Corn - collectionscanada.gc.ca · rnnp ulhere (S. (1) is a rumplete metric space then f has exactly one fied point a E il and Vx E S the sepuence gzuen by: converges to a. NOTE: f

.A t hesis subniitted to the

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in conformit- with the requirements for

the degree of '\laster of Science

Qiieen's h i v e r s i ty

Kingston. Ontario. Canada

January 2002

Copyright @ Andrew Bemat, 2002

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Abstract

Frnctal i rnag~ îumprcssion is an area where wr erpect very high rates of compression

as opposed to .JPEG miiprrssion ivhich prochces high quality images at cornparably

lm- rates of coniprcssionl in ii reiisonable amount of time. The siiccess. in terms

of compression ratio and image qiiality. of 3 fractd encoder Iies in the choice of a

partition scheme.

\te will revicw a forma1 model for a Partition Iteratcd Function System (PIFS) en-

coclrr/clecoclrr togctlier wi t b t hc nccessary mat heniat i d background. .\ description

of a fractal tool follows. This tool implernents the PIFS model and several partition

srhemes. It u s tvritten in C++ by Andrew Bernat (author of this thesis). Reason-

able theoretical boiinds on error and image quality for different partition schemes are

difficult to obtain: empirical studies are the only alternative. The fractal tool serves

t his purpose. 1 t can help to determine ivhich kinds of partition schemes yield higher

compression rates and larger signal-to-noise ratios (SSR) for particular images. We

discuss t his in the paper.

' W e will make precise the meaning of hi& and low.

Acknowledgement s

1 have had many cr~joyalle and sad moments while in Kingston. I'd iike to thank

d l rny fricnds for heing around throughoiit those tinies and alloiving me to be with

tiirm. Thanks .-\nibrlr. .\ncirew. Chris. Jeff. .Jcn. J~rcrny. Marta. Nitch.

I'd also likc to t haiik rny friends nway from Kingston whom 1 talked with over the

phone for niany hotirs about ttiesis work and lifr. Thnnk you Elena. Tuny. Also, niy

stipen-isor who ,pave me the opportunity t o work on anything that was cool.

SIy family has a l w q s supported me and they rarely have the opportunity to

rweive my gratitude. but the! know it is there. Thank o u klorn. Dad, Nami. Kati.

anci Bob.

Contents

1 Background Information 1

1 Introduction 3

1.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 3lotivntian 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Problem 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Thesis 5

1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Review 7 .i. . . . . . . . . . . . . . . . . . . . . . . . . 2.1 hlathernatical Background I

2.2 Application To Image Compression . . . . . . . . . . . . . . . . . . . 11

2.2.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . II

2.3 Self-Similaritu Of Real- Worid Images . . . . . . . . . . . . . . . . . . 12

2.4 Partitioned Iterated Function Sqstems . . . . . . . . . . . . . . . . . 13

2.4.1 Informa1 Intuitive Treatment . . . . . . . . . . . . . . . . . . . 13

2.4 . 2 Forma1 Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.3 Partitioning Techniques For 2D Images . . . . . . . . . . . . . 26

2 .4 .4 Domain-Range Vector Matching Techniques . . . . . . . . . . 27

vi CONTENTS

4 Evaluating Partition Schemes . . . . . . . . . . . . . . . . . . 28

II F'ractal Tool And Partition Schemes

3 Fractal Tool 33

3.1 The Cornpress Progrnm . . . . . . . . . . . . . . . . . . . . . . . . . 33

R.2 The Decornpress Program . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Gericral D~ta i l s For Block Basecl Partition Srhemes . . . . . . . . . . 36

3.3.1 Oiitpiit Of h p p i n g s . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 Finding Range Blocks . . . . . . . . . . . . . . . . . . . . . . 42

3.3.3 Block ClCusification . . . . . . . . . . . . . . . . . . . . . . . . 45

. . . . . 3.3.4 Quantizntion . . . . . . . . . . . . . . . . . . . . . .- 50

3.4 PartitionSchrrries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Sample Test Data 55

5 Cornparison of Partition Schemes 57

5.0.1 Scheme 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.0.2 Scheme 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.0.3 Scheme 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.0.4 Scheme 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

LI- 5.0.5 Scheme 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i a

5.0.6 Compared to JPEG . . . . . . . . . . . . . . . . . . . . . . . . 77

CONTENTS vii

III Discussion

6 Interesting Questions 81

6.1 Son-Uniform Quantizer . . . . . . . . . . . . . . . . . . . . . . . . . . $1

. . . . . . . . . . . . . . . . . . . . . . . 6.2 Re-corripressing The Oiitpiit 82

6.3 Adnptiw Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

. . . . . . . . . . . . . . . . . . . . . . . . 6.1 Target Conipression Ratio 83

. . . . . . . . . . . . . . . . . . . 6.5 Fos t-Processing For Deconipression 93

. . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Lossless Encoding Phase 83

. . . . . . . . . . . . . . . . 6.7 Silnibcr Of Iterations For Deronipression 81

7 Future Work 85

. 4 . 1 Frnctal Aiitlio Compression . . . . . . . . . . . . . . . . . . . . . . . . 85

7.2 Fractal TooI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8 Summary And Conclusions 87

Bibliography 89

A List of Symbols and Abbreviations 93

B Calculat ions For Comput ing Range-Domain Mapping 95

B.O.l blapping for ri . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.0.2 Mapping for rl . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.0.3 Mappings for other range blocks . . . . . . . . . . . . . . . . . 98

C Range Block Results 99

viii CONTENTS

D Compression Ratios

E SNR

Vita

List of Tables

4.1 Location of Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . 56

. . . . . . 5.1 Niimber Of Distinct Alappings for Images Cnder Scheme 1 78

Rnnge Blocks For Schenie 1 . . . . . . . . . . . . . . . . . . . . . . . 100

Range Biocks For Scheme 1 . . . . . . . . . . . . . . . . . . . . . . . 101

Range Blocks For Scheme 1 . . . . . . . . . . . . . . . . . . . . . . . 102



Range Blocks For Scherne 3 . . . . . . . . . . . . . . . . . . . . . . . 105



Range BIocks For Scheme 4 . . . . . . . . . . . . . . . . . . . . . . . 108

x LIST OF TABLES

List of Figures

3.1 Bit L q o i i t Of -4 Range-Domnin 4Iappiiig . . . . . . . . . . . . . . . 37

. . . . . . . . . . . . . . . . 3.2 Divicle image into perfect square regions 43

3.3 Breakclown the perfect square regions into sizeable blocks . . . . . . . 45

3.1 Examine each block and t~ to Bnd a mapping for it . . . . . . . . . . -16

3.5 Algorithm to compute the class of block .A . . . . . . . . . . . . . . . 49

3.6 SIightly modified algorithm to compute the class of block .-2 . . . . . 49

3.1 Quantization code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.8 Dequantization code . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Spatial Transformations Of Domain Blocks . . . . . . . . . . . . . . . 66

Part 1

Background Informat ion

Chapter 1

Introduction

1.1 Thesis Overview

This t liesis iliscusses several variations on the quadtree partition sclieme[3]. These

paritioii s t h r n i t ~ s are rrtipiricalIy r~strd on various test images arid their resiilts are

reportrd in Cli i ipt~r 5 .

This thesis describes a forma1 mode1 of a partitioned iterated function -tem.

This mode1 is explained in the context of image compression. The necessary sim-

plifications to build a fractal cornpression/decompression systeni in software are also

discussed.

We discuss how to execute the fractal tool. There are many options for each

partition schenie that result in different behaviour. The details can be found in

Chapter 3.

Information about the test suite can be found in Chapter 4.

Experimental data can be found in the appendices.

4 Introduction

Figiirtl 1 . 1 : Sirrpinski Gasket

1.2 Motivation

Considcr t hc following systcrn of fiinctions:

t I j ( r . ) = ( 7.r. - :y) - 1 ( ) = (:.r + :. - A!)) -

\vhcw n-r let F : [O. 11' - [O. 11' (notat ion for [O. l] x [O. 11 is [O. 112) be:

Let Sn = {I : .? E [O. 11'). \fi. pirk P randomly. Let SI = F ( & ) = { F(f) 1 V.? E Sol.

Let Sn = F(S,_J . If me plot tlir sets SI. S?. S3.. . . on [O. lj2 then the plotted image

will converge to the Sierpinski Gasket (sliorvn on Figure 1 . 1 ) . We cal1 the Sierpinski

Gcasket t he attractor of F.

F can be described by writing out the coefficients in front of each term for each

f,. In general. a function /, looks like: /,(x. y) = ( a i l + b i y + t , ~ + U i g + ui ) . TO

describe a single /, ive must m i t e out 6 numbers. Therefore. a description of F

requires 19 numbers. Csing only 18 numbers. we can describe the Sierpinski Gasket.

By changing the parameters a,. b,. c,, t,. o,. u,: we can obtain different types of detailed

irnases. If we can find functions f,.. . . . jiy for arbitrary images such that the attractor

of F = JI u . . - u f,v is close to the original image then we can expect to have very

good compression rates.

1.3 Problem 5

1.3 Problem

T h systmi of fiinctioris wtiicli wr mil F = u:,/, is an ~ s a n i p l e of an iternted

fiiriçtioii systrni [2]. Iri a prrfwt ivorld. for any image we coiild find a system F

ivli~st. ;it t rnctur wotild t )cb 'close' to t tie original image. C'nfortunately, real-world

iriiiipt~s ;in3 not i-or~qdiwly sdf-siniilar (like the Sierpinski Gasket); so a more clever

solution is neecld. Chaptcr 2 will describe an iterated niodcl which is more practical

for r n t ~ i r d I ~ I ~ P S .

1.4 Thesis

Thr stuclu of fractnl sets is usually an area associated nith pure mathematics. How-

<.ver. in 1993 Arnaud .lacqiiin piiblished a paper that bridged the knowledge of fractal

sets to image compression by automatic computer analysis. The revolutionary corn-

ponent of his publication was about breaking the image down into a partition and

npplying an iternteci function system to a portion of the image. Since the publication

many partition schcmes have been created; however, few have been impiemented and

nnalyzed from an cmpirical viewpoint. We present several partition schemes based

on the qiiadtree partition that operate on several test images taken h m standard

image compression benchmark libraries. We make conclusions about these partition

schemes as they relate to particular images or error thresholds.

6 Introduction

1.5 Contributions

Ttit. (~iiiidtree partition s ch~n i e is well known: hoaever. the ides of using a local

iioiglil)oiirli00(1 ;iroiintl a range Mock whm scarching for a matching dornain block is

I . This tlitlsis rontribiitrs ro tiir arra of fractal image compression by exarnining

t tltl r f f w t s ~f 1~ca1 and rariclorxi neiglibourhood searches and ~Iassificatiori schemes.

\\il iilso riiake a n a t t rn ipt tu proride a fraitieaork for future partition schemes which

mn br tcsted on an image test suite.

Chapter 2

Review

2.1 Mat hemat ical Background

\\é introduce some niaterial which we will neeci to justify the use of iterated function

systmis for image compression. Most of the iollowing defini tions and t heorems can

IR lotirid in an- iinalysis test book.

Definition 1 .4 rnetric space is a space .Y together with a function d : S x X + R

suçh that the following conditions hold:

1. d ( r . y) 1 0 Qx. y E S

The function d is called the metrie, and the metric space ts denoted (X, d ) .

7

8 Review

Definition 2 .4 ntnp f : ,Y + S is contractive ouer the m e t r i space (.Y. d ) if:

( f . ( ) S . ) Kr. 9 E .Y

ichttrp O < s < 1 rs n d l d tlrr contmctivi ty i$ f.

Definition 3 -4 sequence { I ~ } F = ~ in .Y ts said to convetge to some x E .Y where

(S. d ) is u rnet7-i~ spuee if Vc > 0 3 .V > O such that:

Definition 4 A s e q t r i r i w {.K. }:=, t r r S i s (1 Cauchy seqaenre if Yc > O 3 .V > O

surh thtzt:

( ) < V n . m > . V

Definition 5 .4 complete metric space (Sr l ) is a metric spoce uhere every

Cm~h!y sequence conctcrges tu some r E S.

Definition 6 a E .Y is a f i e d point of the function f : .Y + S zf f(a) = a.

Theorem 1 Contmction Mapping Theorem: If f : .Y + .Y zs a contractive

rnnp ulhere ( S . (1) is a rumplete metric space then f has exactly one f i e d point a E il

and Vx E S the sepuence gzuen by:

converges to a .

NOTE: f "(z) denotes f composed uiith itself i tirnes.

2.1 Rlathematical Background 9

Proof:

Let O 5 i < j and suppose f hifi contrnc:tivity S.

O l m r v ~ :

d ( f m ( r ) . faJ(r ) ) 5 s d ( f " - l ( x ) . f u j - l ( r ) ) f is contractive

< - . s ld(r . /"-'(1)) f m is contractive

5 s l [ ( f ( ) + . . . + d ( / O ( ) . f O J ( ) ) ] A Iriequality

= ~ ' C ; ; ~ l ( f ~ ~ - ~ ( x ) . ~ ~ ~ ( x ) ) -

I - 1 Qk-I 5 -7' &l d ( - ~ I ( 1 ) ) f0'-' is contrnctivc

= s l d ( r . f ( r ) ) ~ ~ ~ ' 1 s" ' l - s J - k + l

= .yzd(x. j(x)) -

< s l d ( = . f ( x ) ) &

Now. pick an? c > O. Since O 5 s < 1 3 .V > O such that:

Thercforc. by the definition of Cauchy seqiience, the sequence given by:

is a Cauchy sequence. Iloreover, since the metric space is complete then every Cauchy

seqiience must converge to a point in the space. So, given any x E .Y, the sequence

of repeated applications of f on x converges to a E X.

We need to show that a is a fixed point of f and that it is a unique fixed point.

Observe:

d(a . f ( a ) ) 5 d(at f o k ( x ) ) + d( f (z), f ( a ) ) A Inequality

= d(a. fok((r) + s d ( f o k - - ' ( x ) , a ) contractivity of f is s

10 Review

If we k t k + .x then me know that Jok(x) + a . This implies t h a t the riglit hand side

of ttie ;tl~ovc rc1ii:ition ciln bc broiiglit down to O (a more rigoroiis argument c m l x

miide by coritriidiction). Therefore d(n . f ( c i ) ) <_ O. Cl~arly. this implies tha t n = ! ( a ) .

Hm(-P. (i is i i fisrcl poirit of f .

Sow. ive rriiist show that tr is thp imiqtw fisecl point of f . S I I ~ P O S P 3 two fixed

points of f. Narnely: n anil b siicti thiit f ( ( 1 ) = (1 and f ( b ) = b. Therefore:

d(o. b ) = d( f ( a ) . f ( b ) ) 5 d(«. b)

If s = O thrn cl(( i . h ) 5 O which iniplics ci = 6 . If s f O ttien d(n. I ) = O hivniise s < 1:

tlierefore. wc have t l iat (L = 6.

So. V r E S ttie scqiirrire of itrriitecl appliratiuris o f f on x converges to the unique

fixecl point of f . A s rrqtiired. O

This theorpin (2 . 171 is really n corollary of the Contraction 5Iapping Theoreni.

Theorem 2 Collage Theorem: I j (S. d ) r s (1 cornplete metric spoce and / : .Y + S

is a rontr<it:titw mrip with fixed point r i E S theri Yx E .Y

1 d ( x . a) < -d(r. f (z))

l - s

where s is the contractivity of f .

From the proof of the previous theorem we can see that:

1 4x7 i O k b ) ) 5 -4G f (X I ) l - s

2.2 Application To Image Compression 11

The rcçirlt follows. 0

The relevance to image corripression nia- not bc clear with the way tlie trvo tiieo-

rciris 11;n-e becri stat cd. Coiisicirr i i r i alterriate s t i i t~nimt of tlir Contraction Mapping

Ttieortm: Let .Y be the space of ining~s (which Iiiipperis to be complete). If we have

a contractire map f that operates frorii A' to X then t h e r ~ is a spccial image (called

the attractor of f ) such tliat if ive start rvith an- randoni image in X and apply f

iterativrly then tlie secluence will converge to the nttractor of j . This theotem ex-

plains why the motivational rs;irriple worked: the attrnctor of F (in the example) is

t t i ~ Sicrpinski Gaskct .

\\i. rridly want t o solw the followiiig probleni (cdled thr inverse problem(2l):

Civcn ;i target i m a g ~ S. fiiid a contractive niapping f siich that the at-

tractor of f is close to S.

If LW <.an firitl / for a given S t h n by ttlic Contraction Llapping Tlieorrni Ive can use

/ to approximate S by repeated applications of f to a random image.

2.2 Application To Image Compression

2.2.1 Justification

An image is a grid of different coloured pixels; eacli (z, y) position has a colour

associatecl with it. We can think of an image as a function of the position of a pixel

to its colour value. Pixel positions and colour values are discrete, so an image is a

function of the form f : ZC x Z+ + Z+ LI {O}. The domain of f is finite. There are

12 Review

sewrnl w1ys to measiire the distance tictw-vn 2 images. Example:

Tlic proof thii t d, is a metric cari br foiirid in ariy analysis test. Notice that d,

is a nirtric on the space o f iniagcs ivhidi is a finite space.

Kt. \vil1 ilse the following rrietric for tlettmiining the distance betweeri two images:

The fiirictiori i f is ;i nirtric o n tlic spacr of iiriiiges. Tlic proof can be found in ariy

;mil!-sis test.

Lemma 1 cf ( / . g ) is il mrtnr nherrl f. g W P jlrnrtion rrpr~srntcitions of images.

\Vc will h o t e the set of imng~s bu J I . LVe have shown that ( M d ) is a metric

spict.. Is (JI. d ) cornplete? Obsrrvo that if ivti fix the niauinluni size of images and the

niinihrr of ïolours allowcd thrri -11 is finite. In a finite spnce. rvcry Caiicliy sequcnce

converges: lience .\I is conipiete. TIiiw estra assumptions that WC have made are not

too restrictive. Real-aorld images are not arbitrarily large; moreover, the number of

allowd colours is also finite for cornputer applications. If we construct a contractive

map f : .\! + .\f then we know that f will have a unique attractor by the Contraction

XInpping Theorem. Sow. WC construct f such that its attractor is close to a target

image.

2.3 Self-Similarity Of Real-World Images

Vnlike the Sierpinski Gasket. real-world images are not cornpletely self-similar. Recall

that the Sierpinski Gasket, denoted S, is the Lxed point of a rnapping F = u:=, fi

2.4 Partitioned Iterated 'Function Systems 13

(sep ctiapter 1 for detnils). More forriially: F ( S ) = jl (S ) U f2(S) U h(S) = S. Each

fiirirtim f, takrs S ;u: its input m t l rnanipulates S in some w q so that the union of

the ft's is S. It is wry ciifficult to create such a mapping F for real-world images.

Corisirlibr titking a picturc of j-ourself. trnnsforming several copies of yoiir pictiire and

pnsting the traiisfornicd copies b x k together t o get a picture of ourself.

Instwii . ive create ftinctions jt whicli arc restricted to part of the entire image.

This rvorks becituse real-worltl images are partially self-similar; that is. some parts of

ari iniagc. iiri. siniilar to otlier parts of an image.

This is the idea of Partitioned Iterated Function Systems (PIFS) [6]. .A clear

rsp1;iniition follows.

2.4 Partitioned Iterated Function Systems

2.4.1 Informal Intuitive Treatment

Our goal is to find a contractive riiap II' whose fixed point is close to the image we

ivish to compress ( p ) . The mapping W* will be constructed by taking the union of

several fiinctions. w, which exploit partial self-similarity of p.

\Ce construct IV as follows:

Divide I L into disjoint blocks r,. . . . , r,tfr, cdled range blocks, so that they cover

p. Call the set of range blocks R. hlr is the number of range blocks. i.e.

.\Ir =J R 1. See Figure 2.1.

Divide p into domain blocks d l , . . . . d M , . Dornain bloch are larger than range

blocks. but they are not necessarily disjoint. Call the set of domain blocks D.

Md is the nurnber of domain blocks.

14 Review

Figure 2.1: Range block partition rl to rd , ,

Figure 2.2: Domain block partition di to dbId

2.4 Partitioned Iterated Function Systems 15

0 For each r , E 'R. find a contractive mapping w, and a domain block dm, E 2)

siich t hat

Theorem 3 11. = u:.~ io, u contractiue where each ioi is contractiue.

Lrt v he the fixed point of II'. Sincc there ~ x i s t s a fiinction tu, that maps a p

prosimately ont0 each r,. where r, is n part of p. then W ( p ) = p =+ d(p , lY(p))

is sniall. Tlierefore. by the Collage Tlieorerri we kiiow tbat d(v. p) is small. Since

t h ~ tixnf point of Il' is dose to the original image thm ive know thnt CV is a good

rrprcsentation of the originai image. II' is a good description of p. Writing out CI;

( t m h I C ~ ) in fewer bits thnn the original image is the goal.

2.4.2 Formal Treatment

\Ve present a general matbernatical formulation of PIFS [3. 61. CVe will use 1 dimen-

sional signals; however, the technique generaiizes to higher dimensions.

Encoding

Given a vector (or block) p E P. we want to find W such that the h e d point of I.V

is close to p. The mapping IV must satîsfy:

IV must be contractive

16 Review

Lrt W t ) t b the set of rriappings I I * wiiich satisfy the abow two conditioiis. Fi~rtherrnore~

tvp rrqtiire that the fixeci point of I I ' . f,. has the property that:

Essmtially. the relation nt~ove is a forma1 description of the inverse problem.

Finding the optimal mapping I I * is difficiiit. We are ailling to sacrifice optimality for

Encoding Example

We aallow IV to be a system of .\Ir functions in, wliere ul, : RD + RB. D is the size of

a domain vector and B is the size of a range vector. Both range and domain vectors

are selected from the original signal. We want r, z w i ( d m ) for each i = l..Mr. .An

example is given below.

Let p = (8. 10.23,25,. . . . 1 3 . l 5 , l i . 19,. . .) E P.

Let rl = (8.10) and ds = (13,lL 17,19). See Figure 2.3

Suppose that al1 range blocks are of size 2 and domain blocks are of size 4.


O a,, 6, are scalar parameters.

Thr fiinction d tnkes a clornain block and arerages out adjacent clernents in the block'.

This is how wr get w, to niap orito range blocks. i.r. By spntiiilly contracting the

doniain block by a factor of one t d .

Notice that @ ( d 6 ) = (14.LS). By inspection. we c m see that (S. 10) = $(14,18) + 1 ( 1. 1). So, r l = utI (&) = i ~ ( d ~ ) + l (1 .1) . In this example, ive were able to niap

n domain block esactly onto a range block. So. the wror is O. This is not true in

SirnplSying Assumptions For PIFS

Observe that given v E Rn then u = Ilr(v) = uZ, wi(&J where d, is a domain

rector in u that best matches the range vector, ri. in if. The output vector u is made

of .\Ir range blocks. Since I V : Rn + Rn then n = MrB.

We make the following assurnptions to make analysis of PIFS easier:

0 n is a power of 2 where n is the size of the vector.

B is a power of 2 where B is the size of a range vector.

The notation d(&, ) ( j ) rneans apply d to the domain block ci,,,, and take the j th ccoordinate of the resulting vector.

18 Review

D = 2 B wliere D is thr sizc of a rloniiiiri wctor.

Dh = B wlirrtl Dh is t h tiorizoritiil stiift distance ht~twen conscciitive domain

I)l»cks. Let c E P. Ttiv j t h iwordinntc of the clornain block of a is given

t y

Tlierefore. the niimbcr of dornnin blocks. dld7 is giwn by:

, b, are paranieters of ru,.

- I is â wctor of s ix B of 1's.

- d(dml) is defineci to I)e sonip spatial contraction from a domain vector to a

range vector. \\é define the j th . for j = 1.B. coordinate of a transformed

domain block (by O ) to be:

6 just takes the average of adjacent domain coordinates to produce a vector

that is the same size as a range vector. as in the previous example.

The proofs of the following 2 lrrnrnas are presented for completeness. The mapping

b t.akes a block of size D x D and outputs a block of size f x O.

Lernma 2 o ïs contractive.

2.4 Partitioned Iterated Function Systems * 19

Proof:

L N rl,,,, ; m l d,,' h~ E D (ie. they ;ire two tloniain rectors of size D).

~ ~ ~ = i ( f ) ? [< lm, (2k) + d,, (Yt - 1) - (&(2k) + dm2(2k - 1)12

Cliwly. 4 is coiitrnrtiw with contractivity of !.

Lernma 3 L d r be a range vector (E 72) O/ size B. Let w(r) = ar + b ( l ) B where

( 1) 1.. a range urct or. O / l 's and ci. 6 are scolnrs such that O 5 a < 1. Then w is

iwntractiue.

Let d l ) and ri2' be two range vectors.

Observe:

= &Li [a(dl))(i) + b ( 1 ) ~ - ( a ( r ( 2 ) ) ( 2 ) + b ( l ) B ) ] 2 definition of w

= JC,B=~ d[(r ( l ) ) ( i ) - ( r ( ? ) ) ( i ) ] *

Since O 5 a < 1 then clearly w is contractive with contractivity a.

20 Review

C \ é need u: t o be contractive so that II' = ~ : , , w i is contractive which means

t h svqtitariw of rt.pr;it.ed it~rations of IIw will converge. Q is contractive to gire the

iniagc detail. \\'ittiout it being contractive. each image in the sequence would appear

griiiriy 131.

Encoding a vector IL by PIFS is done as follorvs:

I . Partition Ir into range blocks. The jt" coordinate of the ich range block is givcn

t )y:

2 . Partition Ir into doniniri blocks. Recall that the j th coordinate of the m:h domain

t~lock is gi~pti by:

(LI, (1) = 14(m - W h + j )

rn, = l..Md and j = l..D.

3. For i = l..ill, do: find ai,bi,mi such that d( r i , wi(dm,)) = d(ri, ai#(&,) + b i ï B )

is minimized. Store the parameters: a, ,biomi.

Finding n,.b,.mt (as in step 3) may not be obrious. The most straightforward

method is to erhaustively search the domain vectors (m, = l..bk) for each range

vector. r , where i = 1 ... II,. For each range-domain vector pair, finding the values for

a, and bi to minimize d(a,o(dmt) + bi , r,) is a least squares problem with our chosen

met ric.

2.4 Partitioned Iterated F'unction Systems 21

Least Squares Solution For A Domain-Range Mapping

For a range vector Z = ( r l . . . . . r s ) and a spatially transformecl (by $) cloniain vector

d= (dl . . . . . d B ) . ae wniit to find a and b to mininiize:

which is the same as mininiizing:

This is a le.?st squares problem 131. We can find optimal a and b by taking the partial

d ~ r i v n t i w s and setting theni to O as follows:

We gct that:

When nfe are done. we espect tha t W ( p ) - * d(CCr(p), p ) is small. Therefore,

by the Collage Theorern. d ( p . A,) is also small where f, is the fixed point of I V . This

means t h a t p is close to f, so 11.' is a good representation of p.

Encoding and Decoding Example

An example tvill help illustrate the theory. Consider the vector:

22 Review

Figure 2.4: Eiicoding/ Drcoding Esam plr

Tihle 2.1 : Range-Dorniiin Block Nappings

KP rirr<l to find a matrhing tloniain block for rl = [23 21 17 191. \Ve will exhaiistively

search the clornain block space. We start with di = [23 21 17 19 11 9 15 131. Notice

that there exists 1 range vectors and 3 domain vectors.

Observe that o(dl) = [22 18 10 1-11. We Vewant to find al and bl such that:

Range Block (r,)

is niinimized. The optimal values for al and bi are: al = f and bl = 12. We must

also compare rl with the other 2 domain blocks. However. it tums out that dl is the

best match. So s e write out (i. 12.1) where the 1 is the index of the best matching

domain block. For this example, see Table 2.1 for al1 the mappings.

Scale (a,) 1 Dornain Block 0.5 j [ z 3 a i ; i 9 1 1 9 1 5 1 3 1

Offset (b , ) 1 Error (rms) 12 O


Decoding

\\é can pick any mctor in ng" alid apply II' to it iterativeiy iintil the clifference

betwecn siiccessive iterations is negligible. The resiilt will be a vcctor close to the

origiiiitl vector that ae cnrodecl (p). I I w can be desrribcd as a seqtirnce of a,. I I , . and

m, values. For i = 1. we know that domnin vector dm, rnaps onto range vector r l .

We also know the pnrnmrters a l and b l . So. we know eeractly how IIv operates over

e x h clorriain block: the iiriion of al1 the transformcd doniain blocks produces a vector

in Rn.

In Table 2.2. w~ h w r drcoclcd the vertor froni the prrvioiis rsnrnplr. O h s ~ r w

that after cach itrration. the resulting vector cornes closer to the original vector p

that we encoded.

Zooming

Siippose iw hiive ri rniipping II" : Rn i Rn rvith a fised point 1'. Let 11'1 be the

same mapping as II*' except that B+ = kB1 (recall that B is the range block size). -

same. CVe can see that I V : IP? -t I P ~ .

Table 2.2: Decoding A Vector

Iteration O 1 3 - 3 4

lrector

[ O O O O O O O O O O O O O O O O ] 112 12 12 1 2 8 8 8 S O O 0 0 4 4 4 4 ] (18 18 16 16 8 8 10 10 4 4 O O 10 10 8 81 (21 20 16 17 10 8 13 12 4 5 2 O 13 12 8 91 [22.2520.2516.5t8.2510.259.514.2512.254.56.252.250.514.2512.258.510.25)

Error (rms) 53.8145 26.8328 12.3288 5.8310 2.7272

24 Review

Theorem 1 Given a PIFS code /or I I " rchrr-e the <ittrwtor /or I I v 1 r s f ' theri the

iittrrictor fit- Il-f . d r n o t d f f . is yicrn by:

Proof: 1 1

\\il m n t to show tliat I I * j ( j : ) ( l ) = f f ( 1 ) for i = 1 . . $ - Recitll:

f 1 ( j ) = j' ( ( 1 - 1 1 B1 + k). i.e. The j t h mortlinatc. is in the i t h range tdocgk at

coorriinate A-.

W1( f ' ) ( j ) = W1( f ')((i - 1)B1 + k ) = (~,o(ii,,~) + h,. i-e. Ne only necd to look

at the ir, for the i t h range block.

Observe:

( ) ) = 1 ( f i - 1 ) + ) Coordinate k in ith range block. k = l..$

= a , p ( d k , ) (k) + 6, 1 t

= a, (d& ( 2 k ) + d$* ( 2 k - 1)) + 6, = ?(a ,db ( 2 k ) + b, + 4&, (5k - 1) + b,)

= $[%fi((, - 1 ) ~ i + 2 k ) + b, +a,f i ( (mi - 1 ) ~ : + 2 k - 1) + b,]

At this point, it is important to stress that the domain and range blocks are taken

frorn the vector fi. ..\lso. notice that we can f i in terms of f l . We continue

2.4 Partitioned Iterated Function Systerns 25

Su j ! is t h Asrd poin t of II*! ;is r t y i i i r t d . O

Thc syinmetric forni of this theormi which goes the other kay is stated below:

Theorem 5 Lrt f f be the attrnctor of l l w j . Then f ' is the attructor of W1 where

Proof:

We want to show that ilv( f ' ) (Z ) = f ' ( 1 ) for 1 = l..n.

W(f t ) ( l ) = W( f l ( ( i - 1)B1 + j)) = aid(dL,)(j) + bi = ai:[f1((mt - l)D$ + Z j ) + fL((mi - 1)Dh + 2 j - 1)) + b;

= ai f !((mi - i)f$ + j ) + bi Proven above

= f ((i - 1 ) + j ) Given in statement of theorem

26 Review

\\'lii~t do t licstl theoreiris rrie:tn'.' Tliey tcll ils how ive can zoom in and out of the

fisecl poii~t of I I - . I f wc tliink in tcriris uf irrxiges thcn the concliision of the second

tlitwrtm is qtiiro nice. It tells 11s how to zoom irito an image. This iipproach generatcs

artifiriiil <irt ; t i i iit mcli zooni Ievrl. Esperiirients revcal that zoorning in with fractal

ri)inprrww(l iiiiiigrs looks good wlirn zooniing irp to i~boiit 8 tirnes. Aft~rwards. the

art ificiid ( i ~ t ni1 is inenninglcss. Conversely. an iniage corii pressed wit h .J PEC requires

pisrl tlriplicat ion when zoorning in. This l e i h tu blockiness in the zoomed image.

2.4.3 Partit ioning Techniques For 2D Images

So fiir. wr-r 1i;ivr-r dividrd I r into unifc)rnily-sized range blocb. Suppose that for a

particiilnr rnngc block. the bcst dorniiin-rarige niatching yields a large error (the

rlistanw is big). If \i.c niake tliat pnrticirlar range block smaller then it is more

likely that a p will find a matching doniain block such that the distance is smaller.

ffuwrvrr. i f 1i.r iisc snialler range blocks then we will need more w, transformations.

This rrrlric~s the compression ratio. but improves the image quality.

.Jko. the size of oiir domain pool. .DD. affects the runtime and image quality. A

large dornain pool means that more domain blocks need to be searched for each range

block. Howewr. this improves the likelihood of finding a good match. Matching

techniques are mentioned briefly in section 2.4.4.

The choice of partition scheme has a significant impact on the level of compression.

Several partition schemes have been proposed: some give more weight to image quality

than others. \\è are not so concerned about image quality as we are worried about

compression rates.


Quadtree Partition Scheme

Tliis piirtirioning schcrnc [1. 3. 9. 151 is clesigned to improve the image quality of

rtir coriipressed imagr by wrying the size of range blocks until a desirable error is

c i . .A qii;icitree partition is a representation of an image using a 4-child tree

structure. The root of the trce represents the initial image. The root node has 4

rliililrr~i. Each child node represents one quadrant of the parent node. The algorithni

is iriitinlized with the tree Iiaving a minimum depth along each path so that the size

of r;ic*ti qii;i(ir;int iri r fie Irai-rs is t lit1 initial range block size. For a given quadrant

r i t o c ) . i f thr I~rsr niatcliing domain block is not good eriough (distance is

large) tlirn dividc the q i i n h n t iiito 4 çhildren and t v again. This process cannot

coritiriiic inclefinit~ly: otherwise. n-e would be encoding each pixel separately which

woiild cause an explosion of bits. Therefore. we enforce a maximum depth on the

t ree. 'i'liis translates into a rnininium sized range block.

Tliis type of partitioriing scherne does complicate the encoding process because ive

ncrd to spccify the size of the range blocks in the output for the decoder. LIoreover,

the spatial transformations of domain blocks also reqiiires careful attention.

2.4.4 Domain-Range Vector Matching Techniques

For ench range block. we must examine al1 the domain blocks to find a good match;

this is a slow process. If ae wish to have reasonable image quality then we need a

large domain pool. tVe want good image quality and fast searching.

Classifying dornain and range blocks is an effective method of improving the

matching process. Each domain and range block belongs to 1 of N classes. We

knuw how earh w, affects the class of a block so we only need to search a particular

28 Review

class of rii~main blocks. There are several dever techniques for clnssifying blocks [Ki).

Orw cl:~~sification scheme involvrs:

1. Briuking ~ ; i c h block irito -1 sub-blucks.

'1. C'onipii ting the average piscl valrie of each sub-block.

Tlir ordrring of the intensity of average pixel values for a siib-block determines the

r-lxss of the block. In this case. t h e are 4! = 2-4 different classes.

Anor lier cliissificat ion sçhenie irivolms annlyzing propertics of blocks to deterniine

if r l i ~ I do& is smoot h o r if it h a ari cdge in it or it could have some other properties[6].

2 A.5 Evaluating Partition Schernes

-4s rntlnt ioned earlier. the partitioning scheme affects the image quality and the corn-

pression ratio. Therefore. stiidyirig the properties of a particular partition scheme

i l 1 t i f 1 Cnfort iinatrly. it is difficult to make strong mat hematical arguments

for a partition scheme. A good partitionirig scheme wiil adapt to different t-es of

images. However. we tvould like to know if one partitioning scheme produces higher

quality images. or higher compression rates, than another scheme. and on what kind

of images does one method 'out-do' another. We have chosen to empirically examine

various partitioning schemes by running thern on various test images and analyzing

the resiil ts.

To try out a partitioning scherne, we need a fractal tool which will accept a

partitioning scheme and an image as input to encode the image. This tool would

allow ils to compare partitioning schemes on specific images. It would also tell us if

particular method works %etter' on certain kinds of images; the word 'better' can

2.4 Part itioned Iterated Function Systems 29

niean sereral different things. It could niean higlicr compression ratio. lowcst average

pixel crror. or highest signal-to-noise ratio.

Review

Part II

Fractal Tool And Partition

Schemes

Chapter 3

Fractal Tool

. I Itwninghil thcorrtical resiilts about a partition schcme arc difficult to obtain; instead.

ire try to rcason eriipirically. The fractal tool serves tbis purpose. I t provides a

franiework for csernining partition schernes. Image partitioning techniques can be

pl i iggd into the framework and tested on sample data.

The fractal tool contains a compression and decompression program. Each par-

t i t ioning nicthod hm different program options which control the behaviour of the

pürtitioning algorithm. Cornpressing an image is a very slow process. For a small

image (128 x 1 2 ) , the runtime of the compressor is under 5 minutes. However, for a

vcry large image the compressor can take several hours.

3.1 The Cornpress Program

Run the cornpress program as follows:

compress C-e error-threshold) [-d dom-level] [-SI

I [-1 nbhd-size] [-r rnd-pts] 3 inf ilename . bmp outf ilename. f rac

34 F'ractal Tool

Typing: ~cornprcss gives a b r i ~ f tlescription of each option and its default value. .A

more (Ir tidecl explnnnt ion follorvs:

error-threshold: The error-threshold is the rnauimiirn error (distance given bu:

<l(<~d+hl,,,, 1 . ) ) ive are willing to accept between a range block and its inatching

transfornied ciorriain blork. >[ore precisely. suppose we have a range block

and a domain block d =

4 n . l . . d2n.2n

The prror 11t~t~rvet.n the range block and the transformeci domain block is com-

piitivl as follows:

Since it i s faster to cornputc Error2. oiir goal is to find range-domain pairs

tvticre Error2 5 error-threshold. That is, the average error of the red, green.

and blue planes must be less than error-threshold. Use 100 for high qual-

ity. 5000 - 10000 for high compression with low quality, +lOOOO for very high

corn pression.

domlevel: Specifies the spacing between adjacent domain blocks. Let D x D be

the dimension of a domain block; where D = 2* for some k 2 2. The spacing

between adjacent domain blocks is given by: hl; where dom-leva1 is O or

1. The number of domain blocks affect runtime image qudity, and compression

ratio.

3.1 The Cornpress Program 35

s: Turns on statistic reporting. The following statistics are reported for each range

block size:

0 Niimber of range blocks without a niatching doiriain block. Constant

colotired range blocks cio not need a matching dornain block.

Number of range blocks with a rnatching dornain hlock.

Average niapping error squared for red. green, blue planes and the variance.

0 Masimiirn and minimum error squared for red. green. blue.

Average distance betwen range and domain blocks.

. - \ v ~ r a g ~ ciifference for qiiantizrd and non-qiiantized error sqiiared calcula-

tion.

a Yiimber of domain blocks searched for ench range block size.

Orientation of matched domain blocks.

.A count of the different qiiantized levels for contrast and brightness.

nbhdsize: Applicable to particular partition schemes. For each range block, we

only search those domain blocks which are in a local neighbourhood of the

range block. If nbhd-size is N then we search an area of 32iV x 32N pixels

around the range block. This takes advantange of local self-similarity in an

image: if it exists.

rnd-pts: For particular partition schemes. For each range block, we search the local

neighbourhood and we also search rnd-pts random neighbourhoods of the same

size as the local neighbourhood.

36 Fracta1 Tool -. - - -

3.2 The Decompress Program

Run tlic derompress prograrn as follows:

decompress [-i num-iterations] infilename.frac

Typirig: .(i~cornpress' gires a brief description of each option and its default value. A

riiorr drr ailcd ~splariat ion follows:

num-iterations: Xuniber of iterations to perform with the iterated function system.

Given a set of rnappings. called F . in the fi le infilename. frac and iinu initial

image pi. We iteratively apply F as fdlows:

The oiitput is pnumilmarimr. The default value for mm-îterations is 7.

3.3 General Details For Block Based Partition Schemes

Block-based encoding schemes behave similarily to achieve high compression rates.

They use similar techniques for writing out range-domain mappings and finding range

blocks.

3.3 General Details For Block Based Partition Schemes 37

3.3.1 Output Of Mappings

A range-clornain mapping is writtcn to the fractal (compressed) file as shown in Figure

red green blue red b lue contrast contrast contrasi brighinrss Kikness brightness orientation dornain id 1 I u u ü ü u u u 4 bits 4 bits 4 bits 6 bits 6 bits 6 bits 3 bits

Figure 3 . 1 : Bit Layout Of -4 Range-Domain blapping

Orimtatiori is only iised for tliose schemes that allow dornain blocks to bc rotated

and r d l w r ~ d . There are 4 possible rotations and 2 retlections for a domain block: 8

possible orientations in total. The domain id is a number that uniquely identifies a

clornairi block. The niimber of bits for the domain id depends on the size of the image.

Diiririg the decompressioii pliwe. the dornain id is used to recoristriict the position of

the domain block. Let:

0 D be the side length of a doniain block.

imgrow be the number of rows in the image.

imgcol be the number of columns in the image.

domrout 5 irngrow - D be the starting row of the domain block; indexed from

O.

d m c o i 5 irngcol - D be the starting colurnn of the domain block; indexed from

O.

rlmnspace = .,,,z,,,, be the spacing hetween adjacent domain blocks tvhert.

dornJecel = 0.1.

The obvious twy to conipiite the domain id is as follo~vs:

wiiere .VDB is the niirriber of domnin blocks in a row. If domain blocks are spaced

1 pixel apart then :VDB = irngcol - D + 1 . However. domain blocks are spaced

tlornspticr pixels apart. Herice the number of doniain blocks t h fit on a row is

rmqrnl- D iirrrnspocc + 1 . Thereforc. we have that:

Observe that timnrou: mod domspace = O and domcol rnod domspace = O because

dornain tilocks are (lonrspnce pixels apart. Thercfore:

(dom r ow ( irngcol - D moddomspace = O (*)

domspace

Recnll that domspace = , , ~ e v e i and that D is a power of 2. Therefore domspace

is also a power of 2. Le. domspace = 2' for some integer k. Therefore (*) implies that

the 1st k bits of domrou, 'zzit ( ( + 1 + domcol are always 0. It is not necessary 1 ) to write out those bits. So Ive compute the domain id as shown by Equation 3.1 [12].

domspace

The maximum domain id is giveo by Equation 3.2.

(imgrow - D) ( tmgcd- D domspac~ + (imgcol- D)

maxdomid = dumspace (3-2)


Equation 3.3 gives the number of bits needed to represent the domain id.

The following lernnia rigorousiy j (1st i lies why wr cari use Eqtiation 3.1 to iiniqiiel~

ideiitify a domain block.

Lemma 4 Giuen a D x D domain blocbk starting ut position ( r , c) in a imgrow x

irngcol image. The domid (as in Equation 3.1) is unique umong al1 D x LI domain

biocks.

Kecail that domain blocks are incrernented by domspace: therefore. r mod domspace = O

ancl c mod dornspnce = 0.

.Assume domid is not unique; therefore, there exists distinct D x D domain blocks

at ( r , , c l ) and ( r 2 , c2) having the same domid. Notice tha t r l , r . ~ 5 imgrow - D and

c l . c2 5 imgcol - D (t). Therelore:

- - imgcol- D + l) + dm?pace

=2 - C l domspace durnspace

(dom?pacr - r f L p a c e ) (~"p: + l ) - - domspace - dornspace ($)

If rl = r2 then clearly cl = cz which contradicts our assumption about distinct

domain blocks. Suppose rl > r2. Since rl, rz mod domspace = O and r , > r2 then

Therefore, by ($)

40 Fkactal Tool

wliich irript ies t hat

(-2 - cl > irrtgcol - D + domspace

Howcwr. c2. CL <_ irngcol - D by (t) nhich implies that c? - cl 5 imgrol - D. Hence

w b;iw ir contradiction; t herefore. rl = r2 and cl = c2. So the domain id is unique

for doniain blocks of sizc D. 0

T h size of the domain block. D. is not specified in Figure 3.1. The details of the

piirtitiori sdierne will clarify how ivtl determine the size of a tlomain hlock.

Choosing a Domain block over 3 planes

The fractal tool chooses a single partition scheme for al1 3 planes as opposed to

part itiorii ng t lie red. green. and blue planes independently. The frac ta1 tool minimizes

the awragc niapping err& within the error threshold over the 3 planes when looking

iit rangedomain pairs. If the 3 planes were partitioned separately then the image

qiiality woiild certainly be iniproved. However, the compression ratio would sufier in

most esamples. Lemrna 5 provides some justification why we take the average of the

3 planes.

Lemma 5 Using a fized error threshold. let:

.YR be the number of mappings for the tzd plane

.k be the number of mappings jor the green plane

YB be the number of mappings for the blue plane

Vd4 be the number of mappings for the average of the 3 planes

3.3 General Details For Bkock Based Partition Schemes 41

Proof:

\\'r M i n e the partition sclienies for rach plaiie. let:

PR bfi the partition scticrne of the red plane

Pr: be the partition schenie of the green pliiiie

PB be tlic partition schemc of ttie blue plane

P.., be the partition srlimir of the average of the 3 planes

Let r , t ~ e a k x k range block in P.., ancl d, is t h r corrcspoiiciing 21; x 2k matching

rloninin t~lock for r,. Therefore. e.., = d(r, . a,(!, + b, lk, k ) < errer-thr~shold. Since e.4

is t lie average errer of the 3 ylaries then some other plane nitist have a niapping error

of a t l e s t the sanie size i i s e+.l for ttiat range bIock; or ttiut range Llock is'broken

down into 4 quadrants on some other plane. So. for r , there are eithcr the same 3

ïilrigc t>locks iii PR, PG, PB or at l ~ a s t one of the partition schemes ail1 have that

riinge block broken down into 4 quadrants.

Clearly XR + .VG + XB > X.+ 0

Lemma 5 says that the niimber of mappings for the average of the 3 planes is

always less than the sum of the nurnber of rnappings for the individual planes. I t

does not make a general statement about the compression ratio for taking the average

mapping error. Consider the following contrived example: Suppose we have an m x n

image made of 3 planes: , 1 , p . Where pR[ i , j] = c and pc[i, jl = d Vi < m

a n d V j < n where co d are constants. Every range block in the red plane would have

maximal size and the rnapping would look like: O&) where O indicates O bits for

tlie con t ra t and c(q) is the brightness which is the quantized value of c. Sirnilarily,

mappings for the green plane would look like O&). However, suppose the blue plane

is very complex and it aeeded many range blocks; hence many mappings of the form:

42 Fractal Tool

r f h f r i , where n l indicat~s the contrast for the l t h range block and B intiicates the

hliie plarir: h l is the hrightriess. Let N R . Xc. and :VB be the number of mappings

on eacti plarir and 'i, is tlie n imber of mappings for the average of the planes. We

; i l n d y kriuw t h .YB >> .YR. &. If ttie mors for the Mue plane are ver- Iiigh then

.Vg of these mappings which means we are writing out 0c(q) and 0d(4) many more times

t han necessaru. Ir1 t his case, comprrssiiig t hc 3 plancs inclependent ly would resiilt in a

Iiiglier conipr~ssioii ratio. Also. tlie nuriil>er of range-domain mapping computations

is less if the plitries are coiiipressed indepentlently. This probleni only occiirs wben

one or two of the plancs cari b~ coniprmscd iisiiig vcry few range blorks iirid nnoth~r

plane n~ecls rnany range blocks.

3.3.2 Finding Range Blocks

Deterrnining an efficient organization of range blocks for an image is the hard part of

any partition scheme. One technique is the qiiacitree partition. mentioned in section

2.4.3. A pseudecode algorithm for perfect square range blocks (side length is a power

of 2) is given in Figures 3.2-3.4.


DivideImage(int image-start-rou, int image-start-col, int image-num-rous, int image-num-cols)

if image-num-rous ! = O AND image-num-cols != O then ( p = smallestPowerQf2( min(image-num-rovs, image-num-cols) )

TraverseImage ( RangeBlock(image-start-rou, image-start-col , p) )

image-start-col,

P) image-start-col + p, image-nu-cols - p)

Figurr 3.2: Divide image into perfmt square regions

Lemma 6 Div ide Image (Figure 3.2) breuks mi arbilrarily sized image into several

disjoint rtyiorts. Each region is a perfect square. The union of the regions covers the

mtire image.

Proof:

By induction.

Show the result is true for a 1 x 1 image.

Clearly. TraverseImage is called on a range block starting at position (0,O)

with 1 row and 1 column. So the image is covered by a single region that is a

perfect square.

Hence. the result is true for a 1 x 1 image.

Assume the result is true for al! images with dimensions smaller than r x c.

..\ssurne that TraverseImage is called on disjoint regions where each region is

2k x 2'. and the union of the regions cowrs the image.

44 Fkactal Tool

O Proz.~~ the rcsirlt ~ . s tnrr fin- mi r x c mage.

()hsrrvr that p = ~ i l l > ~ ? i m l n ( r + c ' ) ! < - ~ n l n ( r , C )

So. we c;ill TraverseIrnage startirig at (0.0) with p rows and p columns: hence

n . ~ Ii;iw rovrrt4 a prrfert sqiiare region of the image.

\fé nrr I ~ f t witti 2 srrialler disjoint images:

- The imagc starting at (p. O) with r - p rows and p columns.

- The i n i a g ~ stsring at (O. p) with r rows and c - p columns.

By t h p induction hypotlit%sia. ive c m cover the 2 smoller images ni th disjoint

square r~gions. Htm-e tlir entire imagc can bc covered by disjoint square regions.

K t \ stio~wd the result truc for 1 x 1 image. and when we assume the result

triie for i i l l images a i t h dimension smaller t han r x c then we showed the result

t rue for any r x c image. So. by the Principle Of Y at hematical Induction, the

rrsiilt is trtie for al1 images.

O

Clcarly. since the range block passed into Traverse Image (Figure 3.3) is a perfect

sqiiare t hen ench recursive cal1 of TraverseImage also passes in a range block with

prrfrr t sqiiare diniensions. TraverseImage cornpletely covers its region by perfect

squares. Sloreowr. each covering square will be the same size.

Observe that ExamineRangeBlock (Figure 3.4) outputs a O bit when it breaks

down a range block. This is how the decornpressor keeps track of the size of range

blocks and the corresponding domain blodrç (always twice the size of the range block).

The FindBestMatchingDomainBlock function searches through dl possible domain


TraverseImage(RangeB1ock rb) if rb.side,length <= MAX-SIDELENCTH then <

ExamineRangeBlock(rb) ) else (

Divide rb into 4 evenly sized quadrants; cal1 them rbl, rb2, rb3, and rb4. TraverseImage (rbf ) TraverseImage(rb2) TraverseImage (rb3) Traverse Image (rb4)

Figure 3.3: Breakclown the pcrfect square regions into sizeable blocks

th- l is to fincl the hrst match for the given range block. Best match means lowest

tbrror. Swtbn 3.3.3 disciisscs irnprovenients to FindBest Mat chingDomain0lock.

3.3.3 Block Classification

Srnrrhing al1 possible domain blocks for each range block is a very time consuming

prowss. For an m x n image. the nurnber of domain blocks. of side length D. nhich

n e d to be searched is giwn by:

( m - D + l ) ( n - D +L) dmnspace domspace

Notice that r n a x d m i d = (e + 1) (e + 1) - 1; where mazdomid is

defined in Equation 3.2.

;\ssuming range blocks are half the size of domain bloclis and we cover the image

with a uniform range block size then the total number of range blocks is given by:

46 Fractal Tool

ExamineRangeBlock(RangeB1ock rb) db = FindBestMatchingDornainBlock(rb) if MappingError(db, rb) c error-threshold then (

Output a 1 b i t t o indicate tha t a mapping follovs. Write out the mapping

) else ( Output a O b i t to indicate re-sizing of range block. Divide rb into 4 evenly sized quadrants; cal1

them rbl, rb2, rb3, and rb4. ExamineRangeBlock(rb1) ExamineRangeBlock(rb2) ExamineRangeBlock(rb3) ExamineRangeBlock (rb4)

1

Figure 3.4: Examine each block and try to find a mapping for it

Thrrrfnrc the total number of domain blocks searched using only 5 x 4 range

If we xssiinw a qiiatltree partition thcn range blocks are not al1 uniformly sized. It

is possible to hnw a range block and not find a reasonably good matching domain

block. In this case, we break the range block into 4 even quadrants and perform

nnother domairi hlock search for each nen range block. The number of domain blocks

to search is incrrnsed by a constant factor of 4 per range block. A range block may

need to be divided several times before a good match is found: the constant can be

qiiite expensive.

Searching through al1 possible domain blocks is time consuming. Instead, we

classify r~mge blocks and domain blocks by their pixel characteristics. Given a range

block in a certain class, we only need to search those domain blocks belonging to the


snme class. Rather tliiin having one long list of domain blocks we have ari array of

lists of doniain blorks mhere each list contains domain blocks belonging to a particular

rlnss. We ctepend on a random distribution of block classes for an efficient search. It is

siniiliir to the way a hash table depends on a good hash function to evenly distribute

k c ~ s . Our hlock classifier is analgous to a hash function.

I.hfortunatcly. sonie blocks can be mis-classified which means the optimal match-

ing ilornain block for a particular range block may be in another class. It is up to the

dassifi~r on Iiow to Iiandle this problem. One solution is to alloa blocks to belong to

rriiiltipk cnli~sscs. or rhoosr a better classifier.

Soriie classification rnethocls are mentionert in section 2.4.4. The fracta1 tool clas-

sifirs blocks ncrording to the ordering of the sum of pixel values of each quadrant

[ 151. More prrcisely:

Givcn a block B. whose quadrant p ~ ~ e l sums are indexed by B[O], . . . , B[3] .

We define the following function:

( O otherwise

The class of a block B is given by:

class(B) = 6[f~(O.l) + f~(0,2) + /~(0,3)]+ ?[fl3(1,2) + f ~ ! l * 3)]+

f~ (273)

Lemma 7 Given two blocks -4 and B.

class(.-l) = class(B) o (.q[z] > A[j] o B[z] > B[j]Vi, j )

Proof:

If .-i[i] > A[j] e B[i] > B[ j ] V i , j then

48 Fracta1 Tool

j,.\(i. j ) = 1 .4[2] > A[;] * B[i] > B [ j ] * f ~ ( 1 . j ) = 1.

fq4( i . j ) = û - A [ i ] 3 .i[j] * B[i ] 3 B[jl * j&. j) = 0.

Ttierefore. it is clear tliat clnss(.-l) = class(B).

Iwratise every number from O to 23 can only be written in exactly one way 6((0.1,3,3))+

? ({O. 1 . 2 ) ) + I({O. 1))-

()k)servc:

Sincc 3) = fB(2. 3) then .-@] > A[3] B[2] > B[3]

If /.-l(l. 2) = O and f s ( l , 2 ) = 1 then it must be that jO4(1.3) = 1 and fB( l ,3) = 0.

Therefore. .4[1] # .4[2] but B [ 1 ] > B[2] and A[l] > A[3] but B[1] < B[3].

Hmce A [ 2 ] > .4[3] * f&3) = 1 and B[2] < 8[3] * f B ( 2 , 3 ) = O. This is a

contradiction. Hence it cannot be that fA(l? 2) = O and f B ( l . 2 ) = 1 .

Sirnilarily. fa&. 2) = 1 and f B ( l , 2) = O is not possible. They are both O or both 1 .

This means that A[1] > A[2] B[1] > B[2] and A[ l ] > 431 B[1] > B[3].

The same is true for A [ O ] > .1[1] o B[O] > B [ l ] and so on.

Therefore, A[i] > Ab] B[i] > B(j] Vi, j O


Let f a c t o r COI =6. f a c t o r (il =2. and f a c t o r [2] =l. Therefore:

r~ln.s.s(.-l) = fnctor[O] -[I..l(.i(O.i) +/ . . \ (0 .2)+ f.-,(0.3)]+

factor[l] . [f,-\(l . 2 ) -+ f..\(i. 3)1+

f tzctor [2] . [Je4 ('2.3)]

= fntfor[O] - f . . l ( O . l ) + fnctor[o]- f.4(0.-)+ffzctor[0]. fa4(o.3)+

factor [il . ja4 ( i ? 2) + /uctw[l] /f( ( 1.3) + f rrctor[Z] . f,@. 3)

A n qu ive lan t ;ilgoritlirniç cornpiitntion of c~lass(.-L) is sfiown in Figiire 3.5.

temp = O f o r (i=O; i<=2; i++)

f o r ( j = i + l ; j<=3; j++) temp = temp + f a c t o r Ci] * f -{A) (i , j) ;

c l a s s = temp; return c lass ;

Figiire 3.5: Algorithm to cornpute the class of block .A

temp = O for ( i = O ; i<=2; i++)

for ( j= i+ l ; j<=3; j++) i f ( A r i ] > A [ j ] ) then temp = temp + factorCi];

c lass = temp; return class;

Figure 3.6: Slightly rnodified algorithm to compute the class of block .4

Figure 3.6 is slightly different. but still equivelant to Figure 3.5. The fractal tool

cornputes the class of blocks using the code from Figure 3.6.

50 Fracta1 Tool

3.3.4 Quantizat ion

Tlit. ~.oritrnst ( ( 1 , ) and brightness ( 6 , ) vi i lu~s for a range-domain mapping are floating

point riunilwrs: rach one rcqiiircs 3'2 bits of storilge. Therefore. to dwxibe any range-

doniain rrinppiiig woiild reqiiire 3 - 32 + 3 . 32 = 192 bits plus some cstra bits for the

cloniaiii id (see Figure 3.1. Sincr 1 pixel takcs 24 bits tlien a range block niust have

at least [F] = 8 pixels to nchieve compression. Sirice the side lengths of blocks

niiist t,r ;\ powr of 2 ( i t i the fractnl tool) then we would have to use range blocks

witli 16 pixrls in ttierii to avoid explosiori. Ttiis woiild resiilt in low qiiality images.

IF wr qiiantizc* th<. roritriut and hriglitnrss [12] v;ilues so they reqiiire 4 and 6 bits

r ~ s p w t i d y t hrn a single niappirig woiiltl tiike 3 - 4 + 3 6 = 30 bits plus the domain

id. Tu tir-hirve comprt~ssion. ü range block riiust have at l e s t 2 pixels. Qiiantizing

allows the partition scheme to use smaller range blocks whilc still cornpressing the

data for eaîh range block.

Cnfortunately. this estra çonipression does not corne for free. Larger errors per

range-doniain mapping are a side-effect of quantization. ie. d( r , , quant (44 + quant ( D i ) l,,,) >

d ( r i . ci,d, + bi ln,,). The fractal tool performs uriiform scalar quantization on contrast

and brightness values as shown in Figures 3.7 and 3.8.

9'4-1 3 return q

Figure 3.7: Quant ization code


dequantize(qvalue, max-value, num-bits) dq = (qvalue/(2~(num-bits) - 1)) * max-value return dq

Figure 3.5: Deqiiantization code

Tiie qiiantizcr breaks iip the intemal [O. mur-rcrluil] into 2nUm"t' + 1 equally spaced

values; lience 2"U'nmts intervals. The intervals are cnunierated starting froni 0' so the

tiighcst riiirrlbcrcd interval is z " ~ " ~ ' ~ " - 1. \;dues get mnpped to tlieir enclosing

ir~trrvd i~icles.

Quant ka t ion Wit hout F'ractals?

One rriight woiid~r if i t makes more sense to simply cluantize individtinl pixels and

&op al1 of this fractal stuff. Consider a 512 x 512 image using 4 x 4 domain blocks.

By Equation 3.3. the number of bits needed for the domain id is 14 (doniain levei

O ) . Therefore. by Figure 3.1. a single mapping would require 48 bits. If we quantized

every pixel to be 12 bits (ie. 4 bitslplane) then the average pixel error in one plane

L 256 would be given by: 5 (T - 1) = 7.5 units. So, if an image is broken into only 2 x 2

range blocks (hence only having 4 x 4 domain blocks) then the compression rate would

be the same as a 12 bit/pïuel quantizer. However. the fractal scheme does have a

lower average pixel enor for the compressed image.

Although? specific examples do exist where a simple quantizer does better than

a fractal encoder rising the partition schemes which we have chosen to examine. An

image consisting of ma- vertical lines one pixel apart is an example of this problem.

52 Fracta1 Tool

3.4 Partition Schemes

Tlic fractal tool is p;ick;tged witii four quad t ree-li ke partition scliemes:

schemel llost simple techiiiqii~. Range t~locks haw power of 2 side Itjngtiis. aiid

riorriairi blocks tiaw side lerigtli rxactly trvire the size of their rnatching range

thck . Other notable featurcs are:

a Dornain blocks have only one orient. t' ron.

a For a givt!n rarigc block. the entirta imagta is sr;irchcd for domain blocks.

Blocks are classified nccorcling to the ortlcr of the sum of their pixels in

e:irli quadrant. Thertl are 2-4 hloc*k classes (set. Section 3.3.3).

scherne2: Slight ly more üdcinred t han the previous niet hod. Allows for different

transformations on a ciutriairi block. .

Domain blocks have 8 possible orientations. Reflections and rotations are

al lowd .

For a Aven range block. the entire image is searched for domain blocks.

Blocks are classified according to the order of the sum of their pixels in

each quadrant. There are 24 block classes (see Section 3.3.3).

scheme3: L'seful for examining local similarïty in an image. Xlso allows us to ex-

amine the effect of the classifier.

Domain blocks have 8 possible orientations. Reflections and rotations are

allowed.

For a @en range block. CVe search neighbourhoods for domain blocks.

3.4 Partition Schemes 53

Tliere is only 1 block chss.

scheme4: Examines loral siniilarity witli the use of the classifier.

Dornain blocks have S possibk orientations. Reflections and rotations are

al 1owec.i.

0 For ;i given range block. \Ve scarch neigtiboiirhootis for dornain blocks.

e Blocks are classified nccording to the ortler of the siim of their pixels in

cilch quaclrnnt. Tliere are 24 Mock classes.

sclieme5: Examiries loi~il siniilarity like schemc -4. hi i t ;illows bloclis to bclong to

more than one r-las.

Doiriain b1oi:ks have 8 possible orientations. Reflections and rotations are allo~vcd.

For a given range block. We senrch neighbourhoods for dornnin blocks.

Blocks which are close to another block class can belong to both classes.

54 Fractal Tool

Chapter 4

Sample Test Data

The sarnpie images whirh the esperiments are based on corne from various standard

image rcpositori~s fotind ori the InternetL and from ciistom-made images. Al1 test

images are 24 bits per pixel bitmaps. Xone of the test suites cover a broad range of

sample data: in pnrticuiar. there are fea cornputer generated images in the standard

test data. Xlso. thert. is no emphasis on symmetry and local similarity in the test

data. Howevcr. the stanclnrd tests do highlight typical real-wvorld data. ie. images

one might take with a cariiera. Using sample images. we will identify whicb partition

scheme is best suited for a pa,rticular type of image.

The location of the test images is given in the table below:

'Repositories can be found at:

a Waterloo Fractal Image Compression Project. (http://links.uwaterIoo.ca)

a JPEG Test Images. (htt p://~w.geocities.com/SiliconVdley/Lakes/~/test-images)

University Of Southern California Signal and Image Processing Institute. (ht tp: / / s ip i .usc .ec iu / sert - ices /databVe.html)

56 Sarn~le Test Data

arc t ichare tiorisr

t iii1)Oori Q X t d l

('itt

Icna pool w a k r fruits 2.1.0:3 2.1.1 1 2.2.11 " '1 1.5 -.-.

4.1.01 -1.1 .O3 4 2.05

hi i i chg 1 briilding;3 biiilding-l building5 building6 p m ~ n 1 pers0112 person3 blostein degg

PePPers Serrano frymire monarch

sail tulips

ht tp://aww .geocities.com/Silicon\.'alley/lakes/6686/ test-images lit t ~~://sipi.tisc.eclii/s~rvices/datsb~w/Database. html tit tp:/ /waw .geocit ies.coin/Siliconl*alley/Lakes/6686/test-images lit tp://liriks.uwaterloo.m ht tp://links.uwaterloo.ca http://Iinks.iii~atcrlon.ca k t p://links.uwaterloo.ca lit t p://links.uivatcrloo.ca Iit tp://links.ti~vaterloo.ca lit t p://sipi.usc.~clii/si~rvices/database/Database.litiril lit t p://sipi.iisc.~clii/scrvices/<iatab,uc/Database. htnil lit tp://sipi.usc.rclii/st~r~ices/rlatnb.?se/Database.html lit t p://sipi.usc.t~dii/s~rvices/clatnbnse/Dati~bi~e.html ht t p:/ /sipi.usc.ed~/scn.ices/database/Database.hti~il ht t~~://sipi.iisr.eciii/srwic~s/database/Database. html http:/ /sipi.iisc.cdti/sen.ires/<latabase/Dat~e.htnii ctistorn custorn custoni custom ciistom custom custorn custoni custom ht tp://links.ii~vaterloo.ca ht tp: / / ~ v ~ w . g e o c i t i e s . c o m / S i l i c o n v a l l e ~ http://links.uwaterloo.ca http://links.uwaterloo.ca ht t p://a-nnv.geocit ie~.com/SiliconVal1ey/Lakes/6686/test-images ht tp://nwtv.geocit ies.com/SiliconValley/Lakes/6686/ test-images ht tp://wiiw .geocities.com/Siliconvalley/Lakes/6686/test-images

Table 4.1: Location of Test Data

Chapter 5

Cornparison of Partition Schemes

Comparing partition schernes empirically requires tools to compare the *likenessY of

2 iningrs and to quantif? compr~ssion. Measuring compression is quite emv; it is

conip~itecl as a percentage of the original file size. b'e can measure compression

ratio as "omprrssed 'le This tells us how rnuch space a compressed file uses original file size

cornpar~d to the original file size. We could also compute compression ratio as

1 - This tells us how rnuch srnaller the cornpressed file is corn- original file size

pared to the original file. \Ve use the latter measure throughout the thesis. Several

tools esist for measuring the similarity of two images: signal-to-noise ratio (SNR),

Peak SSR (PSYR), average pixel error. or the distance between tao images (as de-

fined by some metric). If r, is the original signal (image) at position i and yi is the

compressed signal at position i then:

58 Cornparison of Partition Schemes

Tlie wrrage pisel m o r nieaslires only the noise in the compressed signal. SXR

rntwslirrs t hc r~ l i l t i w niagnit ut de of the sigrid compared to the iinc~rtainty in t hat

signal (rioisr) on a per pisel basis. The PSKR rneasures the noise relative to the peak

signal ! 161.

h f o r t iinntely. one mensiire t iiat considers both compression ratio and image simi-

lari ty does iiot esist . Honever. the tno rneasures are related: we can trivially cornpress

;in irniigt. niaking it al1 I~lack (y, = O) . but the SYR would be very lowl. For fractal

i.oiiil)riwit ,ri. t 1 1 ~ type of imagc cmmbine(1 wit li n particrilar partition schrrrie and the

;il1o\v;iIilt~ Ior;il ( m o t (irt~rniines the rate o f rumprrssion and the comprcssed images

l i k m r s ri) l l i ~ original image. Lifortunately. cletermining a çliiss of iniagrs which

I>r~liavr siniilarily iinder a particiilar partition scheme is elusive.

.\Il tiw sc-iicines liehawd similarily whcn it cornes to compression ratio. The

difff3rmc.c br t w e n the best compression ratio and rvorst compression ratio r a s less

than 5% for al1 images: for most images it \vas less than 2%. The compression ratio

wrsiis the rrror thresliold curve has some interesting patterns ahich depend on the

image.

For ver' high quality compressed images (local error threshold is O to 1000), al1

partition schernes must break the image into many small range blocks2. This implies

that d l partition scliemes will use the same number of mappings: therefore. they will

have similar compression ratios. Furthermore, at this level of detail, domain level O

(full-width domain spacing) has a better compression ratio than domain level 1 (haK

rvidth domnin spacing) despite the fact that domain level 1 h a a larger domain pool.

This is because the local error threshold is so low that even at domain level 1 large

'The S9R wouId be 1 'Cornpressing any image using a local error threshold of 100 wilI remit in a partition consisting

almost entirely of 2 x 2 range blocks.

Cornparison of Partition Schemes 59

range blocks cannot be rnatched so the partition sclieme must divitle the image into

niinirnally sized range blocks3. However. writing out the id for a doniain block using

rloni;iin lei-el 1 rcquires at rnost 2 extra bits wer (loniain lerel O (this ran be seen by

relerring to Eqiiation 3.1 and observing that domspace is equal to D or 2). Since

t hc nuinber of mappings for domiiiii level O and levd 1 is the samr (sep .\ppendix

C) then it is clear that doniain level 1 will require more space. However, we are also

concernrd with iniagc quality. At t his error thresiiold. there is an abiindance of smail

range blocks at cfoniain lewl 0; thereforc, the sim of the domain pool is large which

implies the likelihood of a good match% high. Hence. the compressed image quality

at doniain Ievel O is close to tlie image quality nt domain level 1 (see Appendix E):

~Itbspite the fart thii t domain lrvel 1 dors have 4 timrs ns niany (lotnain blocks as

dumain level O.

For liigh quality compressed images (local error threshold is 1000 to 7000) the

clifference betwcen doniairi level O and domain level 1 is more significant. We can

see that the niimber of range blocks used by dornain level O is gea ter than tlie

niimber used by domain level 1 (see Appendix C). Therefore domain level 1 uses

fewer mappings; this directly affects the compression ratio. However, domain level 1

requires more bits per mapping for the domain id. These 2 properties of the different

domain levels seem to cancel each other out; leaving ils with domain level 1 and

domain level O having approximately the same compression ratio (see Appendix D).

The SNR varies depending on the scheme and the image being compressed (discussed

later).

Low quality images (local error threshold greater than 7000) cornpress equally well

3For this application, minimally sized range blocks is 2 x 2. IWe define a good domain-range match to be a rnatdùng whose error is below the local error

t hreshold.


for domain level O as they do for domain level 1. The rcason is the same as before.

Doniain level 1 uses feaer range blocks thnn domain level O. Oiie might wondcr if it

is worth the extra work (coniparing 4 times as rnany blocks) to use domain level 1 as

opposed to domain level O. If the SNR for domain level 1 is higher than dotnain level

O then the extra aork c m be justified.

This news is a little disappointing. In terms of compression, domain level O is a

bctter choicc than dornain level 1. The SKR value depends on the partition scheme.

LVe tlisciiss t his for each partition schcme separately.

5.0.1 Scheme 1

Compression Ratio

Domain level. 1 and domain level O are almost eqiiivelant in terms of compression

ratio. as discussed earlier. However. we should observe that some images compress

noticeably better than others using this scheme. The images with the worst com-

pression a t the highest error threshold were: serrano, frymire, clegg, and building3.

Three of those images (serrano. frymire. clegg) are cornputer generated (not wry

natural scenes). Further tests reveal that images with text and many straight lines

do not compress well: or if they do then the quality is terrible5. This is because

straight lines do not align nicely across range block boundaries and perfect lines do

not agree with our mode1 of self similarity. In particular, scaling a domain block by

half does not produce crisp lines. Other images did compress well. At the highest

error threshold, compression ratios of 95%6 and higher are common. At the lowest

'The SNR is low compared to other images. 695% means that the compressed file size oceupies 5% of the original size. ie. 1 - '-!r;':$f


rrror rhrcshold. compression ratios of 80% are typical. Images which compress very

wrll soriietimcs have dimensions which are a power of 16 (an implementation detail).

Thcy ;ilso have properties of symmet ry embedded in themselves. Furthemore, they

do iiot have many quick transitions betaeen colours (edges); instead, there are smooth

t r;irisit ions.

Irnagrs whirh have dimensions that are a power of 16 can be divided up in large

tilocks aithout having 2 x 2 blocks left over. This is partly why the irnplementations

which WC prrserit compress those images well. Images with symmetrical shapes (like

Imiltliiigs) or reflective properties (rnirrors. water) seem to have high self-sirnilarity

propcrties. .-\lthough. riot every synimetrical image compresses well. Those images

ttiat agrcr witli Our definition of siniilarityï compress well. For example, an image

with wenly spaced vertical lines is symmetrical but it does not cornpress well with

our rhoicr of transformations. An image with many sharp colour transitions will not

necessiiry compress poorly; however. if many of those transitions occur on an odd

pixel boundery then that transition will lie inside of a domain blocka. The problem

is that when ae shrink the domain block by half, we take the average of adjacent

pixels. This causes sharp transitions to become bluny. Moreover, it often incurs a

large error aloiig sharp edges which means that part of the image must be broken

down into srnaIl range blocks. On the contrary, very smooth areasg compress well.

By looking at Appendix Dy we see that this partition scheme does not have a

'based on the definition of our choice of contractive map. "ecause domain bloclcs have dimensions that are a power of 2 and images must have even

dimensions El

%mooth means uniform colour acros the block. ie. z, for any j, where zj denotes the j t h pixel in the block.

62 Cornparison of Partit ion Schemes

ver? high compression ratel0 when the local error threshold is lowLL. However, if we

i t l low i;i high loriil error threshold tlien we can achieve a rate of conipression that is

desirable. At tliis level, ive are only concerned that we can out-perform .JPEG1' and

t liat the image cari still be recognized. With this partition scheme, the highest error

thr~stiold still produces a recognizable image.

SNR

The SSR is n o t so simple when we compare it to domain level 1 and domain level

11 ~ t i tiic sarne partition scheme. For many images. the SNR is nearly equivelant

whct t i t~ ive iisc? domain level 1 or 0 (see Appendir E). However. there are some

itiiiigvs whtw tlie two ore not the sarne. For instance. if we look at arctichare. we can

o t ~ s ~ n - e a significant difference between the SNR for domain level 1 and O at higher

error thresholdst3. At low error thresholds, the size of range blocks is small whether

w~ use domain lewl O or 1. so we are likely to find good matches. Hence. there

is l i t t le differcnce between SNR of dornain Ievel O and 1. As we increase the error

threshold. the size of the range blucks and the corresponding domain blocks is large.

This means that the supply of domain blocks is less abundant; so anything we can do

to increase the might create better15 range-domain matches. The distance

between the SNR curve for domain levei O and domain level 1 increases as the error

threshold increases for some images (buildingô, house). It is difficult to determine

' o ~ ~ % is soniething we expect fiom JPEG; when we think of fracta1 compression we should expect 90% compression

' Belon- 7000 12Higher compression ratio and SNR l 3 It is interesting to note that the dope of the curve is almost O as we inuease the error threshold

beyond 7000 for this image, but the dope of the compression ratio curve is sirnilar to other images. 14Such as use domain level 1 LS Better in this case means Iower local m o r threshold for the range-domain rnapping


which images will have a higher SNR with domain level 1 as opposed to domain level

O

There were a few surprising results regarding SXR. For instance. there were a few

images wliere the SNR for domain level O was higher than domain level 1 (frymire).

These anomalies are rare. The most surprising resiilt is that some images (2.2.15,

22-11) had an increasing SXR (curve had positive slope) as the error threshold in-

rreased! This is also rare and difficult to explain. Both images are satellite images

and at a first glance appear random. They are not very structured images.

Some images do yield higher SNR using this partition sclieme than other images.

For instmce. images involving people and faces have the highest SNR (41.03, blostein,

personl. person2, person3, 4.2.05. cat). The fine details of the face may not be

arctirate. but most of the rest of the image is faithful to the original. The relationship

between SNR and compression ratio is not obvious. images with very low SNR do

not necessarily compress well or poorly in al1 cases. For instance, some images which

did compress well (pool) had relatively low SNR compared to most other images.

However, there were quite a few images which did not compress well and they had

the lowest SNR ( fiymire. serrano, baboon) .

The majority of images behave equally well with domain level 1 as they do with

domain level0. For this partition scheme, justihring the extra work needed for domain

level 1 is futile. Only on rare occasions d o s it seem useful (only in terms of SNR) to

use domain level 1. Unfortunately, it is difficult measure this beforehand.


5.0.2 Scheme 2

For a particular range block. scherne 2 considers more possible matchings than scheme

1. This is because schenie 2 allorvs a domain block to be rotated and reflected; each

domain block has 8 possible orientations.

Compression Ratio

Scheme 2 is sirnilar to scherne I in thnt domain level 1 and O are almost equi~pelant

in terms of compression ratio (sec Appendix D). We can sce that domain level 1

reqiiires f e w r range blocks than domain level O (as before - see Appendix C), but a

rnappine, with domain levrl 1 requirrs more bits.

SNR

Schcnic 2 behaves quite differently in terms of SNR cornpared to scheme 1. At

particular local error thresholds. the SNR for some images witli scheme 1 is slightly

higher than scheme 2. Since scheme 2 finds more matches for larger range blocks than

sclieme 1 then in some instances the local error for that rrgion of the image to be

better in scheme 1 than scheme 2 because scheme 1 would have broken the range block

down into smaller quadrants. An example will help to illustrate this point. Suppose

there exists a range block, r, in scheme 2 whose rnatching dornain block must be

rotated. In scheme 1, no such matching exists for r; therefore, r is broken down into

4 equally sized sub-blocks: r!'), ri1), ri1), and ri1). It is also possible that sorne of riL)

will be further broken down into rl(f), r!:), r,(?),rif) . Since scheme 1 requires a refined

partition then it is reasonable to expect that it may have a higher SNR than scheme

2: particularly a t higher local error thresholds (see Append~u E).


At lower error thresholds. the SXR for scherne 2 does at least as well as scheme

1 for almost ail images. This is because both schemes must use a fine grid for the

partition schemc: and since scheme 2 hi= more dornain blocks to choose frorn then it

is more likely to find a better match.

Some images siich as personl. pool. fruits, arctichare. building3 have an improved

SNR using scheme 2 as opposed to scheme 1. It is interesting to note that these

images use many domain blocks that are rotated 180" and reflected horizontally (see

Figure 5.02 as an esaniple, O-NONE. 1-ROT90. 2-ROT180, 3-ROT270. 4-REFHOR.

5-REFJER. 6-ROT90REF-VER. TROTSOREFAOR): the siim of the iiumber of

rotated and reflccted blocks used is greater than the number of non-rotated and non-

reflected blocks used. It would seern as though some images have a higher degrce of

t ransformed self-sirnilarity t han ot hen. This affects the qciality of image produced

!>y scheme 2.

Comparison To Scheme 1

The compression ratios for scheme 2 and scheme 1 are almost identical for al1 images;

this is clear by looking at Appendk D. Neither scheme differed from the other by

more than 5% compression. However, scheme 2 seemed to irnprove the compression

ratio of domain level O compared to scheme 1. We can compare the figures for the

compression ratio and see that the distance between the domain level 1 and O curves

for scheme 1 is larger than the same distance for scheme 2. The domain pool for

a range block in scheme 2 is 8 times larger than in scheme 1. Representing these

rotations and reflections requires 3 bits per mapping. At error thresholds > 1000,

scheme 2 compresses marginaily better than scheme 1 for most images. For lower


Histogram of Domain Block Orientations that get maped onto 2x2 range blocks in pool, error threshold 1000 1 200 I 1 1 I I I I 1

- O 1 2 3 4 5 6 7

Orientation

Histogram of Domain Bock Orientations that get rnaped ont0 2x2 range blocks in personl, emr threshold 1000 1200 1 I 1 I l 1 I I I

u

O 1 2 3 4 5 6 7 Orientation

Figure 5.1: Spatial Transformations Of Domain Blocks

Cornparison of Partit ion Schemes 67

error t hresholds. the type of image determines the differeiice in compression ratio

b~tweeii the two scliemrs. Images like: 2.1 .O3. sail. 22-11. buildingl, biiilding3,

frymire. serrano. baboon did not conipress as ive11 nt lower error thresholds with

scheme 2 than scheme 1. However. images like pool. watch. 2.2.15, personl, ra ter ,

4.1 .O3. person3 compressecl slightly better with schenie 3 at lowr error thresholds.

1s it wortti examining rotated and reflected domain I~locks'! Not from a compression

ratio point of view. I t is clifficuit to justify that much work for 1 percent improvernent.

The SNR for scherne 2 cornparecl to scherne 1 depends on the image and the

rrror threshold used. For instance, i l we look at the SNR for biiildingl. we can see

that scherne 2 has a higher SNR at error threshold 1000. and a lower SNR at error

t hresliold 22000. We expect tliis to be true at larger m o r thresliolds for niost images.

Scheme 2 is able to find a match for larger range blocks which means the local error is

quite high for a large region of the image. Scheme 1 is more likely not to find a match

so it must break down the range block; hence the local error for that same region of

the image may bc snialler. This means that scherne 2 is more likely to produce an

image that is faithful to our local error threshold requirements. Unfortunately, we do

not get a significant improvernent on the compression ratio in these instances.

Domain Block Rotations And Reflections

It is interesting to observe that the most frequently transformed domain blocks are:

not rotated or reflected. rotated 180°, and reflected horizontally. It seems as though

rnost natural images have a strong self-transfomability with respect to these three

operations on blocks. Scheme 2 attempts to take advantage of these characteristics

To improve compression, it would be interesting to only tly to use the two, or four


most comnion spatial t h & transformations. This woiild conserve t ao (one) bits in

tlie description of earli ritapping resiilting in iiripromd compression ratios.

5.0.3 Scheme 3

Schmie 3 does not ilse tlie blork classifier. Without the classifier, searching for a

niatrtiing domain blork is r e r y slow. For an m x n image, searching for a matching

clom;iin block For a single range hlock is O(rnn). More detailed analysis is found in

srvtioti 3.3.3. -4s a rcsiilt. sclicriir 3 does not s ~ a r c h the entire domain space for a

niatrhing doniain blovk. Ins t~ad it only searches a local nrighbourhood around the

range blork. Neverttiolts. wit hoiit the block classifier. this scherne is still painfully

slow? Scheme 3 optmtes on the suspicion that images have a high degree of 10-

r:il similarit': therefore. it is not worthahile to searcfi the entire image space for a

nintching domain block.

Compression Ratio

Li ke previous schemes: the clifference in compression ratio between domain level 1

and O is small. In fact- the sliape of the compression curve is identical to scheme 2

for most images. This is remarkably interesting because it indicates that images do

have a natural local similanty1'.

I6Images compressed using scherne 3 take at Ieast twice as long to cornpress compared to any O t her scherne.

"Csing Our very limited mode1 of similarïty baseci on our contractive maps.


SNR

Since schcme 3 does not use a classifier, the difference between domain level 1 and O

is not as signifirant as schenits 2. Therc are many dornain blocks to choose from for a

given range bl«isk. wwi at domein level O. The SNR behaves much like scheme 2 in

ternis of values af SSR and patterns for different images at different domain levels.


The rlassification scliernc is vital For rpasonable speed performance. However, it is

cwentinl t h t t lie hlock dassifier corrcctly classifies blocks. Suppose we are given a

range I~ lork r thrii 3 an optinial matdiing dornain hlock ralled d . h prrfect classi-

fication woiilci have range blocks r and d in the sarne block class. Ensuring optimal

matches is a Iinrd probiem [[13]]. The classifier used for the Fractal Tool is described

in Section 4.3.3. The compression ratio resulting from scheme 3 is quite similar to

schcme 2. Most images (such as buildingl, ttilips, fruits, etc . . . ) have the sarne

compression ratio with scheme 3 as they do with scheme 2; see Appendix D. There-

fore. the suspicion is that these images are locally self-similar because we were able to

find a reasonably good matching dornain block in a neighbourhood of a range block.

However. it would be interesting to see how well an image would be compressed if we

searched the entirc dornain space instead of a local neighbourhood. Unfortunately,

the amount of time this would take for a 128 x 128 image would be prohibitive. Some

images (building4 clegg, watch) did cornpress worse using scheme 3 compared to

scheme 2. However. the difference was not significant. It is likely that those images

did not posses as high a degree of local similarity as some other images; instead they

have global similarity. Pictures of buildings which are very symmetnc seemed to fa11


iiito this catcgory.

Soiiir irnagcs rornpressd bet ter wi t h scheme 3 t han scheme 2 (personl, person2,

hlost~in. pool). This is because of mis-classifieci blocks in scheme 2. The following sce-

nario ivill help to illiistratr: Consider a range block r . Suppose the optimal matching

doniain Idoçk. 11. is in a local neighboiirhood of r. However, the block-classification

algorithm assigiis r to be in block class Ci and d in block ciass C2. According to the

algorithiri of scheme 2 . block d ail1 never be searched as a possible match for block

r . It is ~~ossiblr t tint ttie oniy dornain blocks in the sanie class as r do not match well

P ~ O I I ~ ~ : hmce sclienie 2 rniist break r into -1 quadrants. On the ot her hand. scheme

:3 will match d with r bccausc scherne 3 does not use a classifier.

Theri. ;ire only a IPW instances where the SNR for scheme 3 is higher than scheme

2 (blostein. pool. personl). However, the increase is minor. Most images have a lower

SNR with scheme 3 than scheme 2. This could be the result of having too small of a

local neighbourhood. At liirger error thresholds, scheme 3 does worse than scheme 2.

This is because the range blocks hecome so large that the number of possible domain

blorks in a local neighboiirhood is small: so the best match is not that good despite

the fact that we are looking at al1 possible domain blocks in that neighbourhood;

as opposed to domain blocks in the same block class. For most images, the SNR

with scheme 3 is lower compared to scheme 2; however, the difference is quite small

( 1 S-VRxhtme2(image) - S~VR~&m3 (image) (5 i)

5.0.4 Scheme 4

Given a range block. scherne 4 searches for domain blocks in a local area of the range

block and in random neighbourhoods (optionai). Scheme 4 uses the same kind of


block transformations and classification method as schenie 2. We can determine how

wrll the block classifier does its job by comparing the results to schenic 3.

Compression Ratio

The compression ratio follows the same pattern as previous schemes. The difference

in compression between domain Ievels O and 1 is small. The shape of the compression

curve for any image is the same as other schemes. Schenie 4 permits using random

n~ighhourhoods when s~arching for a domain block. The change from O random

rieighbourlioods to 1 randoni neighbourhood iniproved the compression ratio more

significantly thnn the change from 1 random neighbourhoocl to 2 randorn neibtiour-

iioods. Appendix D demonstrate this observation. In particular. observe that images:

4.1.03. 4.2.05, arctichare. buildingl, building-l. building5. cat. person3, pool, watch.

water have an improved compression ratio when one random neighbourhood is used

ks opposed to none. The rnost pronounced improvement with raridom neighbour-

hoods occun at lower error thresholds where the number of domain biocks is high

in a neighbourhood. It is not so advantageous to use 7 random neighbourhoods as

opposed to 1. However, some images demonstrate a slight improvement in cornpres-

sion iising 2 random neighbourhoods, for instance: arctichare. blostein, cat, watch

(compressed poorly with scheme 3), water. Some of these images compressed poorly

with scheme 3. Images which improved using 2 random neighbourhoods over 1 saw

a greater irnprovement from using 1 random neighbourhood over O.

These images are more likely to have a poor rate of compression with scheme 3.

The difference in compression ratio between choosing 0,1, or 2 random neighbour-

hoods is not significant (< 5%). It may not be worth the extra work to search a

Cornparison of Partition Schemes

random neighbourhood in the Iiope of finding a better match. It would be interesting

to sec i f searching a loçitl neighbourhood is necessary for good compression.

SNR

The effect of randorn neighbourhoods is quite clifferent with scheme 4 compared to

scheme 3. Sorne images have radically shifting SNR values at different error thresh-

olds mhen using random neighbourhoods. See Appendix E. For instance, 1.2.05,

fruits. arctichare. This could be as ri result of the randomness: withoiit iising randon

neighbourhoods, the SNR ciirves are quite sniooth. The improvement in SNR for

iising a t least 1 random neighbourhood is sipificant. Since we are using a classi-

fier. it makes sense that we either increase the neighbourhood size or use more than

just the local neighbourhood when searching for domain blocks. Scheme 4 is also

unusiial that there is still a noticeable improvement in most images in the SNR when

iising 2 random neighbourhoods as opposed to 1. In particular, domain level O shows

t tic greatest improvement when usirig 2 random neighbourhoods. Unfortunately, two

different people cornpressing the sarne image with scheme 4 using random neighbour-

hoods can get significantly different results in terms of compression and SNR. For

some images. it is necessary to use random neighbourhoods, ie. cat, fruits, frymire

images. These images have a higher degree of global similanty than local similarity;

compare their SNR with scheme 3.

It is necessary to use at least 1 random neighbourhood to obtain good image

quality. One might wonder if the cwt of searching random neighbourhoods is worth

the savings obtained using a block classifer. Obviously, this depends on the size of

our random neighbourhoods. It is difficult to determine what the best neighbourhood


size shoiilrl be; however. if ive assunie that it is srna11 in cornparison to the image size

tlien for a given range block r. the cost (in terms of running tinie) of searcliing range

blocks in randoin neighboiirhoods ttiat have been classified is less than the cost to

search an unclassifiecl local neighbourhood of blocks. This can be seen by observing

the running tinie for scheme 3 and scheme 4. Houwer, this is an argument based on

implcmentation. Since both scheines use fixed size neighbourhoods, one could argue

that both schemes are O(1) to search for an inclividual domain block. Quantifying the

constant factor depends on the block classifier. If we allow our neighbourhood to be

nri n x n region theii for a k x k range block. the riumber of domain blocks that need

t« be srarched withoiit t h e use of a block classifier is given by (- + l)?. If ive

assiinie that our block classifier distributes blocks uniformly among the 24 possible n- Ik + 1l2

classes t hen searching a single neighbourhood requires looking a t ( J"" '~ ' dornain

blocks. Therefore. tve can search upto 24 neighboiirhoods using a block clessifier and

have a tower constant than if we did not use a block classifier. Recall. that this rests

on the rtssumpt ion t hat our classifier distributes blocks uniformly among the different

cl asses.


The difference in the compression ratio between scheme 4 and scheme 2 is less than

5% of the total file size. Since scheme 4 differs from scheme 2 by only doing local

searches then we expect the compression ratio to suffer; however, by experimental

tests we observe that the magnitude of the difference in compression ratio is not

large. This gives us confidence for using neighbourhoods. Furthemore, if we only

use a local neighbourhood then the number of bits for the domain id could be reduced


significaritly this wvould increase the compression ratio for scheme 4.

\Vitil regards to SNR. sclienie 4 behaves similarily to scheme 2. This is exciting

n w s : instead of seardiing the entire domain space. we only need to s~arch locally

witti a few randorn neighbourhoods. Using at least one random neighbourhood seems

to be ncccssary Slany images such as: 2.1.1 1. arctichare, 22.15 have an improved

SNR using scheme 4. However. these images have a lower compression rate usirig

scherne 4. So, the problem is that a large range block is not usually rnatched with

scheme 4 as opposed to scherne 2. Therefore. the range block is broken down into

snialler picces whrre ;t lower error mapping c m be found: hence we espect better

q~ialiry. \lé (-an s w this by looking at .lppcndis C: wc see that the nuniber of 2 x 3

range blocks is higher for scheme 4 than s r h m e 2 for most images.


Scheme 4 behwes similarily in terms of compression ratio to scheme 3. This instills

confidence in oiir choice of block classifier: it seems to produce results similar to a

local scheme without a classifier. hloreover, the amount of time it takes to cornpress

an image with scheme 3 is much higher compared to scheme 4.

In terms of SNR, scheme 4 seems to break down range blocks into smaller pieces

whicli means Ive expect a slightly worse compression ratio, but a higher SNR. For

some images, such as: 4.2.05, arctichare. blostein, and buildingl, the SNR for scheme

4 is lower than scheme 3. T h i s could be as a result of the classifier. In these images,

scheme 4 uses more range blocks; which implies smaller range blocks, but still has a

lower SNR. However, most images have nearly the same SNR using both scheme 4 (2

random neighbourhoods) and scheme 3.

Cornparison of Par t i t ion Schemes 75

Sclicme 4 seems to do rernarkably aell considering its limited domain block pool in

cornparison to other schemes which have a much larger domain pools. BIocks which

arc mis-classified are close to belonging to more than 1 block class where the meaning

of close depends on the choice of block classifier. With the classifier that Ive chose,

close means that given the siim of pixel values of 4 quadrants of a block: ql , q2, q3, q4

nt l e s t 2 of those sums are within 5 of each other. ie. 1 qi -q, I<_ 5 for i # j . Allowing

these blocks to belong to multiple classes might solve the probleni of mis-classifying

blocks.

5.0.5 Scheme 5

Scheme 5 behaves like scheme 4; hoivever. blocks can be put into nt most 2 classes in

srheme 5. A block is put in 2 classes if it is 'close' to another block class. This method

nttempts to correct any mistaken block classifications made by the block classifier by

slightly irirreasing the domain pool search size.

Compression Ratio

The compression ratio curve follows the same pattern as al1 the other schemes. Do-

niain levels O and 1 behave like previous schemes; ie. the curves a re close to each

other. Using random neighbourhoods only slightly improves the compression ratio on

a few images (ie. arctichare, buildingl, cat). The effect of random neighbourhoods is

not as pronounced as it was with scheme 4 possibly because the domain pool is larger.

We can see this by looking at the difference in compression ratio for scheme 5 with O

random neighbourhoods and 2 random neighbourhoods as opposed to the same dif-

ference with scheme 4. The classifier in scheme 5 seems to misclassify fewer blodrs.


Hoivewr. images w hich comprcssed poorly \vit ti sclicme 3 also cornpress poorly wit h

schcrne 5 (local neighbourhood search only).

SNR

Rancioni neighbourhoods improve SSR. We sas this with schenie 4 as well. The

rnodifird classifier has some effect on the SNR of t he final image; sec cornparison to

srht!mc 4. It is iriteresting to note t h the SNR For sorne images changes abniptly

ncrording t« the local thresholcl when random neighbourtioocis are heing iised (see

hiiilding-l. liiiilding5. cat. friiits). Without using random neighboiirhoocls. thesc im-

;igrs I in~c r e p large drops in the SNR at certain error thrcsholds. So. it woiilrl seem

thnt tlic self-sirnilarity of an image also deperids on the error threshold.

Cornparison To Scheme 4

Therc are man- blocks which are close to several block classes. One might wonder if

it is useful to allow a block to be in multiple block classes. If block 6 is in class cb,

but is close to being in class c, then adding block b into class c. is useful if there is no

block in c. that is close to cb. When we are dealing with low error thresholds, there is

an abundant supply of range blocks in each block class. It is very likely that if block

b is dose to class c, then there is a block a in c, that is close to class cb. It is not

so useful in these cases to add block b into class ca or vice versa. However, at larger

error thresholds where the pool of domain blocks is smaller there is an advantage to

increasing the domain pool by allowing blocks to belong to multiple block classes.

Comparison of Partition Schemes 77

5.0.6 Cornpared to JPEG

1s fractal imagr coriiprrssiori coriipnrnble to .JPEG? For Iiigh quiility representation.

the ;insiver is no. .JPEG coiripresses reasoiiably w l l (70-90%) with a high SNR. For

low c p l i t y reprcsentation. tiic c-ompressiun rate for JPEG is about the same as the

liactally conipressed files: ho\vcver. tliis is an unfair comparison. JPEG allows a loss-

1t.s encoding phase after its lossy compression. The fractal image cornpressor does

riot do this at the moment. To ohtain niasimal compression ratio for fractally corn-

prcssed images. it is riecessaru to i i s ~ a lossless encoclcr that rinderstands the format of

rriappings. Uany of the niappings froni ciornairi blocks to range blocks are the same.

Therrfore. a (.lever lossless m c o r l ~ r c m represent cornnion niappings with fewer bits.

\\é cari see froni Table 5.0.6 that the number of distinct mappings is frequently much

less than the total number of mappings used. refer to Table 5.0.6. This indicates an

rsce1ler.t opportunity for a dictionary style lossless coinpression algorithm. If many

mappings are common among images t hen a well known initial dictionary might be

admntageoiis. This coiild have a significant impact on compression.

By iising a standard JPEG compression tool, we managed to cornpress JPEG im-

ages smaller than the corresponding fractal image (wit hout a lossless encoder phase).

The SNR of the JPEG compressed image was higher than the fractal compressed

image. This is only slightly disapoint ing nem. Wit h an intelligent lossless encod-

ing phase a t the end of the fractal compression phase, ive expect compression ratios

higher than 95% which is what we currently have. At such high levels of compression,

we don't expect JPEG to perform well.


I ni agr

2.1 .O;3 '2.1.11 2.211 2.2.15 4.1.01 -1.1.03 4.2.05

iirc, t ichre h b 0 0 n l)lostt4n

t~uilcli~igl hiiilding3 hui1dir~g-l tmiltling5 hiildirig6

cat degg friii ts

frymire hoilse lena

nionarch PePPers person 1 person2 person3

pool sail

serrano tiilips watch ira t er

Error T hreshold

dom 1 5419 10781 $736 0-0

rai6 3647 1190 9326 3620 30316 1 ri6 11456 1 1 15-4 15419 11236 15164 15701 29197 7813

25122 1880s 12483 18702 14768 2767

Table .5.l: Number Of Distinct Mappings for Images Under Scheme 1

Part III

Discussion

Chapter 6

Interest ing Questions

6.1 Non-Uniform Quantizer

The range of quantized values for contrast and brightness are not uniformly dis-

tributed. In particular, for most images the brightneçs values are densely packed in

the center of the quantized range using any scheme. Therefore, using a non-uniform

scalar quantizer for the contrast and brightness values would improve the SNR, and

possibly improw compression. Obviously, the quantizer should have finer gradations

in the center of the quantized range than at the eàges. One rnight wonder how the

quality and compression ratio of the compressed image might improve. Early in-

vestigation seerns to indicate that the qualit? of the compressed image will not be

significantly higher using a non-uniform quantizer.

82 Interestine; Questions

6.2 Re-compressing The Output

Giwri an iniage. 11. ive cornpress the image to obtain pc. We then deconipress the

image to obtaiii /id. We hope that p d and p arc close to each otlier: hence the SNR

is higli. Consider what happens if ive conipress the image pd into p4. The image

,ud is already in block form (because we do not have a smoothing process) so we

ivoiild cspect a tiigher SNR and a better conipression ratio compared to p~ and 14.

The average pixcl crror I~etween p d and p woiild nlso be higher conipared to p~ and

\\é espect this lmause is /id an iniage that is the result of being rnodelleci by

fractals. Cornpressing is similar to re-constructing the mode1 that ic have already

biiilt. Simple trsts siipport tliis conclusion. If WC repeat this proccss. ac ivill get

c w n hettcr compression and higher SNR. Hoivever. the SNR is only in romparison

to the prerioiisly used image: not the original.

6.3 Adaptive Error

Certain parts of the image are not so important to represent as accurately as other

parts of the image. Determining which regions of the image are important is a difficult

problem. If ive solve this problem then adaptively modibng the error threshold to

be lower at the important regions would improve the image quaiity. The compression

ratio would not suffer because less important regions would have a high error threshold

and would have a high compression ratio.

\\é could try a very simplistic heunstic. For instance, the middle of the image

cotild have a lower local error threshold then the edges. The justification being that

the focus of the image is in the center and the rest of the image is not so important.

6.3 Target Compression Ratio 83

Obvioiisly. numerous coiinter-esaniples esist where this simple heuristic woiild fail.

6.4 Target Compression Ratio

Obtaining a target compression ratio. cl,,,,, . is an interesting project. Initially, we

ran stiirt with n large grid of range blocks. This is the minimum coinpression ratio

that ive c m achieve. cal1 it cm,,. If cm,, < c,.,,, then we can procwd to break up the

range hlock wit h the worst matching domnin block' . This gives us a new compression

ratio. cm,. We continue to break itp the the range block with the worst matching

doniaiil I~lock and iipdate cc,, ;ippropriately iintil c,, > s,,,,t[3].

6.5 Post-Processing For Decompression

Csing higher local error thresholds leads to noticeable blocks in the decompressed

image. Thcse blocks can be elirninated by adding n post-processing phase to the

dccornpression process to smooth the transition between adjacent blocks. Since the

decompression phase is reasonabl y fast and since the fractal file contains information

about the blocks then smoothing the transition between blocks should not be a costly

process. Smoothing only needs to occur at the edge of blocks; not inside them.

6.6 Lossless Encoding Phase

.A lossless encoding phase after the fractal compression would improve the compression

ratio. Unfortunately. the extra compression obtained by adding a lossless phase using

lWorst matching domain block means the mapping error is larger than the mapping error for any other rangedomain block pair

84 Interesting Questions

ciirrent tools is stnall. Current tools look at patterns of bits. In this case. it would

be ;idvantagcoiis to look at patterns of mnppings for lossless compression.

6.7 Number Of Iterations For Decompression

Several papcrs [[SI] have addresseci the issue of fast convergence for decompression.

Alost images need 7 iterations before the SNR becomes stable2. However, some images

nerd more iterations. Obviously. the easiest way to decornpress an image is to take

the ciiffwmcr hetwecn successive itcrations and when that difference is small enough

thv process ciin stop.

?Stable means that more iterations do not significantly irnprove the SNR

Chapter 7

Future Work

7.1 F'ractal Audio Compression

Aiidio, music in particular, has mmy repetitive patterns. Focusing on the self-

similarity aspect of music could be fruitful for fractnl audio compression. Using

tecliniqiirs that have been mentioned earlier. it might be possible to compress music

fractally by looking for self-similar patterns.

7.2 Fracta1 Tool

AIany improvements to the fractal tool can be made. The current colour model is

RGB. It may be advantageous to use HSB (hue, value. chroma) or some other colour

model. Furthermore, rnany components of the fractal tool can be manually optimized

to improve speed performance. Also? the ability to plug in new partition schemes is

not easy; the software engineering aspects of this problem can be explored hirther.

The decompression phase could also be improved by having a block smoother

86 Future Work

which woiild smooth oiit tlie pixel differences betwen adjacent blocks. This wotild

rliniinate t lie blockiness of deconipressed images.

Chapter 8

Summary And Conclusions

The rvholc idea of taking one block and mapping it onto another block for the purpose

of image compression secms liidicroiis; -et i t wor ks. The implementation provideà

with this thesis demonstrates that thcre are sereral methods to make this style of

image compression work.

Domain lerel 1 uses n lnrger domnin pool thnn domain level O. However. there

is no significant advantage over using domain lerel 1 because the compression ratio

is about the sanie as domain level O. Also, the SNR is nearly the same. It is dso

interesting to note that the rnost common spatial transformations are: none, rotate

180". and reflect horizontnlly out of the 8 possibilities. We may be able to improve

the compression ratio for many images by taking advantage of tbis fact. The speed

would definitely be improved if we only consider 3 possible spatial transformations.

One method for finding domain blocks is local neighbourhood searching combined

with random neighbourhoods. This style of domain block organization seems to be

effective. The compression ratio is aearly unchanged and the SNR is oniy a little lower

for some images. The block classification technique is more effective at lower error

88 Summary And Conclusions

tliresliolcls conipared to tiigher error thresholds; which is the result of an increased size

in the tloniain pool. At higher error thresholds. we observe that the SNR is increased

~vtitw ive allow t~locks tvhicli are close to more than one block class to belong to

riitiltiple t~lock rliissrs. However. since the clifference in SNR is not significant then

n.tb may roncluclt? t liat the classification scheme which ive have chosen is reasonable.

In conipirisori to .JPEG. c w have not clcnioristratetl that fractal image comrpession

is superior. However. the providecl implementation does not have a lossless encoding

ptiase like .JPEC;. \t'cl <lenionstr;ited that since the number of distinct mappings is

siti;iilcr tiian t lie total riuiriber of niappings used t hen it wvould be possible to achie~e

goo(l lossless coniprcwion rates with an intelligent encoder. .-ln intelligent encoder

riiiist iincierst;iri(l ttic? riotion of rnappings. If w .?dd this feature then it is quite

possible to aciiieve Iiighcr compression rates than .JPEG together with higher SNR

values.

Bibliography

[l] Al i . kfaariif and Clarkson. T.G. Suruey of Block Based Fractal Image Com-

prrssion and its ..lpplicotions Proceedings of the 2nd Conference of Information

Technology anci its .-\pplications. ITA'93

[2] Barnsley. SI. Frnctals Evenjwhere. Aradernic Press, 1993.

[3] Fisher. 1'. Fractal Image Compression: Theory and Application. Sprînger-Verlag,

1995.

[-II Forte. B. aiid Yrscay. E.R.. Theonj of generalzzed fractal t m n s f o m . Proceedings

of the NATO ;\SI Concerence. Fractal Image Encoding and Xnalysis, Trondheim,

July 1995. Berlin Heidelberg, 1998. Springer-Verlag.

[SI G haravi-Alkhansari, hl. and Huang, T.S. A fiactal-bmed Image Block-coding

.4 lgorithm University Of Illinois

[6] Jacquin. A. Fractal Image Coding: A Reviev Proceedings of the IEEE. Volume

$1, Xumber 10. October 1993

[ï'] Johnson. D.S. A Theoretician's Guide to the Ezperimental Analysis of Algorithm

ht tp://mnv.research.att .corn/ dsj/papers.html

89

90 BIBLIOGFMPHY

[SI Kominrk. J. Conuergence of Jrnctnl encoded images Proceedings DCC'95 Data

Compression Conference. .J. A. Storer. hl . Cohn (eds.), IEEE Computer Society

Press. h r c h 1993.

[9] Kominek. .J . ;Idvances in Fractal Compression for fbhllimedi~ Applications. Uni-

versity O f Waterloo. http://links.uwi.itrrloo.ca. April 1996.

[ I O ] Lii. Y. Fmctal Irringing. Academic Press. 1997.

[ 1 11 Ilarscltm -1. E. and Hoffmnn R I . J . Elementaq Classical Analysis. 2nd edition.

K . H . Frccwinri and Company, 1993.

[ l?] Ndson. Mark and Gailly. Jean-Loup. The Data Compression Book. 2nd edition.

UkT Books, 1996.

[13] Reusens. E. Padztzoning complexity issue for iterated funclions systerns based

image coding Proc. of VI1 EUSIPCO, Edinburgli, September 1994

[ I 41 Saiipe. D . Fractal Image Compression via Neorest Neighbor Search Proceedings

of the NATO .AS1 Conference. Fractal Image Encoding and Analysis, Trondheim,

.Jiily 1995.

[IS] Saupe. D. and Hamzaoui, R. and Hartenstein, H. Fractal Image Compression:

.4 n Introduct ory 0 u e n n . e ~ Revised version of Chapter 2 from Fractal Models for

Image Synt hesis, Encoding and -4naIysis.

[16] Sayood, Khalid. Introduction To Data Compmssion. 2nd edition. Morgan Kauf-

mann Publishers, 2000.

BIBLIOGRAPHY 91

[li] \,*rscay, E.R. A Hztchhiker 's Guide to 'Frartal-Based ' Function Approrirnation

and b n a g e Compression. University Of Waterloo. http://links.uwaterloo.ca. Au-

gust 1995.

92 BIBLIOGRAPHY

Appendix A

List of Symbols and Abbreviations

Ir : Ckially rrprrsrrits iiti image or ;i vector that ive wnnt to approximate or apply a

furict ion t o.

O: Spntially contractive hnct ion that oprrates on domain blocks. It may rotatelreflect

the cfomain block.

5: Sidp length of a range block.

d(x. y): lletric function.

D: Side length of a domain block.

Dh: Horizontal distance between consecutive domain blocks.

/(x. y) and g(x. y): Usually represent images at coordinates x, y.

F(S) : Mapping that operates on a set.

A&: Number of domain blocks.

94 List of Symbols and Abbreviations

.\Ir: Siimber of range blucks.

1 , : .-\ niapping from a clornain block to a block tliat is the sanie size as a range block.

I17(p): The union of al1 the to , rnappings. I I* operates on an image.

PSNR: Peak Signal r o Soise Ratio. Defined in Chapter 5.

SNR: Signal to Noise Ratio. Dcfined in Chapter 5 .

Appendix B

Calculat ions For Comput ing

Range-Domain Mapping

This appendis refers to Table 3.1. For each range block, we examine each domain

h l d and cornpute the optimal paramcters. as descnbed in section 2.1.2.

Let:

96 Calculations For Computing Range-Domain Mapping

Ke linw foiind n perfect matching. ie. a range-domain pair such that the mapping

yields an mror of O. I t is not necessary to test rl against the other range blocks. rl

matchez trith d l such that al = ! and b1 = 12

B.0.2 Mapping for r2

Xow. we move onto TI>.

Calculations For Computing Range-Domain Mapping 97

Observe for r2 and dl :

- 2 - -- 5

K e rcquire that the contrast (a2) be in [O. 1). Therefore, we must force a2 = O . Hence

t hc calculation for becomes:

= f(11+9+15+13)

= 12

Xow. ae compute the error:

Observe for rz and 4:

Thcrefore. the brightness woudld be:

Hence, the error would be the same as it was before:

Observe for rp and d3:

98 Calculations For Computing Range-Domain Mapping

The error is:

\\Q have found a perfect rnatching. r2 rnutches with d3 such that a? = h and b2 = 8

B.0.3 Mappings for other range blocks

Finding tlir mappings for r3 and r~ can be done similaril.

Appendix C

Range Block Results

100 Range Block Results

Iiriage File

arc t ichare hause

baboon \vat ch

cat lena pool \vater fruits '1.1 .O3 2.1.11 2.2.11 -7 - - W . -7 15

41.01 4.1 .O3 4.2 .O5

building1 building3 building4 building5 biiilding6 person 1 person2 person3 blostein

~ 1 % lena

PePPers Serrano -mire

monarch sail

t ulips

Nuniber Of Range Bloch Of Each Side Length 1000

Total 8499 4.3009 51697 31410 30477 31621 5115 5 17-14 16849 '24613 26791 63088 55393 1861 3346 23527 2 1298 15127 29928 21648 37296 5895 7345 11526 4015 44561 31621 38353 25710 60657 49431 56736 56571

Table C.1: Range Bloch For Scheme 1

Range Biock Results 101

Image File

arct ickiare hoiise

baboon watch

cat lena pool

water fruits 2.1.03 2.1Sl 2 2 . 1 1 '2.3.15 4.1.01 4.1 .O3 4.2.05

building1 building3 building4 building5 building6 person l person2 person3 Dlostein

c h % iena

PePPem serrano frymire

monarch saiI

tulips

Surnber Of Range Blocks Of Each Side L e n d l

Total 3126 11416 17737 14475 13269 6877 3201 29541 6097 5230 7747 15766 13'217 2206 1318 6988 10999 10149 Il919 9132 1-1454 2598 2650 4614 1-177

26447 6877 9844 13278 43533 12381 17211 15621

Total 3270 10777 155'14 12804 12105 6394 2856 36'163 5347 4954 7177 15637 11926 1966 1186 633 1 9673 9343 10761 8190 13122 2253 2263 4188 1321

24584 6394 9154 11754 40611 11199 15957 14331

Table C.2: Range Blocks For Scheme 1


Irriage File

arctichare house

baboori watch

ciit lena pool

water friii ts 2 . l .On 2.1.11 2.2.11 2.2.15 4.1.01 4.1 .O3 4.1.05

building1 building3 building4 building5 building6 personl person3 person3 blostein

clegg lena

PePPers Serrano frymire

monarch sail

tulips

Yumbcr Of Range Blocks Of Each Side Length 22000

Total 2733 19'75 Il389 10101 9393 4438 '2637 18540 4072 3895 4351 9559 4600 1327 961 4786 TG65 7630 8364 6264 9483 1899 1753 3207 1132 15947 4438 5941 8262 33330 7692 12180 9960

Tot al 2451 7486 11014 8964 5358 4174 2331 m i 2 3577 3667 3922 8566 4420 1246 826 4339 6157 6868 7656 5607 8727 1635 1582 2901 991

14948 4174 5416 7503 29775 7125 11514 9228

Table (2.3: Range Blocks For Scheme 1

Range Block Results 103

arct ichare ho use

baboon vatch

cat lena pool

~ v ; ~ t er fruits 2.1 .O3 2.1.11 22.11 .> -.-. 9 15 -4.1.01 4.1 .O3 4.2.05

building1 biiilcIing3 building4 biiilding5 building6 person 1 person2 person3 blostein

ch% lena

Pe PPers serrano kymire

monarch sai 1

t ulips

Total 8973 43135 51853 32208 3 1344 31669 5289 81635 11104 24853 27208 63034 e55-435 7930 3388 23848 21289 15142 30006 2 1720 37659 5610 7426 11463 4060 -44546 31669 38242 25647 61014 49419 57162 56634

Image File

Total 8133 42133 50578 28602 28161 30220 4872 76512 15805 23689 25096 6201 1 5361 1 7579 3145 22369 20371 15001 28170 20247 34812 5193 6889 10515 3796 44504 30220 37048 24975 59868 47385 55053 54585

Number Of Range Blocks Of Each Side Length 1000

Table CA: Range Blocks For Scherne 2

- 7-


arcticfiare hoiise

baboon \vatch

t-zit

lena pool water fruits 2.1 .O3 2.1.1 1 2.2-1 1 2.2.1.5 -4.1.01 4.1 .O3 4.1.05

building1 building3 biiilcling-l building5 building6 person 1 person2 person3 blostein

clegg lena


monarch sail

tulips

Tot al 3348 10792 15796 13170 1276 6324 2982 '31039 5518 4957 1213 15634 L 1896 '2017 1177 6508 9664 9391 10716 81 15 13203 2229 2386 4116 1330

24239 6424 9418 11796 41199 11340 16182 14526

Total 3039 10369 14131 11835 Il337 6088 2694

24993 4954 4771 6694 15508 11608 1855 1078 6076 8686 8632 9912 7440 12150 1977 2107 3858 1255

22601 6088 8710 10584 37845 10578 15294 13632


Range Block Results 105

Image File

arct ichare 11oi1sc

b;thoon watch

vat lena pool

ivatcr friii ts 2.1 * 0 : 3 2-1.1 1 2.2.1 l 2.1.1'1 -4.1.01 4.1 .O3 42.05

building1 building3 building4 building5 building6 personl person2 person3 blostein

clegg lena

PePPers serrano frymire

monarch sail

tulips

'iiirnber Of Range Blocks Of Each Side L e n ~ t h

Total 2490 7639 11053 9165 8670 4171 2454 11505 3703 3100 3961 8123 4420 1237 517 4408 6748 6919 7692 5625 8814 1524 1621 2943 1027 14714 4171 5488 7590 30564 4221 11562 9387

Total 2259 7195 10741 8253 7749 3946 2277 16377 3298 3526 3646 7839 4273 1177 748 4057 6121 6301 7140 5196 8151 1368 1492 2616 925

13490 3946 5056 7008 27162 6786 10902 8769



Image File 1) Yuniber Of Range Blocks Of Each Side Length

iirct ichare hoilse

baboon m t c h

cat

lena pool

water fruits 2.1 .O3 2.1.1 1 2.2.11 2.2.15 4.1.01 4.1 .O3 -1.2.05

building1 l>uilcling3 building4 biiilding5 building6 person 1 peson2 person3 blostein

Lena PePPers Serrano fryrnire

monarch sai 1

tulips

Total 1 O662 44638 53983 41046 37245 33721 5790

100377 19384 26092 i30640 64090 57688 8170 3553

25702 22258 15235 33561 24807 42573 5904 7945 12480 4144

44573 33721 40090 26268 62112 52320 62970 61311

Total 9555 43540 52510 35604 32841 32179 5187 91131 17545 24895 27829 63373 56296 7735 3265

24025 21418 15109 31302 23779 39330 5403 7255 1154'7 3880 44561 32179 38806 25707 61461 50259 59220 58626


Range BIock Results 107

arc t ichare lioiise

haboon watch

cat letia pool

\rater fruits 2.1 .O3 2.1.11 2.2.11 '2.2.15 4.1.01 -1.1 .O3 1.2.05

building1 building3 biiilding-l building5 building6 person 1 person2 person3 blostein

clegg lena


monarch sail

tulips

Niimber Of Range Blocks Of Each Side Lendh

Total 3969 12004 19000 15972 14208 7066 3150 32994 6346 5239 8050 15865 12235 2182 1237 7237 11 182 9898 12855 9621 15192 2529 25'72 461 1 1372

26405 7066 10083 13401 45222 12540 18612 17109

Total 3366 11353 16450 14142 12891 6490 2817 29490 5470 4957 738 1 15130 11956 1942 1120 6694 9772 9151 11346 8691 13794 2160 2212 41 10 1285

24773 6490 9310 11823 42465 11406 16992 15243



image File

arct icliarc ~iousc

baboon [vat ch

(:a t Iena pool

wiit,cbr

fruits 1.1 .O3 '2.1.11 22.1 L $1 '1 15 W . - .

-1.1.01 -1.1 .O3 -1 -2.05

h i l d ingl building3 biiilding-l building5 building6 person 1 person2 person3 blostcin

clegg lena

PPPcrs serrano frymire

monarch sail

tulips

Stiniber Of Range Blocks Of Each Side Length

Tot al 2877 8485 11527 11259 10080 4564 2556 201'24 4192 3841 4639 10099 4699 1333 595 4891 7510 7426 8769 6645 10035 1896 1723 3213 1066 15962 4564 6214 8361 35388 7995 12744 10728

Total 2529 7987 1 IO86 9900 9087 4264 2349 18504 3661 3715 4244 9115 4468 1198 784

4417 6667 6817 8040 6006 9144 1578 1540 2883 955

15020 4264 5659 7551 31638 7317 11967 9702


Appendix D

Compression Ratios

110 Compression Ratios

Error Threshold x 10'

1'3 0 dornain level O Scheme '

d , +-+ domain level1 Scheme ' 1 1 1.5

Enor Threshold

a

3- 0 domain level O +-+ domain level 1

Enor Threshold 10'

aï . . . . . . . k0.g31 € .!jj- 1 + . - .O domain level O Schern 1

8 0.92 ' -+ domain level 1 Schem 1 O

0.91 ' I

O 0.5 1 1.5 2 2.5 Error Threshold x IO'

Compression Ratios 111

cat.f rac

0.82 1 O 0.5 1 1 .S 2 2.5

Error Threshold x IO*

V) Qi & 0.86 E 6 0.84

pool .frac 0.98 i 1

1.'

. ,

1ena.f rac

al Q w

c 0.96 O .- V) cn

O 0.5 1 1.5 2 2.5 Enor Threshold x IO'

1 .

KI . r

/

. 0 domain level O Schem 1 0 -+ domain level1 r i s c h m 1

o. . O domain level0 Scheme

O 0.94 -

O O 5 1 1 .S 2 2.5 Enor Threshold x IO'

V.. Y

O 0.5 1 1.5 2 2.5 Enor ThreshoM x 10'

:' +-+ domain level 1 l Schemq 1


arctichare.f rac 0.99 11

0.92 O 0.5 1 1.5 2 2.5

Error Thres hold x 104

baboon .f rac

'1

Enor Threshold x 10'

0.65 ' l O 0.5 1 1.5 2 2.5


, E + O O domain level O Scheme

domain level 1 I Schemq



0)

0 . 8 5 o domain level0 Schem 2 o 0.84 + -+ domain level 1 Schem 2

n


pool .f rac 0.98 1

0.94 ' 1 O 0.5 1 1.5 2 2.5


2 al O V 6 0.96 -- CO Y,

2 E 0.95

0

1ena.f rac

I l

1 _ 0.8 ,! 0. -0 domain level O Schem t - O i +-+ domain level 1 I Schern

# A3

' f

/

/ .


. + O . .O dornain level0

-+ domain level 1 O o domain level0 +-+ domain level1

0.5 1 1.5 2 2.5 Emr Threshokî x 10'


1 0.5 1 1.5 2 2.5 Error Threshold x 104

O 0.5 1 1.5 2 2.5 Enor Thres hold x 10'

Enor Threshold 10'

watch.frac

O 0.5 1 1 .S 2 2.5 Enor Threshoiâ 10'

Com~ression Ratios 115

O.'' 1 1;' 3. 0 domain leval 0 Çcheme :

+ +-+ domain level 1 Scheme I 6 ln t 0.8 1

O 0.5 1 1.5 2 2.5 Error Threshold x 104

0.95 O 0.5 1 1.5 2 2.5


i o. .O domain level O ' +-+ domain level 1

,$. ,


water.f iac

I l

+-+ domain level 1


arctichare.frac (O Rnd Nbhds)

0.91 O 0.5 1 1.5 2 2.5

Error Threshold 1o4

baboon.frac (O Rnd Nbhds)

0.5 ' 1 O 0.5 1 1.5 2 2.5

Error Thres hold x 10'

1

h

al

% 0.9 c. c a 2 0.8.

O Y

C O . - 2 0.7 2 a 5 0.6 O

house.frac (O Rnd Nbhds) 1 ) 1

Error Threshold x IO'

. 4.- e. -@

6 AY 6'

d t

1. .

watch.frac (O Rnd Nbhds)

3 O domain level O Scherne 4 +-+ dornain level 1 l Schemq


cat.frac (O Rnd Nbhds)

O O. 5 1 1.5 2 2.5 Error Threshold x 104

pool.frac (O Rnd Nbhds) 0.99

6 0.95 / 4,- O domain level0 Schern O -+ domain level 1 1 Schem i

0.94 1

O 0.5 1 1 .S 2 2.5 Error Threshold x 10'

lena-frac (O Rnd Nbhds)

1 - . .O domain level O " ' k-+ dornain level 1


water.frac (O Rnd Nbhds)

0

. . . .

.O domain level O -+ domain level 1



arctichate-frac (1 Rnd Nbhds)

/ + - o o. O domain level 0

+-+ domain level 1

0.9 L O 0.5 1 t .5 2 2.5

Error Threshold 104

baboomfrac (1 Rnd Nbhds) 1

A

0.5 1 O 0.5 1 1 .S 2 2.5


house.frac (1 Rnd Nbhds)

'1

; 0- . o domain level O Scheme 4 . . . ..

d A +-+ domain level 1 1 Scheme 4

Error Threshold IO'

watch.frac (1 Rnd Nbhds)

- .O domain level O -+ domain level 1

0.9 O 0.5 1 1 .S 2 2.5



catfrac (1 Rnd Nbhds)

/ 0- O domain level0 +-+ domain level 1 +

O 0.5 1 1.5 2 2.5 Error Threshold x 1 0 4

pooLfrac (1 Rnd Nbhds)

h

0.94 ' 1 O 0.5 1 1.5 2 2.5


lena.frac (1 Rnd Nbhds)

I : 0- .O domain level O .' +-+ domain level 1

1 0.75 *

O 0.5 1 1 .S 2 2.5 Enor Threshold x 10'

water.frac (1 Rnd Nbhds) 1 ) 1

O domain level O Scheme O ,:-. +-+ domain level 1 I Schemq

Enor Threshold x IO'


arctichare. frac (2 Rnd Nbhds)

N

. 0 dornain level O -+ domain level 1

Error Threshold x lod baboonfrac (2 Rnd Nbhds)

0 domain level O -+ domain level 1



'1

O 0.5 1 1 .S 2 2.5 Enor Threshold x 10'


i 0 .921 ' / * 1 : 3. .O domain level0

6 +: +-+ domain level 1 1 0 . 9 ~ ~ ' .

I

O 0.5 1 1.5 2 2.5 Enor Threshold x 10'


cat-frac (2 Rnd Nbhds)

/ 0. O domain level O Scheme 4

+! +-+ domain level1 Scheme 4 n

I Error Threshold x 10'

pool-frac (2 Rnd Nbhds)

O domain level0 -+ domain level 1

0.94 O 0.5 1 1.5 2 2.5



1;: 0 ; &man level O [Scheme 1 +-+ domain level 1 Schemq 4

0.75 O 0.5 1 1.5 2 2.5


watecfrac (2 Rnd Nbhds)

I l

b /.'o. 0 domain level O Scheme 4 +-+ domain level1 l Scheme 4



arctichare-frac (O Rnd Nbhds)

0.91 I


baboon.frac (O Rnd Nbhds)

O domain level O O -+ domain level 1

O 0.5 1 1 .S 2 2.5 Error Threshold x 10'

house.frac (O Rnd Nbhds)

0.65 ' 1 O 0.5 1 1.5 2 2.5


watch-frac (O Rnd Nbhds) 0.98 1

. .O domain level O O -+ domain level1

0.88 ' 1 O 0.5 1 1.5 2 2.5

Enor Threshold x 104


cat-frac (O Rnd Nbhds)

cn 1 2 +. 0.8 . .

O 3 0 domain Ievel O 0 +-+ domain level 1

Error Threshold to4

pool.frac (O Rnd Nbhcîs)

' /: O domain leuel 0 $chemq ! + -+ domain level 1 Schern f O

.


lena-frac (O Rnd Nbhds) 1 1 I

- 0 dornain level O -+ dornain level 1

0.7 ' l O 0.5 1 1.5 2 2.5

Error Threshold x 104

water-frac (O Rnd Nbhds)

I l

, f. b- 0 d o r n a i n T I ç c h e m s / t 0 -+ domain level1 Schern f

--- O 0.5 1 1.5 2 2.5

Error Threshold 10'


arcticharehac (1 Rnd Nbhds)

'1

O dornain level O O -+ domain level 1

Error Threshold x lod

baboon.frac (1 Rnd Nbhds) 1 1

:0.6/d (t - O domain level0 O -+ domain tevel 1

U.J



I l

Error Thres hold 10'



Conipression Ratios 125

cat.frac (1 Rnd Nbhds)

Error Threshold 104

pool.frac (1 Rnd Nbhds)

r

0 domain level0 b-+ domain level 1


I l

r I

0. .O domain level O Schemer ! +-+ domain level 1 Scheme !

I I Enor Threshoid x lo4

water.frac (1 Rnd Nbhds)

. .O domain level 0 Schem f -+ domain level 1 r Schem el :

Error Threshold x 10' Enor Threshold x IO'


arctichare.frac (2 Rnd Nbhds)

0.9 ' O 0.5 1 1.5 2 2.5


baboomfrac (2 Rnd Nbhds)

0 domain level O -+ domain level 1

0.5 [ l O 0.5 1 1.5 2 2.5



I l

. ,O domain level O Schem 5 0 7 5 i d 0.7 1 -+ dornain level1 Schem 5

0.65 O 0.5 1 t.5 2 2.5

Enor Threshold 10'

watch.frac (2 Rnd Nbhds) 0.98 rrl

Enot Threshold 10'

0 domain level O Scheme 5 O 1:> i-+ dornain level 1 1 Schernq 5 1


cat-frac (2 Flnd Nbhds)

'1 0 domain level0 1 -+ domain level f 4-

1 1.5 Error Threshold


0.94 1 O 0.5 1 1.5 2 2.5

Error Thres hold x IO'

lena.fr;tc (2 Rnd Nbhds)

I l

. .o domain level O O -+ domain level 1

I I ' I

Enor Threshold IO'


$ 0 damain level O [schernj ! . -+ domain level 1 Schem !

+ O 0.5 1 1.5 2 2.5

Enor Threshold x IO'


Appendix E

SNR

130 SNR

O 0.5 1 1.5 2 2.5 Error Threshold 10'

3- 0 Ave. dom O(Scheme ' +-+ Ave. dom. 1 (Scheme

+\ .

. .


house-f rac

O 0.5 1 1.5 2 2.5 Error Threshold IO'

watch.f rac

0 Ave. dom O(Scheme 261P 1-+ A v e . c i o r n . ~ ( S c h Z ~ )


SNR 131

cat.f rac

- - 0 Ave. dom O(Scheme d -+ Ave. dom. 1(Scheme

Error Threshold x lo4

1 . /:o~ve. dom O(Scheme 1) 1. -+ Ave. dom. 1 (Scheme 1


lena-frac

'y O Ave. dom O(Scherne .) %+-+ Ave. dom. 1 (Scheme

0.5 1 1.5 2 2.5 Error Threshold x IO*

u + b 0 Ave. dom O(Scheme ' y. +-+ Ave. dom. 1 (Scheme

. . . . . . ';. . '

0.5 1 1.5 2 2.5 Error Thresbold x lo4

132 SNR

20 .Y 3. 0 Ave. dom O(Scherne \ +-+ Ave. dom. 1(Scheme

19. , i


O +

3 0 Ave. dom O(Scherne +-+ Ave. dom. 1 (Scheme


. .O Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme

21 1 O 0.5 1 1.5 2 2.5

Error Threshoid x IO'

0- 0 Ave. dom O(Scheme . +-+ Ave. dom. 1(Scheme

I O 0.5 1 1.5 2 2.5

Error Threshold 10'

SNR 133

cat.f rac

0 Ave. dom O(Scherne -+ Ave. dom. 1 (Scheme ]

\


O 0.5 1 1 .S 2 2.5 Error Threshold x 104

lena-frac

0 Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme


water.f rac

I 1

0 Ave. dom O(Scheme t -+ Ave. dom. 1 (Scheme P

O 0.5 1 1.5 2 2.5 Error Threshold x 10'

134 SNR

V

O 0.5 1 1.5 2 2.5 Error Threshold 104


house-frac

. 0 Ave. dom O(Scheme Y., -+ Ave. dom. 1 (Scheme


25t+tt O Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme


- -


SNR 135 .

cat.f rac



pool-f rac

0 Ave. dom O(Scherne -+ Ave. dom. 1(Scheme


& o Ave. dom O(Scheme 9k - -+ Ava. dom. 1 (Scherne


o Ave. dom O(Scheme \ -+ Ave. dom. 1 (Scheme b )


136 SNR

arctichare-f rac

o Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme

0.5 1 1.5 2 2.5 Error Threshold 10'

0 Ave. dom O(Scheme

15' 0:5 I

O 1 1.5 2 2.5 Error Threshold x 10'

0 Ave. dom O(Scheme Ave. dom. 1 (Schem


0 0 Ave. dom O(Scheme +-+ Ave. dom. 1(Scheme

O 0.5 1 1.5 2 2.5 Ermr Threshold x IO'

SNR 137

cat.f rac

o Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme t 1


- O Ave. dom O(Scheme -+ Ave. dom. 1(Scheme 41 ) 1

16l 1 O 0.5 1 1 .S 2 2.5


0 Ave. dom -+ Ave. dom.


water.f rac

O Ave. dom O(Scheme -+ Ave. dom. 1(Scheme


138 SNR

arctichare.frac (1 Rnd Nbhds) 9e:

0 Ave. dom O(Scheme


baboon.frac (1 Rnd Nbhds)

0 Ave. dom O(Scheme -+ Ave. dom. 1

O 0.5 1 1.5 2 2.5 Error Threshold 104

house.ffac (1 Rnd Nbhds) b ,

1 3 O Ave. dom O(Scheme

'. +-+ Ave. dom. 1 (Scheme \

h

Error Threshold

watch.frac (1 Rnd Nbhds) G -f- I

lgL O 0.5 1 1.5 2 2.5 Error Threshold x 10'

1 ' 3 O Ave. dom O(Scheme +-+ Ave. dom. 1 (Scheme

1

SNR 139




16 O 0.5 1 1.5 2 2.5



0 Ave. dom O(Scheme -+ Ave. dom. l(Scheme

O 0.5 t 1.5 2 2.5 Error Threshold 1o4


20.5 1 bit . 1. 0 Ave. dom, O(Scheme 1 -+ Ave. dom. 1 (Scheme

20 . . . : 4

Enor Threshoid x lo4

140 SNR

arctict-iare.frac (2 Rnd Nbhds)

O Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme

O 0.5 1 1.5 2 2.5 Enor Threshold x 104

baboon.frac (2 Rnd Nbhds) 22 1

0 Ave. dom O(Scheme -+ Ave. dom. 1(Scheme

16 1 O 0.5 t 1.5 2 2.5

Enor Threshold 10'

house-frac (2 Rnd Nbhds)

-, 3 O Ave. dom O(Scheme . .. , +-+ Ave. dom. l(Scheme

Error Threshold x lo4


r- 1. 0 Ave. dom O(Scheme

-+ Ave. dom. l(Scheme

j9' 015 I

O 1 1.5 2 2.5 Emr Threshold x 10'

SNR 141




pool.frac (2 Rnd Nbhds) 231 +


\

lena.frac (2 Rnd Nbhds) 26

Q',

,

1 K . 4 O AVB. dom O(Scheme 24 -+ Ave. dom. 1 (Scheme

3- O Ave. dom O(Scheme +-+ Ave. dom. 1 (Scheme


water-frac (2 Rnd Nbhds)

0.5 1 1.5 Emr Threshold

142 SNR

arctichare.frac (O Rnd Nbhds)

0 Ave. dom O(Scheme -+ Ave. dom. 1(Scheme

O 0.5 1 1.5 2 2.5 Error Thres hold x 104

baboon-frac (O Rnd Nbhds) 22 - t 1


t O 0.5 1 1.5 2 2.5


house.frac (O Rnd Nbhds)

, 3. . O Ave. dom O(Scheme Ave. dom. 1 (Scheme


watch.frac (O Rnd Nbhds)

IgL 0:s I

O 1 1.5 2 2.5 Error Threshoîd x 10'

SNR 143

cat.frac (O Rnd Nbhds)

1 Q /: 0 Ave. dom O(Scherne 28 , -+ Ave. dom. 1 (Scheme

20 1 O 0.5 f 1.5 2 2.5

Error Threshold 10'

pooLfrac (O Rnd Nbhds) 231 +

\ 1 1

O Ave. dom O(Scheme +-+ Ave. dom. 1 (Scheme

21

vu - --


lena-frac (O Rnd Nbhds) 26 1 1

0 Ave. dom O(Scheme 24 -+ Ave. dom. l(Scheme

J -

O 1 1.5 2 2.5 Error Threshold lo4

water-frac (O Rnd Nbhds) 21 1 1

ZO.S /q,,, 0 Ave. dom O(Scheme $ 1 -+ Ave. dom. 1 (Scheme )

20

-

O 0.5 1 1 .S 2 2.5 Error Threshold x 104

144 SNR

arctichare.frac (1 Rnd Nbhds) 25 i

0 Ave. dom O(Scheme

O ' l O 0.5 1 1.5 2 2.5

Error Threshold 10'

baboon-frac (1 Rnd Nbhds)

0 Ave. dom O(Scheme +-+ Ave. dom. 1(Scheme ) 1

16' I O 0.5 1 1.5 2 2.5

Error Threshold fo4


21 1 O 0.5 1 1.5 2 2.5

Error Threshold x IO'

,

watchhac (1 Rnd Nbhds)

24

3 O Ave. dom O(Scheme

19l I O 0.5 1 1.5 2 2.5


25 1 ; +-+ Ave. dom. 1 (Schernepd

SNR 145

cat-frac (1 Rnd Nbhds)

l 1

- 0 Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme



O 0.5 1 1.5 2 2.5 Error Threshold IO'


q-. - - . ?. O Ave. dom O(Scherne $ x+-+ Ave. dom. l(Scheme

Errer Threshold x 10'


4 => 0 Ave. dom O(Scheme \ +-+ Ave.dom. 1(Scherne

\

' P .


146 SNR

arctichare.frac (2 Rnd Nbhds) 25 5

I i: 0 Ave. dom O(Scheme 20 \ -+ Ave. dom. 1(Scheme


bzboon.frac (2 Rnd Nbhds) 33

- O Ave. dom O(Scheme f +-+ Ave. dom. l(Scheme ) 1

16' 0:s I

O 1 1.5 2 2.5 Error Threshold x 10'


0 Ave. dom O(Scherne -+ Ave. dom. 1 (Scheme



O Ave. dom O(Scherne -+ Ave. dom. 1 (Scheme

O 0.5 1 1.5 2 2.5 Emr Threshold x 10'

SNR 147

cat-frac (2 Rnd Nbhds) 30 1 I


20 1 I O 0.5 1 1.5 2 2.5



0 Ave. dom O(Scheme

21

I V


lena-frac (2 Rnd Nbhds)

1'- 9 0 Ave. dom O(Scheme 24 -+ Ave. dom. l(Scheme

t6I 1 O 0.5 1 1.5 2 2.5


water.frac (2 Rnd Nbhds) 21 1 i

O Ave. dom O(Scheme -+ Ave. dom. 1 (Scheme

O 0.5 1 1.5 2 2.5 Error T hteshold x 10'

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Corn - collectionscanada.gc.ca · rnnp ulhere (S. (1) is a rumplete metric space then f has exactly one fied point a E il and Vx E S the sepuence gzuen by: converges to a. NOTE: f