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N E W W A V E L E T T R A N S F O R M S A N D T H E IR A P P L IC A T IO N S T O D A T A C O M P R E S S IO N
by
IN D E R P R E E T SINGH
M.Tech., Indian In stitu te of Technology, Delhi, INDIA, 1993 B.E.(Hons.), Panjab University, INDIA, 1992
A D issertation S ubm itted in P artial Fulfillment of th e Requirem ents for the Degree of
Do c t o r o f Ph il o s o p h y
in th e D epartm ent o f Electrical an d Com puter Engineering
We accept this dissertation as conforming to the required stan d ard
Dr. P. i^ ^ th o k lis, Supervisor, D ept, of Elect. & Comp. Eng. ______________________________________________ Dr. A. Antoniou, Supervisor, D ept, of Elect. & Comp. Eng.
r. W -S. Lu Ml
Dr. W-S. Lu, Member, D ept, of Elect. & Comp. Eng.
____________________________________ Dr. R. Illner, O utside M ember, D ept, of M athem atics and Statistics
Dr. F. K ossentini, E xternal Exam iner, D ept, of Elect. & Comp. Eng., UBC
© IN D E R P R E E T SINGH, 2000 University o f Victoria
A ll rights reserved. This dissertation m ay not be reproduced in whole or in part by photocopy or other means, without the permission o f the author.
11
S u p e r v is o r s : Dr. P. A gathoklis and Dr. A. Antoniou
A B S T R A C T
W ith the evolution of m ultim edia systems, image and video com pression is becom ing the key enabling technology for delivering various im age/video services over heterogeneous net works. T h e basic goal o f image d a ta com pression is to reduce the bit ra te for transm ission and storage while eith er m aintaining the original quality of th e d ata or providing an ac ceptable quality.
This thesis proposes a new wavelet transform for lossless compression of images w ith application to medical images. T h e transform uses integer arithm etic a n d is very com pu tationally efficient. T h e n a new color image transform ation, which is reversible and uses integer arith m etic, is proposed. T h e transform ation reduces th e redundancy am ong the red, green, an d blue color bands. I t approxim ates th e lum inance an d chrom inance com ponents of the Y IQ coordinate system . T his transform ation involves no floating-point/integer mul tiplications o r divisions, a n d is, therefore, very suitable for real-tim e applications where the num ber o f C P U cycles needs to be kept to a m inimum .
A technique for lossy com pression of an im age d ata base is also proposed. T he technique uses a wavelet transform a n d vector quantization for compression. The discrete cosine tra n s form is applied to th e coarsest scale wavelet coefficients to achieve even higher compression ratios w ith o u t any significant increase in com putational complexity. W avelet denoising is used to reduce th e im age artifacts generated by quantizing th e discrete cosine transform coefficients. T h is improves th e subjective q u ality of the decom pressed images for very low bit ra te im ages (less th a n 0.5 bits per pixel).
The thesis also deals w ith the real-tim e im plem entation of th e wavelet transform . T he new wavelet transform has been applied to speech signals. B oth lossless and lossy tech niques for speech coding have been im plem ented. T h e lossless technique involves using the reversible integer-arithm etic wavelet transform and H uffm an coding to o b tain the com pressed b itstre a m . T he lossy technique, on th e other hand, quantizes th e wavelet coefficients to obtain h ig h er com pression ratio a t the expense of some d egradation in sound quality. T he issues re la ted to real-tim e wavelet com pression are also discussed. Due to th e lim ited size of m em ory o n a DSP, a wavelet transform h a d to be applied to an in p u t signal o f finite length. T h e effects of varying th e signal length o n compression perform ance are also studied for different reversible wavelet transform s. T h e lim itations of th e proposed techniques are discussed a n d recom m endations for future research are provided.
U 1
E x a m in e r s:
Dr. P. i^ ^ th o k lis, Supervisor, D ept, of Elect. & C om p. Eng.
Dr. A. Antoniou, Supervisor. D ept, of Elect. & Com p. Eng. ____________________________________ r. W-S. Lu, k
Dr. W-S. Lu, M ember, Dept, o f Elect. & Comp. Eng. Dr. W-S. Lu, Member.
Dr. R. Illner, O utside M ember, Dept, o f M athem atics and Statistics
I V
Table of Contents
A b s tr a c t ii T a b le o f C o n te n ts iv L ist o f F ig u r es v iii L ist o f T ables x L ist o f A b b r e v ia tio n s x i N o t a t io n x iii A c k n o w le d g e m e n t x v D e d ic a tio n x v i 1 I n tr o d u c tio n 1 1.1 H istorical overview ... 1 1.1.1 Need for c o m p re s s io n ... 2 1.2 C ontributions of the t h e s i s ... 3 1.3 Thesis o r g a n iz a t i o n ... 4 2 O v e rv iew o f W a v e le t-B a se d C o m p r e ssio n M e th o d s 6 2.1 I n t r o d u c t i o n ... 62.2 The transform ation s t a g e ... 7
2.2.1 S p atial decorrelating tr a n s f o r m s ... 8
2.2.2 Spectral-decorrelating t r a n s f o r m s ... 9
2.2.3 H ybrid transform s: W a v e le ts... 11
2.2.4 D efinition of m ultiresolution a n a l y s i s ... 11
2.2.4.I O rthonorm al b a s i s ... 11
Table of Contenta v
2.2.4 3 C alcu latio n o f one dim ensional wavelets coefficients . . . . 16
2.2.4.4 Signal decom position a n d reconstruction by using wavelets 17 2.2.4.5 E x ten sio n of wavelet basis to two d i m e n s i o n s ... 20
2.2.5 M u ltirate filter b a n k s ... 22
2.2.6 C orrelation betw een m ultiresolution analysis a n d filter banks . . . . 30
2.2.7 B iorthogonal w a v e le ts ... 31 2.2.8 Wavelets in im age c o d in g ... 32 2.2.8.1 R eversible w a v e le ts ... 33 2.3 Q u an tizatio n s t a g e ... 36 2.3.1 Scalar q u a n t i z a t i o n ... 36 2.3.2 Vector q u a n t i z a t i o n ... 37 2.4 C oding s t a g e ... 38 2.4.1 S ta tic coding t e c h n i q u e s ... 38 2.4.2 A daptive te c h n iq u e s ... 39 2.4.3 Em bedded c o d e r s ... 39 2.5 C o n clu sio n s... 41 L o ssless D a t a C o m p r e ss io n U s in g an I n t e g e r -A r it h m e t ic W a v e let T ra n s form 42 3.1 I n t r o d u c t i o n ... 42 3.1.1 P re lim in a rie s ... 42
3.2 P roposed wavelet tran sfo rm using in te g e r-a rith m e tic ... 45
3.2.1 Reversible tw o-ten transform in liftin g s c h e m e ... 46
3.2.2 C oding o f wavelet coefficients ... 47
3.2.3 A lgorithm fo r th e c o d e r ... 47 3.2.4 A lgorithm for th e d e c o d e r ... 48 3.3 C om pression of m edical im a g e s ... 48 3.3.1 E xisting m e t h o d s ... 48 3.3.1.1 Lossless J P E G predictive c o d in g ... 48 3.3.1.2 Im proved predictive co d in g ... 49 3.3.1.3 Tt-ansform-based m e t h o d s ... 49 3.3.2 R esults a n d c o m p a riso n ... 50
3.3.2.1 R easo n for th e b etter perform ance o f th e nonlinear T T filter 53 3.4 A n in teg er-arith m etic color-coordinate tran sfo rm atio n for color im age com pression ... 59
Table of Contents vi
3.5 Com pression o f color i m a g e s ... 60
3.6 C onclusions... 63
A M ix e d T r a n s f o r m T e c h n iq u e fo r L o s s y I m a g e C o m p r e s s io n 65 4.1 I n tr o d u c t i o n ... 65
4.2 Overview of vector q u a n t i z a t i o n ... 66
4.2.1 I n tr o d u c ti o n ... 66
4.2.2 Definition of vector q u a n tiz a tio n ... 67
4.2.3 O p tim al vector q u a n t i z a t i o n ... 68
4.2.4 T h e LBG a lg o r ith m ... 69
4.2.5 C odebook in itia liz a tio n ... 70
4.2.6 Com pression o f images using vector q u a n tiz a tio n ... 71
4.3 A new m ixed-transform technique for low b it-rate image c o d i n g ... 72
4.4 Proposed M X D T c o d e r ... 72
4.4.1 Selection o f codebook size and w i d t h ... 75
4.4.2 S tatistical analysis o f th e wavelet c o e ffic ie n ts... 75
4.4.3 E rro r c o r r e c t i o n ... 76
4.5 D e c o d e r s ... 77
4.6 R esults and d iscu ssio n ... 79
4.7 Com pression of color i m a g e s ... 87
4.7.1 Proposed color im age com pression te c h n i q u e ... 88
4.7.2 R esults and discussion ... 88
4.7.3 C o n c l u s io n s ... 91
O n t h e D S P I m p l e m e n t a t i o n o f W a v e le t l ï a n s f o r m fo r R e a l- tim e S p e e c h C o m p r e s s io n 93 5.1 I n t r o d u c t i o n ... 93
5.2 Block wavelet tra n s fo rm ... 95
5.2.1 E ntropy c o d i n g ... 96
5.3 Im plem entation of reversible integer a rith m e tic wavelet t r a n s f o r m ... 99
5.4 Im plem entation issues for th e TM S320C30 D S P ... 102
5.4.1 H ardw are d e s c r ip t i o n ... 102
5.4.2 Im plem enting th e block wavelet t r a n s f o r m ... 103
TM e of Contents v ii
5.4.4 Real-tim e Im plem entation ... 104
5.5 Results and discussion... 105
5.6 C onclusions... 108
6 C o n c lu sio n s a n d S co p e for F u tu r e W o rk 109 6.1 O v e rv ie w ... 109
6.2 Reversible wavelets for lossless image c o m p re s s io n ... 109
6.3 Lossy image c o m p re s s io n ... 110
6.4 Real-tim e compression of s p e e c h ... 110 6.5 Future research ... I l l
B ib lio g r a p h y 113
V III
L ist o f F ig u res
Figure 2.1 G eneral stru c tu re o f transform -based image com pression system . . . 7
Figure 2.2 (a) Uniform b a n d w id th filter banks, (b) Wavelet filter banks... 10
Figure 2.3 An M-fold dow nsam pler... 23
Figure 2.4 Effects of two-fold dow nsam pling in th e frequency dom ain (no alias ing). (a) Spectrum of th e in p u t signal, (b) T h e shifted versions of the origimd in p u t spectrum used to form th e spectrum o f th e dow nsam pled signal. (c) Spectrum of th e dow nsam pled signal... 25
Figure 2.5 Effects of two-fold dow nsam pling in th e frequency dom ain (w ith alias ing). (a) Spectrum o f th e in p u t signal, (b) T h e shifted versions of the original in p u t spectrum used to form th e spectrum o f th e dow nsam pled signal. (c) Spectrum o f th e dow nsam pled signal... 26
Figure 2.6 A n M-fold u p s a m p le r... 27
Figure 2.7 T he first noble i d e n t i t y . ... 27
Figure 2.8 T he second noble i d e n ti ty ... 27
Figure 2.9 A n M -channel analysis filter bank... 28
Figure 2.10 A n M -channel synthesis filter b an k ... 29
Figure 2.11 An 2-channel J -s ta g e wavelet decom position... 29
Figure 2.12 A two-channel m axim ally decim ated biorthogonal filter bank . . . . 32
Figure 2.13 Lifting stage: S plit, pred ict, u p d a te ... 35
Figure 3.1 Block diagram of analysis and synthesis wavelet filters... 43
Figure 3.2 Lifting stages for th e T T transform (Q denotes q u an tizatio n )... 46
Figure 3.3 O btaining th e original signal from th e T T wavelet coefficients using lifting (Q denotes q u a n tiz atio n )... 47
Figure 3.4 T he sam ple p red ictio n neighborhood... 49
Figure 3.5 Tw o-dim ensional block D O T for lossless com pression... 50
Figure 3.6 Various m edical te s t im ages... 54
List of Figures ix
Figure 3.8 C om parison o f com pression perform ance (in bits p er pixel) using dif
ferent algorithm s for various USC a n d M RI im ages... 56
Figure 3.9 Dual-wavelet function for th e 2/14 filter... 58
Figure 3.10 Dual-wavelet function for th e T S filter... 58
Figure 3.11 Dual-wavelet function for th e T T filter... 59
Figure 3.12 C om parison between lum inance com ponents of th e existing Y I Q and the proposed Y ' f Q m odels... 61
Figure 4.1 A basic mem oryless vector q u a n t i z e r ... 68
Figure 4.2 Block d iag ram of the proposed M XDT c o d er... 73
Figure 4.3 T he E rro r C orrection m e t h o d ... 77
Figure 4.4 W avelet-based soft thresholding... 79
Figure 4.5 (a) O riginal FA C E l Image, (b) O riginal FACE4 image... 82
Figure 4.6 (a) Decompressed image using th e J P E G decoder, (b) Decompressed image using the S P IH T decoder, (c) D ecom pressed im age using th e M XDT decoder, (d) Decompressed image using the M X D T l decoder... 83
Figure 4.7 (a) Decompressed image using th e J P E G decoder, (b) Decompressed image using the S P IH T decoder, (c) D ecom pressed image using th e MXDT decoder, (d) Decompressed image using the M X D T l decoder... 84
Figure 4.8 (a) Zoomed FA C E l image, (b) Zoomed decom pressed image using the S P IH T decoder, (c) 55oomed decom pressed im age using th e M X D T l decoder. 85 Figure 4.9 (a) Zoomed FACE4 image, (b) 2k>omed decom pressed image using the S P IH T decoder, (c) Zoomed decom pressed im age using th e M X D T l decoder. 86 Figure 4.10 Block d iag ram o f the proposed w avelet-D C T m ixed transform coder. 87 Figure 4.11 O riginal I m a g e ... 90
Figure 4.12 JP E G com pression (Com pression R atio = 67:1, PSN R — 27.6 dB). 90 Figure 4.13 C om pression using the proposed technique (Com pression R atio = 67:1, P S N R = 30.7 d B )... 91
Figure 5.1 (a) R elation between C P U tim e and d a ta len g th for D W T based on T T transform w ith different softw are im plem entations(b) R elation between C PU tim e and d a ta length for ID W T based o n T T transform w ith different softw are im plem entations... 101
Figure 5.2 Block d iag ram o f the real-tim e im plem entation of a wavelet-based speech coder on th e TMS320C30 D S P ... 105
L ist o f T ables
Table 3.1 Lossless compression ratios of USC i m a g e s ... 51 Table 3.2 Compression ratios for medical test images after pre-processing . . . 52 Table 3.3 Entropies of test images for th e H aar, TS an d T T filter-based approaches 53 Table 3.4 C P U tim e (in seconds) for test images using H aar and TS approaches 53 Table 3.5 C P U tim e (in seconds) for test images using th e T T approach . . . . 57 Table 3.6 F ilte r coefficients for various lin ear biorthogonal f i l t e r s ... 57 Table 3.7 C om parison betw een V (lum inance in V I Q m apping) and V ' (lumi
nance in Y ' Ï Q' m a p p in g ) 62
Table 3.8 C om parison of compression perform ance (bpp) am ong different lossless com pression tech n iq u es... 63 Table 3.9 T otal coding tim es o f color images (V ' Ï Q' ) for different coding techniques 64 Table 4.1 An example showing range correction of th e D C T coefficients . . . . 74 Table 4.2 C odebook sizes an d widths for a typical te st i m a g e ... 75 Table 4.3 C om parison of compression perform ance of various schem es... 81 Table 4.4 Subjective evaluation of various s c h e m e s ... 82 Table 4.5 C odebook sizes a n d widths for th e Y (lum inance com ponent) o f a
typical landscape i m a g e ... 89 Table 4.6 C om parison in compression perform ance betw een th e JP E G stan d a rd
and th e m ixed transform te c h n iq u e ... 89 Table 5.1 C P U tim es (in seconds) using different software r e a l i z a t i o n s ... 99 Table 5.2 Q uantization levels for different s u b b a n d s ... 102 Table 5.3 Com pressed file size for the proposed m ethod an d the coder proposed
in [107] 106
Table 5.4 Com pressed file size for various vocoders. T h e com putation tim es for the proposed technique are given in th e p a re n th e s is ... 107
X I
L ist o f A b b rev ia tio n s
ID O ne dim ensional
2D Two dim ensional
bpp B its p e r pixel
BRD Delayed branch instruction
B W T Block wavelet transform
HVS H um an visual system
C PU C entral processing u n it
C R Com pression ratio
C R E W C om pression w ith reversible em bedded wavelets
C T C om puterized tom ographic (images)
D C T D iscrete cosine transform
D F T D iscrete Fourier transform
DM D aughter m odule
DMA D ual-m em ory access
DP D ata page pointer
D PR A M Dynam ic program m able random -access mem ory
DSP D igital signal processing (processor)
EZW E m bedded zerotree wavelet (coding)
JB IG Jo in t Bilevel Image G roup
JP E G Jo in t Photographic E x p erts Group
KLT K arhunen-Loeve transform
LBG algorithm Linde Buzo G ray algorithm
M RI M agnetic resonant imaging
M SE M ean squared error
N IN T N earest integer (less tham or equal to) P S N R Peak signal-to-noise ratio
Q M F Q u a d ratu re m irror-im age filter
RAM Random -access memory
List of Abbreviations x ii
ROM R ead-only meomory
R P T B R ep eat block in stru ctio n R P T S R epeat sigle in stru ctio n
S + P Said 4- Pearlm an
S P IH T Set p artitio n in g in hierarchical trees
SRAM S tatic random-access m em ory
S T F T Short-tim e Fourier transform T S transform Two-six transform
T T transform T w o-ten transform
USC U niversity of Southern C alifornia
VoIP Voice over Internet protocol
VQ Vector quantization
X U l
Notation
• T he symbols Z , 72., and C denote the sets o f integers, real numbers, and complex num bers, respectively.
• is a real inner product space of m easurable functions such th a t f I f i ^ W d x < + 0 0
J — OC
• T he classical norm of f { x ) € is given by i i / i i c = r \ n x ) \ d x
J — OC • norm is given by
l l / f = f l /W I ^ d x
J — O C
• £ ^ (R ” ) is an inner product space of m easurable square integrable n-dim ensional func tions.
• T he classical norm of f { x i , X2, - - -, x„) G £ ^ (R " ) is given by r o c roo ro o
— I j . . . / | y(xi, X2, Xfj)! d x i d x 2 . . . d x j i J — 0 0 J — O C J — O C
• T he series Vq, Vi, V2, . . . refers to the sequence of subspaces in £^(R ).
• T he symbols ‘©’ a n d ‘®’ refer to th e tensor su m and tensor p ro d u ct of two subspaces, respectively.
• T he sym bol j denotes the q u an tity >/—!.
• Functions of a continuous variable are in d icated w ith round parenthesis, for exam ple, f { t ) w here t € R.
• Functions of a discrete variable are indicated w ith square parenthesis, for exam ple, ar[n] where n € Z .
• Bold face symbols are usually used to represent vector o r m atrix variables. • T he q u an tity denotes th e transpose o f A .
Notation x iv
e T he z -transform o f a discrete sequence x\n \ is denoted as X ( z ) an d is defined as
n=tx3
Zx[n] = X { z ) = ^ 2 " (0.1)
n=—oc
• The inner p ro d u ct of two functions in the continuous variable case is defined as
/
+ 0 C f { t ) g - i t ) d t (0.2)-O C
and in th e discrete variable case as
< f[n],g[n] > = Ÿ 1 (0.3)
n=—oo
• T h e d elta function <5[n] is defined as
= ( = ® (0.4)
I 0 otherwise
e T he n o tatio n f * g stan d s for convolution of / an d g
• T he notatio n [xj denotes th e greatest integer not m ore th a n x ( or equivalently x is rounded to th e nearest integer towards —oo).
• F ilter coefficients are assum ed to be real unless sta te d explicitly.
• T he sam pling period of d iscrete sequences is assum ed to be one unless s ta te d otherwise. • A filter H has a transfer function H{z) . T h a t is, ro m an font is used to name a
p articu lar filter while italics are used to indicate th e tran sfer function associated w ith it.
X V
A c k n o w le d g e m e n t
I would like to express m y deepest g ratitu d e to m y supervisors, D r. P an ajo tis A gathoklis and D r. A ndreas A ntoniou, for th eir valuable guidance and com m ents thro u g h o u t m y grad u ate study. I would also like to th a n k D r. R einhard Illner for all the valuable discussions we had on the m ath em atical th eo ry of wavelets. His grad u ate course on wavelets was really helpful.
I would like to th a n k m y wife, M eher, and m y brother, Jasjeet, for th e ir em otional su p p o rt and encouraging me to com plete this work.
In ad d itio n I w ould like to m ention some im p o rtan t friends who becam e a n integral p a rt o f my stu d e n t life. T h ey include A vijit B hunia, Sandeep Agarwal, R a n g a n a th a n Gu- ru n a th a n a n d S u b ram an ian M uthu, to nam e a few.
Finally, I would like to th a n k my m om and m y d ad for all th e sacrifices th e y m ade for me an d for giving me th e will to continue.
X V I
D e d ic a tio n
Chapter 1
Introduction
T raditional image compression techniques have been designed to exploit the statistical re dundancy present w ithin real world images. The discrete cosine transform is one example of this statistical approach. Rem oving redundancy can only give a lim ited am ount of com pression; to achieve high memory savings, some of th e non-redundant inform ation must be removed. By using m ethods th a t closely mimic th e human visual system (HVS), compres sion can take into account the im portance o f each individual coefficient and code accordingly. Psychophysicists an d visual psychologists have discovered th a t the eye filters th e image into a num ber of bands, eacli approxim ately one octave wide in frequency. Further, in the spa tial dom ain, th e image should be considered to be composed of inform ation a t a num ber of different scales. Wavelets decompose th e image into multiple bands a t octave frequencies sim ilar to th e m ultiple channel m odels of the HVS. New compression techniques based on the wavelet transform are getting extrem ely popular these days. Wavelet technology can help a tta in higher transm ission speeds an d clearer pictures th an th e conventional statistical m ethods by providing higher com pression ratios an d reduced com putational complexity.
1.1
H is to r ic a l o v e r v ie w
Wavelets, filter banks, and m ulti-resolution signal analysis, which have been used inde pendently in th e field of applied m athem atics, signal processing, an d com puter vision, re spectively, have recently converged to form a single theory. T he first wavelet system was constructed by H a ar [1] in 1910. T h e H aar wavelet system , as it is now known, uses piece- wise co n stan t functions as its basis to decompose th e in p u t signal. A lthough wavelet theory has m any close ties to science a n d engineering since th e early 1900s, these linkages were not discovered until th e early 1980s. U ntil th a t time wavelet theory was a disjoint set of ideas th a t lacked a unified framework.
1. Introduction 2
m id 1980s. In 1984, the term ‘wavelet’ was introduced by G rossm an and M orlet [2]. In 1988, a trem endous breakthrough in wavelets was brought a b o u t by Daubechies [3]. In this classic paper, Daubechies introduced a family o f com pactly su p p o rte d wavelet system s th a t satisfy the property o f orthogonality a n d perfect reconstruction. In 1989, M allat [4] presented the theory of m ultiresolution analysis th a t la te r becam e p o p u la r as th e M allat algorithm . T his algorithm provided an easy insight to engineers and physicists in applying wavelet concepts to diverse fields such as signal processing etc. In 1992, C ohen, Daubechies a n d Feauveau [5] established th e theory of biorthogonal wavelet filters com m only known as th e CDF filters. Unlike orthogonal wavelet filters (except for th e triv ial case o f th e H aar a n d other Haar- like transform s), biorthogonal filters allow for sym m etric finitely-supported basis functions. T he sym m etry property can offer significant benefits in m any applications, image d a ta compression being one o f th em . Wavelet transform s have proven to be extrem ely useful for image coding as many researchers have shown (e.g., [6]-[23]). C onsequently such transform s have been used extensively in m any image d a ta com pression algorithm s. In 1993, Shapiro [9] introduced th e concept of em bedded coding. His coding scheme, called embedded zerotree wavelet (EZW ) coding, was based on wavelet transform s. D ue to the obvious advantages of the em bedded property in m any applications, em bedded coding quickly grew in popularity especially as a means o f building unified lossless/lossy com pression systems. In 1995, Zandi et al. [10] proposed compression with reversible embedded wavelets (C R E W ), a reversible em bedded im age com pression system based on some ideas of Shapiro. N ot long after, in 1996, Said a n d Pearlm an [11] introduced a new coding schem e known as se t partitioning in hierarchical trees (SPIH T ), w hich is conceptually sim ilar to EZ W w ith some im plem entation improvements.
A part from th e em bedded coding techniques which involve using highly complex coders and decoders, th ere are o th e r wavelet-based techniques th a t becam e p o p u lar in certain areas of image com pression. O ne o f them is know n as vector qu an tization (V Q ). A lthough the concept of vector q u an tizatio n has been aro u n d for decades [12], applying VQ to a wavelet transform was introduced by B arlaud e t al. in 1989 [13]. M odifications to th e VQ approach were later rep o rted in various other publication [14], [15].
1.1.1
Need for compression
T he need o f com pression em erges from th e fact th a t an y real d a ta gathered through a d a ta acquisition system always h as some degree of correlation. T h e degree of correlation depends on th e type of d a ta an d th e am ount of noise in th e d a ta . C om pression is widely used in
1. Introduction 3
m any areas, e.g., im age processing, speech coding, m edicine and th e In tern et, to nam e a few. Even though d a ta mem ories are g ettin g cheaper a n d faster, th ere will be cases w here com pression of d a ta is necessary. For example:
1. To store a m oderately large image, say a 512x512 pixels, 24-bit color image, requires a b o u t 0.75 M B ytes. A video signal typically has 30 fram es per second.
2. A stan d ard 35 m m p h o to g rap h digitized a t 12 pm resolution requires a b o u t 18 M Bytes. 3. One second of N T S C color video entails 23 MBytes.
T his shows th a t one c an easily find situ atio n s where th e current hardw are is inadequate, e ith er technically or economically. Com pression techniques can reduce th is gap by
• saving storage,
• saving CPU tim e, or by • saving transm ission tim e.
T h e requirem ent for com pression has become even more critical w ith th e boom in In tern et. Various m ultim edia protocols such as H.323 and H.324 have built-in com pression of aud io and video signals to m inim ize th e b an d w id th per channel. Some of th e In te rn e t applications such as voice over In te rn e t protocol (VoIP) use state o f th e a rt audio coders such as G.728 and G.729A [16] [17].
T h e m ajo r requirem ent of com pression is th a t one should be able to quickly sw itch betw een th e original a n d th e com pressed d a ta .
1.2
C o n tr ib u tio n s o f t h e th e s is
T his thesis studies reversible wavelet transform s and th e ir application to lossless com pres sion o f speech and im age signals. T h e ideas are drawn from all of th e previously described developm ents, b u t o f m ost d irect im p o rtan ce are the reversible wavelet transform s proposed by Boliek et al. [10] a n d th e S4-P transform proposed by Said and P earlm an [11]. M any o f the ideas on reversible wavelets are closely related to ideas presented in these two p ap ers. A p a rt from lossless com pression, wavelets have also been used for lossy com pression of im ages. M any existing w avelet-based lossy compression techniques a re based on em bedded coding techniques [9], [11]. In th is thesis, focus is directed towards u sin g vector quantiza tion [12] a n d wavelets for lossy com pression. Some of th e topics addressed in detail are as follows:
1. Introduction 4
— a new reversible transform and com parison w ith existing transform s w ith foctis on medical image compression
— extension of th e reversible transform ations to th e class of color images
— practical issues associated w ith th e im plem entation of transform s such as com pu tatio n al com plexity
• Lossy Compression
— a mixed transform technique based on the discrete wavelet transform and the discrete cosine transform w ith vector quantization technique
— m ethods of m inim izing quantization artifacts using post processing • P rac tic al Im plem entation
— practical issues related to real-tim e im plem entation of reversible transform tech niques
— effects of block wavelet transform o n compression perform ance and com puta tional complexity
1.3
T h e s is o r g a n iz a tio n
The thesis is divided into six chapters. T h e first two chapters provide introductory and background m aterial necessary to und erstan d th e rest of the thesis. T he rem aining chapters present a com bination of fundam entals an d research results in detail.
In C h a p te r 2, concepts underlying m u ltirate filter banks and wavelet transform s are outlined. T h e chapter begins by presenting th e fundam entals of m u ltirate filter banks and wavelet-based decom position. A special class o f wavelets called reversible wavelets [10] is described. T h is is followed by m ethods of quantizing the wavelet coefficients. Two different qu an tizatio n techniques, namely, scalar qu an tizatio n and vector quantization, are discussed. Finally, different types o f encoders are discussed. The advantages o f wavelet coders over other en tro p y coders are also discussed.
In C h a p te r 3, a new reversible transform is proposed. T h e transform is used for the com pression o f m edical images. T h e perform ance achieved is com pared w ith th a t achieved w ith other existing lossless com pression techniques. T h e transform is then extended to compress color im ages. A new reversible color image transform ation to remove sp ectral redundancy am ong th e color bands is also presented. R esults obtained w ith th e new transform are com pared w ith results obtain ed w ith some o f th e existing reversible transform s.
1. Introduction 5
uses wavelets and vector quantization for th e compression of th e wavelet coefficients an d D C T w ith scalar q u an tizatio n for encoding the scaling coefficients. T he results are com pared w ith results obtained w ith s ta te o f th e a rt wavelet coders such as th e EZ W [9] and S P IH T [11] coders. Subjective results are also given.
In C h ap ter 5, th e proposed reversible wavelet transform is used for th e com pression of speech signals. R eal-tim e im plem entation was done on the TM S320C30 DSP. Issues con cerning th e real-tim e wavelet transform (called block wavelet transform) are also discussed. B oth lossless and lossy speech coding are implemented. T he results are com pared w ith results obtained w ith some of th e existing coders.
Finally, C hapter 6 sum m arizes some o f the more im portant contributions a n d results presented in the thesis. T he chapter concludes by p u ttin g forward some suggestions an d directions for future work.
6
Chapter 2
Overview of Wavelet-Based
Compression Methods
2 .1
I n tr o d u c tio n
C oding Is th e process of finding a representation o f a given signal. It is usually used to eith er find a representation w ith less redundancy so th a t fewer bits are req u ired to encode the given signal, or to add red u n d an cy in ord er to facilitate erro r d etectio n /co rrectio n of the signal tra n sm itte d over noisy channels. T h e former coding approach is also called compression. W ith the boom in m ultim edia an d high-resolution video containing a high degree of redundancy, com pression has gained widespread a tten tio n in b o th research and industry.
T here are essentially two basic kinds of com pression schemes: lossless a n d lossy. In the case of lossless compression, one is interested in reconstructing th e d a ta exactly, w ithout any loss of inform ation. Lossless com pression is o ften used for compressing te x t files a n d medical images. Lossy compression, on th e o th er hand, can tolerate errors in th e im age as long as the qucility of th e signal is ‘accep tab le’ after compression. A lthough the te rm acceptable’ is subjective, an acceptable ap p ro x im atio n of th e signal is one th a t is, for p ra c tica l purposes, visually indistinguishable from th e original signal.
T he general stru ctu re o f a n image com pression system is as shown in Figure 2.1. In this diagram represents th e original image. T he transform is applied to th e original image to reduce the sp atial o r sp ectral correlation or both, to generate th e transform ed image y[&, I]. T h e transform ed image is quantized to o b tain a quantized im age f]. Q uantization is used to d iscard any coefficients deemed to be insignificant. T h is results in loss of inform ation content firom th e image a n d th u s the com pression is lossy. In th e case of lossless compression, no q u an tizatio n is perform ed since no loss of inform ation can be
2. Overview of Wavelet-Rased Compression Methods
w t n____________V Ik, u
Forward ; ^ j Cocfficicm |____ J Entropy
Transform Quantizer Coder
Input :_______________________________________________________________________ O u tp u t
im age b it stre a m
F ig u r e 2 .1 . General structure o f transform-based im age compression system. tolerated.
T h e quantized image is encoded to ob tain the final com pressed bit stream 6[m]. T here are various types of encoders and entropy encoder is one of them . Entropy is a measure of the am ount of inform ation in a n ob ject. For example, a speech signal w ith periods of silence will have lower entropy th a n a signal w ith no periods o f silence. Entropy can be expressed in bits. In this form, it is generally referred to as inform ation content. Every non-random signal has some degree o f redundancy or correlation in itself. By an ap p ro p riate prediction algorithm , the redundancy can be removed thus reducing th e entropy of th e signal as well (entropy is directly p roportional to inform ation content). As reduction in entropy reduces th e average num ber o f b its p er sam ple of th e signal, it results in the overall compression of th e signal. An encoder using this concept of reducing entropy to achieve compression is an entropy encoder. T h e entropy is reduced by using th e fact th a t a sym bol w ith higher p robability of occurrence should use fewer bits com pared to a sym bol w ith lower probability of occurrence. At the receiver, the coded b it stream is decoded using an equivalent entropy decoder followed by a n inverse transform ation to o b tain th e decompressed image.
In the sections th a t follow, each o f these two or three stages (for lossless or lossy com pressions, respectively) in a com pression system are discussed in detail.
2 .2
T h e tr a n sfo r m a tio n s ta g e
T h e aim of compression is to remove redundancy firom th e original d a ta so th a t the same inform ation (or slightly less) can be coded in a fewer b its. C orrelated d a ta is characterized by th e fact th a t one can , given a p a rt of th e d a ta, fill in th e m issing p a rt. Several types o f correlation exist. T hey can be characterized as follows:
# S patial correlation: O ne can often predict the value o f a pixel in an im age by looking a t the neighboring pixels.
• S pectral correlation: O ne c an predict one fiequency com ponent by looking a t th e neighboring frequency com ponents. This m eans th a t a spectral transform (Fourier transform , for exam ple) of a signal is sm ooth.
2. Overview of Wavelet-Baaed Compression Methods 8
# Tem poral correlation: In a digital video, most pixels of two neighboring frames change very Uttle in th e tim e direction (e.g., the background).
G enerally a transform ation remove either spatial a n d /o r spectral correlation in an image. Teclmiques th a t remove b o th spectral and spatial correlation are called hybrid coding tech niques.
2.2.1
Spatial decorrelating transforms
A sp atial decorrelating transform removes the correlation from a n im age by using the neigh boring pixel values. A predictor is used to estim ate the current value of th e pixel based on previous pixels. Different types of predictors have been rep o rted in the literature. A compression scheme based o n this approach is called predictive coding. T he current pixel value is estim ated based on th e neighboring pixels and the type o f predictor being used. T h e difference betw een actu al and estim ated value is called th e prediction error (or error residual). For lossless compression, the prediction error is stored w ithout any loss using entropy coding a n d is used to o b tain the final b it stream . O n th e other hand, if the com pression is lossy, th e error residual is quantized and then coded to obtain th e b it stream . T he predictor can also be adaptive in which case its coefficients change w ith the type of signal being coded. The Jo in t Photographic E xperts Group, also known as JP E G , uses a variety of lossless predictors [18] for predicting the current pixel value based on the neigh boring pixels. G enerally th e predicted value of the current pixel is estim ated by using the p a st pixel values in th e rows a n d columns. The m ore the num ber o f neighboring pixels used for prediction, th e higher is th e order of the predictor. A further possibility is to delay the encoding of a pixel until th e ‘future tre n d ’ of th e signal can be observed, an d then take advantage of this trend. T his is called delayed coding [19].
T h e Joint Bilevel Image G roup, also known as JBIG [20], uses a set o f predictors to estim ate the cu rren t pixel value. A lthough JB IG is generally for the coding of binary images, it can efficiently code images w ith resolution up to 4 b its/p ix el. However, beyond th a t, its perform ance tends to suffer compared to other trainsform-based m ethods. T he m ain awl vantage of th e predictive coding technique is th a t it involves very few com putations com pared to spectral-correlation and hybrid coding methods. Various applications where predictive coders are used can be found in [21], [22].
2. Overview of Wavelet-Baaed Compreaaion Methods 9
2.2.2
Spectral-decorrelating transforms
In spectral-decorrelating transform s, th e in p u t image is partitioned into blocks of pixels an d each block of pixels is processed separately. T h e pixels in each block are transform ed in the frequency dom ain (spatial frequency in rows and columns) using a linear-orthonorm al transform ation. A linear transform ation is a n inform ation preserving process. T h e m ain objective of a transform ation is to p ro d u ce statistically independent (or a t least u ncorre lated) transform coefficients so th a t th ey c an be coded independently of each o th e r w ith good efficiency. A nother objective is energy com paction, which m eans concentrating th e energy to a few coefficients and th ereb y m aking as m any coefficients as possible sm all enough th a t they need not be tra n sm itte d . M ost of th e spectral transform s achieve these objectives reasonably well. T he m ost com m only known transform s are th e discrete Fourier tran sfo rm (D PT ), th e W alsh-Had am a rd transform (W H T), the Karhunen-Loeve tran sfo rm (KLT) a n d the discrete cosine transform (D C T ) to nam e a few [19]. A lthough th e K LT achieves th e theoretical lim it of perform ance o f any decorrelation technique (in o th er words, its coefficients are uncorrelated), th e D C T provides a good balance betw een perform ance and com p utation al complexity. Hence it is m ost widely used, and in c ertain circum stances, its perform ance can come close to t h a t of th e KLT [8j. Recently, su b b an d coding h as also been used for image compression. T h is involves filtering th e image th ro u g h a bank of fil ters. T h e o u tp u t of each o f the filters h as inform ation w ithin the passband of the filter. T h e advantage of subband coding is th a t th e q u an tizatio n of each of the su b b an d s can b e done based on hum an perception, thus im proving th e subjective quality o f th e decom pressed image. A specific fam ily of transform s used to o b ta in a type of su b b an d decom position of signals, called wavelets, has received considerable a tte n tio n in recent years [3]. W avelets can be very closely related to subband filters, w ith some additio n al properties, th a t m ake th em favorable for image compression [23]. O ne o f th e m ain features of wavelets is th e ir tim e- frequency representation. Wavelets have higher frequency resolution a t lower firequencies and lower frequency resolution a t higher fiequencies. T h is is a ttrib u te d to the nonuniform band w idth s of the filters used a t each sp atial resolution. Figure 2.2(a) shows th e tim e- frequency representation for uniform filter banks. T h e frequency resolution is c o n sta n t a t different frequencies. T he same also holds for tim e resolution. T his is th e characteristic of th e sh o rt-tim e Fourier transform (S T F T ). C onsider a signal x[t) a n d assum e th a t it is sta tio n a ry over an a rb itra ry window function h{t) o f lim ited d u ratio n , centered a t tim e location r . T h e Fourier transform o f th e windowed signals x{t) h*{t — r ) yields th e S T F T
2. Overview of Wavelet-Baaed Compreaaion Methods 10 time 3T 2T T • • • 0 w l 2 w l 3 w l Frequency (a) time 4T 2TT -• -• 0 wO/2N wO/N wO Frequency (b)
2. Overview of Wavelet-Based Compression Methods 11
as
S T F T ( r ,/ ) = / * x {t) h * { t - T ) d t (2.1) J —OC
which m aps the signal onto a two-dim ensional function in the tim e-frequency plane. How ever, th e analysis depends critically on the choice of th e window h{t). Figure 2.2(b) shows the tim e-frequency representation for nommiform filter banks (e.g., wavelets). T h e fre quency resolution is higher a t lower frequencies and lower a t lower frequencies. O n the other hand, tim e resolution is higher a t higher frequencies and lower a t higher frequencies. The transform ations th a t use b o th frequency and sp atial correlation in an image fall in the the category o f so called hybrid transforms. In the next section, we discuss the basic theory of wavelets and provide details on th e current trends.
2 .2 .3 H y b r id t r a n s f o r m s : W a v e le t s
Tim e-frequency representation, also called midtiresolution representation, is a general m ethod for constructing o rthonorm al bases, developed by M allat and Meyer [24]. Intuitively, m ul tiresolution sUces th e space of square integrable functions C? into a nested sequence of subspaces Vi, where each Vi corresponds to a different scale. The m ultiresolution is com pletely determ ined by th e choice o f a special function called the scaling function.
2 .2 .4 D e f i n i t i o n o f m u l t i r e s o l u t i o n a n a ly s is
M ultiresolution analysis provides a n a tu ra l framework for understanding th e wavelet basis, and for th e construction of new exam ples. T he concept o f m ultiresolution approxim ation of a function was introduced by M allat [4] an d provides a powerful fram ework to u n d erstan d wavelet decom position a n d reconstruction. Before th e theory of m ultiresolution analysis is presented, it is w orthwhile to discuss the concept of orthonorm al basis. This concept is widely used in the developm ent of m ultiresolution analysis.
2.2.
4 .1Orthonormal basis
A
family of elements {u„}, nE I, (I
countable index set), is called orthonorm al if Vn € I , ||u|| = 1 and< Uj,
Uj >=
5ij, Vi, J e I
(2.2)
where < u j > is th e in n er product o f ti< and Uj as given in E quation 0.3. Then {un}nei is called a n orthonorm al set. An o rthonorm al set is complete if it satisfies the p ro p erty
2. Overview of Wavelet-Baaed Compreaaion Methoda 12
of completeness. An orth o n o m ial set of functions, {«„}, n G
I, (I
countable index set) belonging to space, is S8Ùd to be com plete if there exists no functions different from zero in £ ^ (R ) which is orthogonal to all functions Un- A com plete orthonorm al set is also called an orthonorm al basis (or stan d ard basis). Background theory on completeness can be found in [25].As a consequence of this p roperty of com pleteness, for any a rb itra ry function u in £ ^ (R ), if < u, Un > = 0 for all n G
I,
u must be zero. In th is case, u can be represented asU = ^ U, Un Un (2.3)
n e I and
\ M h = E I < « , « " > I" ( 2 .4 )
n € I
E quation 2.3 is th e generalized form of th e Fourier series w ith any arb itrary orthonorm al basis {un}, n G
I .
E x a m p le s
The functions t G I form a com plete orthogonal set in £ ^ [0 ,2vr]. A nother exam ple of orthonorm al basis in £ ^ (R ) is / m . n ( x ) = 2 - ^ / 2 - n ) ( 2 .5 ) where
{
1, 0 < x < 1 / 2 - 1 , 1/2 < X < I 0 , otherw ise 2 .2 .4 .2 S c a lin g fu n c tio n a n d th e m o t h e r w a v e le tA m ultiresolution analysis consists of a sequence of successive approxim ation spaces Vj (closed subspaces) th a t satisfy th e relation
■ V2 C V i C Vq C V - i (Z V- 2 ■■■ (2.6)
w ith
u Vj = £ 2 (R ) (2.7)
2. Overview of Wavelet-Baaed Compression Methods 13
and
n Vj = {0 } (2.8)
iei
If Pj is the orthogonal projection o p e ra to r onto Vj, th en
lim P j f = f V f e C ^ iR ) (2.9)
i->-oo
A n ad d itio n al requirem ent for th e m ultiresolution analysis is
f i t ) G V j f i V t ) € Vo (2.10)
In o th e r words, all th e spaces are scaled versions of th e control space Vq. Moreover,
/(< ) € Vo => f i t — n ) € Vb for aU n € / (2 .1 1) In sum m ary, the five conditions fo r a set o f subspaces VyC£^(R) to be suitable for m ul tireso lu tio n analysis are
1. • • ■ Vb C Vi C Vq C V— 1 C V—2 * • • 2 - U je i Vj = £2(R ) ,
a e i VG = {0 }
3. f i t ) G Vj <=> f i 2 t ) G V j - i
4. T h ere exists G such t h a t Vq = { f \ f i t ) = a,- 0 (t — %)} , 4> is also known as th e scaling function.
5. {4>nit) = <f>it — n ) } is a n o rth o n o rm al basis o f Vb.
T h ese shall be referred as the five axiom s o f m ultiresolution analysis. If any orth o n o rm al basis satisfies all o f th e above conditions, it could be a valid wavelet basis for m ultiresolution analysis.
T h e basic tenet o f m ultiresolution analysis is th a t whenever collection o f closed subspaces satisfies axioms 1-5 o f m ultiresolution analysis, th e n th ere exists a n orthonorm al wavelet basis {t/’j,*; J, & G Z} of L ^(R ), V'j,ifc(®) = — k), such th a t, for all / in Z-^(R),
P j - i f = P j f
+
E
< / ’ V'M > V'jjk
(2.12)
fcez
w here Pj is the orthogonal p ro jectio n onto Vj. Please refer to [3] for details.
Now, if we call W j th e orthogonal com plem ent o f Vj in V^_i, i.e.,
2. Overview of Wavelet-Based Compression Methods 14
th e n Wj contains th e details necessary to go from Vj to Vj-i- Iteratin g E q u atio n 2.13 gives
Vjf_i = W j 0 Wjj^i 0 Wj+2 © ••• (2.14)
B y virtue of axioms 2 an d 3 for m ultiresolution analysis, this implies th a t
L 2(R ) = ® j ^ z W j , (2.15)
a decom position o f Z<^(R) into m u tu ally orthogonal subspaces. In o th er words, a given resolution can be a tta in e d by a su m o f ad ded details.
If a function (f> defines an orth o n orm al basis o f Vq such th a t it satisfies th e five conditions for m ultiresolution analysis, it is called th e scaling fu n ctio n of th e wavelet basis. T hus, if Vo C V -i, then
c/fc 4>{2x - k) (2.16)
k 6 I
where (f>{x) is the scaling function. In teg ratin g b o th sides of E quation 2.16 w ith respect to (w .r.t) z , we obtain roo r o c f <f>{x)dx
=
f ^ ct <f>{2x — k )dx (2.17) J - o o 6 I = cjt (f>{2x - k )dx (2.18) t e l “Assuming <f> 6 £ ^ (R )u £ ^ (R ) such th a t (f>{x)dx = 1, and su b stitu tin g 2x — k w ith x ', E q u atio n 2.18 can be w ritten as
1 = 2 2
r°°(f>ix^)dx'/2
(2.19)k e l
S u b stitu tin g <f>{x)dx = 1 in 2.19 above, we get
2 2 Cifc = 2
(2.20)
k
Also, if <f>{x) satisfies th e orthogonality condition which states th a t,
< <^(x), 4>{x — m ) > = 25om V m 6 I (2.2 1) a n d also satisfies th e dilation equation given in E quation 2.16, it can be shown th a t
22
Cfc Cfc-2m = 2 <îom, Vm € I (2.22)2. Overview of Wavelet-Based Compression Methods 15
where
^ 171 = 0
otherw ise
A nother condition w hich governs the choice of th e c/t’s is the approximation condition, which is as follows: If p is a n atu ral num ber, then
E (-1 )* k ^ C k = 0 (2.23)
k for m = 0, 1, 2, . . . , p — I.
It can be shown th a t the num ber p characterize th e sm oothing function <i>{x) such th a t polynomial functions of degree p — 1 o r less, are linear com binations of <f>{x) and its integer translates [26]. Moreover, the higher th e value of p, th e greater th e num ber o f nonzero c^’s in the orthonorm al basis [27].
If the scaling function (f> is given by E quation 2.16 and th e wavelet basis is defined by E quation 2.12, th en it can be shown [3] th a t th e m other wavelet V’(x) is given by
V’(x) = Z ( - ^ r C i-a <f>{2x - n ) (2.24) amd
iPj^ (x) = 2-^/2 ^ (2-J'x - k) (2.25)
If Pj is the projection on Vj and Qj is the projection on Wj, then
f = P j f -i- Q j f (2.26)
where
Thus,
Q j f — ^ 3 ^ >
V’i.fc
(^) (2.27) it € ITherefore, we can sto re the inform ation in the form of wavelet coefficients < / , V’j,* > for different values o f j an d k. These inner products are called th e wavelet coefficients o f the function f { x ) . So, if we have a n approxim ation of a signal a t th e resolution corresponding to Vj, then a b e tte r approxim ation is obtained by adding th e details corresponding to Wj.
2. Overview of Wavelet-Based Compression Methods 16
T his am ounts to a weighted sum of wavelets a t th a t scale. E quation 2.12 describes this relationship w here < > are th e wavelet coeflScients ( or weights) of the m other wavelet ‘ip{x) a t th e scale j . T h u s, by ite ra tin g this idea, a square integrable signal can be seen as th e successive approxim ation or weighted sum o f wavelet basis {V'j.ikîJi fc € Z} a t finer and finer scales (w ith som e o f these .
2 .2 .4 .3 C a lc u la tio n o f o n e d im e n sio n a l w a v e le ts c o e ffic ie n ts
In this section, th e weighting coefficients for th e orthonorm al D4 wavelet basis are d e
rived [3]. By using th e approxim ation and orthogonality conditions on th e wavelet coeffi cients to g eth er w ith th e su m m atio n condition, th en from E q u atio n s 2.20, 2.22 an d 2.23, we get a set o f equations as
3
53
Cfc = 2 (2.29) fc= 0 3 ^ ^ = 2 (Jom (2.30) t= 053
(-1)*= A:"* Cfc := 0 for m = 0, I (2.31) fceiE xpanding th e above equations, we get
Co -f- Cl -H C2 + C3 = 2 (2.32)
C0C2 + C1C3 = 0 (2.33)
—Cl 4- 2c2 — 3c3 = 0 (2.34)
CO — Cl -I- C2 — C3 = 0 (2.35)
c ^ 4 - c f - l - C 2 - f - < ^ = 2 (2.36) Solving th e first four equations sim ultaneously, we get
I 4- V3
“ 4
3 - V3
2. Overview of Wavelet-Baaed Compreaaion Methoda 17 3 + >/3 C2 = C3 = 4 1 — 4
It can be shown th a t E quation 2.36 is d ep en den t on the rest of th e four equations. Squaring b o th sides of Equations 2.32 a n d 2.35 and adding, we get
2(c^ + 4- 4- Cg) 4- 4(coC2 4- C1C3) = 4
By using E^quation 2.33, we get c0 4 - c f - | - C2 4 - C3 = 2 which is E quation 2.36.
Clearly increasing the value o f p in E q u a tio n 2.23 increases th e num ber of nonzero wavelet coefficients and thus sm oothes th e scaling function [27]. For example, if p = 2, th e n one can reconstruct exactly any factor w hich is a sum o f linear equations. If p = 3, th e n any q u ad ratic curve can be decom posed a n d reconstructed exactly using those wavelet coefficients. B u t, th e num ber o f nonzero coefficients increase as the value of p is increased, th u s adding to th e com putational complexity.
2 .2 .4 .4 S ig n a l d e c o m p o s itio n a n d r e c o n s t r u c t i o n b y u s in g w a v e le ts
T h e decom position of a signal c an be perform ed by using a n algorithm introduced by M allat [4] called th e caacade algorithm. T h is algorithm decomposes the signal by taking th e projections o n the set of subspaces { W j } g enerated by th e seeded m other wavelet {V’j.n}- Clearly, as we increase the value o f j , we sh ift to coarser scales. If
f { x ) = E « n (X ) (2.37)
n w here a° = < / , <t>o,n > , th e n / €
Vq-Moving to coarser scales Vj, j > 0 discards some fine inform ation contained in the wavelet com ponents. Note th a t th e set of subspaces {Wj} are orthogonal to each other,
i.e.,
1 0, otherwia
Wi n W j = { (2.38)
otherwise
In th is way, no inform ation o f th e signal is lost in th e decom position. In general, if we transform from Vq —*■ Vi, i.e.. P i : Vo -4 V i, th e n
P l f = E “n A (^ .n (x )) (2.39)
n
2. Overview of Wavelet-Based Compression Methods 18 However, from 2.16 < ^ Cn—2/ (2.41) (2.42) Therefore. A / — —ÿ= ^1,/ (2.43) v 2 n I — ^ ^ ^ 5 Z ^ (2-44)
Moreover as P i is th e projection o f subspace Vq onto Vi, P i / € Vi. Hence,
P l f = (2.45)
I
where a} =< f , <^i / > , for aU Z E Z. C om paring Equations 2.44 and 2.45, we get
5 Z * ^ -2/ “ n (2-46)
Generalizing this to any transform ation Pj : —► Vj, we get
— -ÿ= 5 Z * ^ -2/ ^ (2-47)
v2 „
where oj = < / , > As th e subspace V) is a coarser approxim ation to V j-i, the transform ation m a trix given in E quation 2.47 is called th e lowpass filter m atrix or the Lf-matrix.
In order to g e t th e wavelet com ponents, i.e., Qi : Vq W i , we can w rite
Q i f = E «n Q i i M x ) ) (2.48)
However, from 2.24
E E V’l,/ (2-49)
n I
2. Overview o f Wavelet-Based Compression Methods 19 Therefore, Ql [0O,n(3?)] -7= H “ n C i_„+a V 'w M V ^ n I ^ ( -1)" Ci_„+2/ OLn where ^ 2 2 ( ~1)" Ci_„+2/ a® (2.51) (2.52) (2.53) (2.54) G eneralizing, we get 2 Z ( ~ ^ ) ” c i_ „+2/ -1 (2.55)
As the subspace W j contains the details th a t were missing in Vj relative to V j - \ , the tran sfo rm atio n m a trix given in the above eq u atio n is called th e highpass filter m atrix or the H -m a trix . C om bining Equations 2.47 and 2.55, we get
aJ = La-'-^ (2.56)
y = H a^-^ (2.57)
In m ost practical applications, is assumed to b e sam e as th e discrete in p u t sequence / (n) where / ( n ) = f{x)\x=n for n € I . In other w ords, th e in p u t sequence is assum ed to be identical to th e finest scale of wavelet decom position. Please refer to A ppendix A for more details.
For signal reconstruction firom th e wavelet com ponents a n d th e signal a t coarsest scale. we use Also where P j - i f = P j f + Q j f = ^aÎ4>j^i{x) (2.58) (2.59) (2.60)
2. Overview of Wavelet-Baaed Compreaaion Methoda 20
S u b stitu tin g value of P j - i f from E q u atio n 2.58, we o b tain ,
^ > + ^ b { < > (2.61)
I I
Using th e fact th a t for any two functions / and g € C?, < g , f > = < g, f >*, E q u atio n 2.61 can be w ritte n as
+ H ] b> (2.62)
D e c o m p o s itio n u s in g w a v ep a ck ets
A nother approach of decom posing the signal is based o n the theory of wavepackets [3]. In this case, th e stream of d a ta is first decomposed in to lower-order scaling coefficients and higher-order wavelet coefficients. Each o f th e higher-order wavelet coefficients is fu rth er decom posed into lower-order and higher-order coefficients. Thus, if th e subspace Vq is split into orthogonal subspaces Vi and W i as given by
Vq = Vi © W l (2.63)
then,
W l = W l © W ^ (2.64)
In E q u a tio n 2.64, W l corresponds to th e coarse d etails in Wi while W ^ contains th e high- resolution com ponents o t W i
-2 -2 .4 .5 E x te n s io n o f w a v e le t b a sis to tw o d im e n s io n s
The orth o n o rm al wavelet basis in £ ^ (R ^ ) can be co n stru cted by s ta rtin g w ith a basis in £ ^ (R ). L et = 2"^/^ ip{2~^x — t ) be th e orthonorm al basis in £ ^ (R ). Its two-dim ensional equivalent is generated by taking th e tensor p ro d u ct
V'ji.ki j , . k , ( r i , Z2) = ^i,.ifc,(a:i) V’j2,Jk2(a?2) (2-65) There exists an o th er b e tte r m ethod in which d ila tio n of the resulting orthonorm al basis controls b o th the variables simultaneously. In th is approach, one considers the ten so r product o f two one-dim ensional m ultiresolution analysis ra th e r th a n two one-dim ensional wavelet basis. Thus, if
2. Overview of Wavelet-Based Compression Methods 21
and
F E Vj F{2^x — Til,2^y ~ Mg) S Vq V n i, ti2 € Z (2.67)
th en V j forms a m ultiresolution la d d er in £^(R ^) satisfying
••• V 2 c V i C V o C V _ i C V _ 2 (2.68) y V j = £2 (R^) , (2.69) J€l n Vj = {0} (2.70) J€l T hus <^j;ni,na(3T, y) — 0j,nt (^) 4*j,Jij{y) (2.71) = 2~^ <f>{2~^x — ni)4>{2~^y — TI2) , nt,Ti2 E I (2.72) V j_ i = V j - i ® V j .i (2.73)
=
(Vj e Wj) ® (Vj e Wj)
(2.74)
= Vj<2> Vj® [(Wj®Vj)©(Vj®Wj)©(PVj®lVj)] (2.75) = V j © W j (2.76)where W j consists o f three p arts given by v>j,n, (r ) <^j,n2(y) for (W j ® Vj), 0j,nj(y) (r ) for (V^ ® Wj), and V’j,ni(a:) V^.naCv) for (Wj ® Wj). This leads to th e th ree wavelets
y) = V'(y) (2.77)
y) = <t>{y) V'(z) (2.78)
y) = V'Cir) V’(y) (2-79)
w here h, v, an d d stan d for horizontal, vertical and diagonal com ponents.
Filtering can be done on rows a n d columns in a two-dim ensional array, corresponding to th e horizontal a n d vertical directions in images, for example. If is an iV x iV array, then applying E q u atio n 2.47 to the rows o f th e image results in an arra y o f size N /2 x N . A pplying E q u ation 2.55 also results in a n array of size N / 2 x N , say T h e transform ation is th e n applied to th e columns of th e array to o b ta in a n d each of size n /2 x N /2 , respectively. T h e sam e tran sfo rm atio n is applied to 6^/^ to o b ta in a n o th er two arrays, 6^’* an d of sizes N / 2 x N / 2 , respectively. T he elem ents of the a rra y a* are called the scaling coefficients of th e image while elem ents of arrays 6^’”, a n d b \ ^ a re called th e wavelet
