basic 2 concepts of a trous algorithm are presented as well. Lastly, this chapter reviews the conjugate gradient accelerated algorithm, which is used to reconstruct an image from its WTMM. Chapter 6 provides the results from the additional experiments and concludes the thesis. In this chapter, the collected data in terms of compression rate, RMSE, and PSNR values as well as image quality provides for comparative study. 3 2. Wavelet Transform Wavelets are mathematical functions, which decompose the signal into components of waves or frequencies and then analyze each component with a resolution to be matched to its scale. They provide solutions to important signal processing tasks such as noise removal, image enhancement, and compression. This chapter deeply reviews the principles of wavelet analysis, which includes continuous wavelet transform, discrete wavelet transform, and multiresolution analysis. Mallats decomposition and reconstruction algorithms and two-channel filter banks are presented in this chapter as well. Wavelet transform decomposes a function over dilated and translated wavelets of various frequency components. Suppose that xp(t) is a wavelet function G L2(R). It must satisfy zero average condition [1, 2, 3, 5, 6], The zero integral of ip is written as Co.'Pi^dt = (2.1) At a scale of a or dilation variable and translation of b, the function ipab (t) is defined by *0(0 = ^(v) (12) where is the normalizing factor. Figure 2.1 shows the graphical plot of Morlet wavelet function, xp{t) = e~f2/2cos (5t). 2.1 Continuous Wavelet Transform (CWT) Given a wavelet function ip satisfying equation (2.1), the continuous wavelet transform of a function / G L2(R) is Wf(ja, b) and given by Wf(.a.b) = jSSZ,mr(SL)dt (2.3) 4 where ip*is the complex conjugation and a, b are real signals. In short, the equation (2.3) is written as Wf{a, b) = ra,i,(t)dt (2.4) This equation shows that function f(t) is decomposed into a set of function defined in equation (2.2) that forms an orthogonal basis of L2(R). - ' 1 1 m i\ J - i \ i \ - A / 1 ^/ \ 1 A / \ / \ ^ 4 - - 11 - 1 w \j 1 T J I ,ti_________________________i________________________i_______________________i ________________________i________________________i________________________i________________________i________________________i -4 -3 2 l 0 1 2 3 4 Figure 2.1: The Morlet Wavelet Function. For any function / Â£ L2(M), the function /(?) can be obtained by taking the inverse continuous wavelet transform under admissibility condition (2.5) where c* = M is called the admissibility. The ip(co) is the Fourier transform of The above equations must satisfy zero average condition, which states i/>(0) = 0 and > 0 as a) -* oo plunging rapidly to make < oo. The equation (2.4) does the analysis decomposition) while equation (2.5) serves as the synthesis (reconstruction) of the signal. 5 2.2 Multiresolution Analysis (MRA) Multiresolution analysis is the important concept in wavelet analysis. It decomposes a signal into subsignals of different size resolution levels. Thus a signal can be implemented in multiple resolutions. It could be said that the signal representation of a wavelet function is coarse in overall approximations and detail coefficients, which have the effect on the function at various scales. The different signal scales represent the distinct resolutions in the same signal. For these reasons, multiresolution analysis is a better way to perform DWT as well as reconstruction of wavelet function. The multiresolution analysis is defined as a sequence {Vj: j E Z] of closed subspaces of L2(M) satisfying following conditions [1,8, 27]: i. The {Vj\ j E Z} in L2(M) is spanned in the subspaces that are nested in increasing scale. Vj c Vj+i V/ E TL ii. lim^.oo Vj = flJL-oo Vj = {0} and lim,-^ Vj = U=_oo Vj is dense in L2(M). As the resolution approaches zero, the approximation function contains less and less information until it converges to zero. On the contrary, as the resolution increases, the approximated signal converges to the original signal in L2(R). iii. f(x) EVj - /(2x) Â£ Vj+1, Vj E Z The space of approximated function can be written from one another by scaling each approximated function by rationing their resolution values. iv. There exists a unique function (p(x) E L2(R) such that for each j ElL the set { that {4)j,k(x) = 'f2Jip(2Jx k): j,k E l] is an orthomormal basis of L2(R). The scaling function
6 k(x)} is obtained by binary dilations and dyadic translations and
along the x-axis while scale j determines the width of k (x) how broad or narrow
functions is defined by scaling and translating the wavelet function i/>(x) in subpaces
basis in Vx
The equation (2.12) can be written as
filter coefficients h^ (n) are obtained as
Figure 2.3 (b) is the Haar wavelet function ip(x).
fix) = Zk cjo (k)(phik(x) + Zk Tf=jo dj(.k)xpjik(x) (2.28)
Ik lT=h dj(k)Vl2ip(Vx k) (2.29)
expands into a 2-D scaling functionary); however, the wavelet function xp(x)
xpv{x,y) = (p(x)xp(y) (2.37)
becomes
- '/2'Lmh(p(m 2k)(p(2j+1x m) (2.44) and wavelet function ip(x) = V2 'Znd^{n)(p{2x n). According to these equations, the iterative a trous decomposition has the following form c;(fc) = Sn hq, (r^Cj-^x + 2;_1n) (5.9) Figure 5.1 shows holes between samples. The sampling distance is increased by a factor of 2 for scale j 1. It also shows that the signal difference, (c; (/c) Cy+1(/c)} containing information between the two scales, is the discrete set associated with the scale function 43 -4-3-2-101234 Figure 5.1: Levels 0,...,2 of an A Trous Decomposition. The associated mother wavelet is therefore computed as ip(x) =
Once the set of approximations Cjo(k)..., Cj(k) have been found, the computation of
the wavelet coefficients is straight forward. Instead of applying the similar steps
described above, the wavelet coefficient can be found from the difference between
the adjacent approximation levels
djik) = cj.^k) cj(k) (5.11)
Note that for each decomposition level, its not necessary to subsample on the
convolution because the low pass and high pass wavelet filters are obtained by just
inserting 2J 1 zero between each filter coefficient, where j is the current
decomposition level. This creates holes (A trous in French). All filters are circularly
convolved instead to prevent the discontinuity near image boundaries. If n is the
number of decomposition levels, then a trous algorithm outputs n + 1 images of the
same size a single approximation image plus n detail images. Beside, a trous
algorithm is also translation-invariant; meaning a shift in the input simply shifts the
coefficients. Without applying any interpolation, the discrete transform values are
44
exactly identified at every pixel location. Furthermore, the correlation across values
can be achieved from its inherent structure [25], If ones would reconstruct a signal by
using a trous algorithm, the reconstruction has a form of
where n is decomposition level, dj(k) are detail coefficient matrices, and cn(k) is the
approximation.
When compressing an image using WTMM, there is always occurring the
changing in the edges. To prevent this changing, the mother wavelet should be
symmetrical in order the edge location matching the WTMM at a given scale. For this
reason, the quadratic spline filters are chosen. They are used to create holes instead of
Daubechies and other filters. These filters are non-orthogonal filters and have been
shown to be optimal filters for image processing. Mallat [1] suggested the quadratic
spline filters are designed as follows:
Suppose that 0 and 0 are the Fourier transforms of the scaling function (p and
wavelet functions 0 respectively. The 0 of a box spline with m degree, which is a
translation of m + 1 convolutions of 1[0 ^ with itself, is given by:
Cj0(k) = c^ + 'Z^djik)
(5.12)
This box spline is centered at t = i for m even and t = 0 for m odd. The low pass and
high pass filters in the neighborhood of co = 0 are designed as
The Fourier transform of the resulting wavelets is obtained by
(5.16)
45
with a box spline of degree m+2 at center of t = . Figure 5.2 shows graphical plot
of quadratic spline wavelet function on the left and scaling function on the right.
Figure 5.2: Quadratic Spline Wavelet (left) and Scaling Function (right)
If this spline supposes to be reconstructed under the perfect reconstruction, the dual
scaling function (p and wavelet function ip, as well as the associated low pass filter h,
and high pass filter g should be redefined as ip, xp, h, g respectively. For ip and xp to be
spline, we choose h = h. As the consequence under the condition, the cp is equal to ip.
Thus, the g is computed as
-iV2elT5in-jSn=o(c057) (5-17)
The quadratic spline filter coefficients are shown in Table 5.1. Here h(ri) and g(n)
are the lowpass and highpass analysis filters while h(n) and g(n) are the lowpass and
highpass synthesis filters respectively.
_ 2|/i(a))| _
46
n h(n) h(n) g(n) g(n)
-2 -0.0442
-1 0.1768 0.1768 -0.3094
0 0.5303 0.5303 -0.7071 -0.9723
1 0.5303 0.5303 0.7071 0.9723
2 0.1768 0.1768 0.3094
3 0.0442
Table 5.1: Spline filter coefficients used in the algorithm a trous.
Shensa [39] showed that Mallat algorithms are relatively inherent from fully
sampled a trous algorithm of same filter bank. As stated earlier in this section, the low
pass and high pass filters are obtained by inserting 2; 1 zero between each sample
of h(n). For any filter h(ri), we denote hj(n), the filter obtained by inserting 27 1
zero between each sample of h(n), whose Fourier transform is the equation (5.14),
then j is the scale parameter, (j E Z). The Figure 5.3 shows the decomposition filter
banks for the two dimensional a trous algorithm while Figure 5.4 is the reconstruction
[1]. In decomposition processes, the approximation is generated by first shifting the
columns of the input signal c;(n) to the left 27_1 times where j is the current level of
decomposition. The shifted columns are filtered by circular convolution using
lowpass filter hj (n) and the rows are shifted to the left 27-1 times and then filtered
with hj(n) to the approximation at level j. The horizontal details are calculated by
first filtering the columns of cy(n) using the high pass filter, Â§j (n) and then shifting
the columns, 27 times while the vertical details are first filtering the rows of c;(n)
using the high pass filter, Â£;(n) and then shifting the rows, 27 times.
To reconstruct the original image, reverse the order from the decomposition
procedures with the exception that the shifting is altered. At each level of
reconstruction, the filters are decimated by inserting 27 1 zeros between each
coefficient. Note that j decreases at each level of reconstruction. The outputs of the
47
horizontal detail coefficients and high pass filter, the vertical detail coefficients and
high pass filter, and the sum of the approximation and low pass filters, are added up.
The results are then multiplied by 1/4.
Columns
Rows
Cj(n)
Figure 5.3: Decomposition Filter Bank for the Algorithm A Trous.
Rows Columns
c/n)
Figure 5.4: Reconstruction Filter Bank for the Algorithm A Trous.
Let hj(n) = hj{ri) and gj(ri) = gj(n). The proposition [1] suggests that for any
; > 0, the cascading convolution formulas to compute the dyadic wavelet transform
are given as follows:
48
(5.18)
(5.19)
Cj+iin) = Cj(ri)*hjhj(n)
dJ+1(ji) = Cj(n)*gjS(n) and df+i(n) = C;(n)*5Â£,(n)
and the inversion is obtained by
Cjin) = cj+1(nyhjhj(n) + d]+1(n)* g}8(n) + df+1{nY 8 g j(ri)) (5.20)
where hj(n) is low pass filter, and gj (n) is high pass filter. The hj(n) and Â§j (n) are
time reversed version of hj(n) and cjj (n) respectively. Note that 5(n) is discrete
Dirac to separate filters such as xy(n, m) = x(n)y(m).
5.2 Reconstruction from Wavelet Transform Modulus Maxima (WTMM)
For the 2-D WTMM of an image / to be recovered, the set of wavelets {ipj,n(x>y)}
that corresponds to the maxima locations forms a frame of subspace L2(T) if there
exists two constant A,B > 0,A < B < oo such that
A\\f\\2 |