CHAPTER 3

FRACTAL TECHNIQUES IN IMAGE COMPRESSION

3.1 Introduction

Data compression has become an important issue in relation to storage and

transmission of information. Specifically, digital image compression is important due to

the high storage and transmission requirements. Uncompressed (bit-mapped) graphic files

are often very large in size. A 512 pixels by 512 pixels, though not very high resolution,

grey-scale image takes up over 260,000 bytes and much more if the image is in color.

Also, the network bandwidth of an average computer is very less. Various compression

methods have been proposed in recent years using different techniques to achieve high

compression ratios. In almost all these methods, the compressed images will be the

approximate of the originals. There are two general categories of compression methods ­

lossless and lossy methods. A lossless method will produce an image that, when

decompressed, is identical to the original image down to the last bit. A lossy method, on

the other hand, will produce an image that closely resembles but is not an exact duplicate

of the original image. The major drawback of lossless methods is that they cannot

achieve very high compression ratios. Hence, for image compression applications where

the input files are usually measured in megabytes, and where the losses of graphic details

are not critical, lossy methods are mostly used. Lossless compression are used in

applications, such as the compression of text files or executable codes.

One of the lossy image compression methods currently available is the method of

fractal image compression, developed by Michael Barnsley (1993) and his associates

( Barnsley, 1993). The method is a proprietary technology of Iterated Systems, Inc., a

firm co-founded by Barnsley. Other examples of lossy methods are JPEG (the Joint

Photographic Experts Group standard), and GIF (Graphics Interchange Format, an image

compression standard originally developed by CompuServe on-line service).

Image compression methods can also be classified as either symmetrical or

asymmetrical. For symmetrical methods, the compression and the decompression

processes take roughly the same amount of time/effort. Fractal image compression is an

example of asymmetrical method. Asymmetrical methods take more time to compress an

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image than to decompress it. The idea is to do most of the work during the compression,

thus creating an output file that can be decompressed very quickly.

The theoretical foundation for fractal compression algorithms is given by fractal

theory (Fisher, 1995; Jacquin, 1993). Fractal image compression finds coincidental self­

similarities between arbitrarily chosen square image segments. The use of fractals for

compression is well-established in the area of signal processing The representation used

by fractal image compression is called partitioned Iterated Function Systems(IFS). The

basic principle is that an image can be reconstructed by using the self similarities in the

image itself. When encoding an image, the algorithm partitions the image into a number

of square blocks ( domain blocks). After this, a new partition into smaller blocks

(range blocks) takes place. For every range block, the best matching domain block is

searched among all domain blocks by performing a set of transformations on the blocks.

The compression is obtained by storing only the descriptions of these transformations

(Frigaard et al., 1994). A number of algorithms have been proposed by various authors

(Muhlmann and Hanslmeier, 1996; Fisher, 1992; Jacquin, 1992; Saupe, 1994; Polvere

and Nappi, 2000 ; Ruhl et al., 1997 ; Harnzaoui and Saupe, 2000; Mukherjee et al.,

2000 ; Hart 1996 ; Saupe and Hamzaoui, 1994) to increase the efficiency of this method.

3.2 Basic concepts and theory on fractal compression

An affine transformation in Rn is a function consisting of a linear transformation

and a translation in Rn• Affine transformations in R2

, for example, are of the form

W(x,y) = (ax+by+e,cx+dy+f) ..........(3.1)

where the parameters a,b,c, and d form the linear part, which determines the rotation,

skew, and scaling; and the parameters e and f are the translation distances in the x and y

directions, respectively.

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An affine map is said to be contractive if its contractivity factor s is less than 1,

i.e. it "reduces" the size of the domain images. Given a finite collection of contractive

affine maps Wi, W2, ... ,Wm this collection forms an iterated functions system (IFS). If B is

a compact, nonempty subset of Rn, then the map

WeB) = ..•.•...(3.2)

is a contractive map on H, the (complete) metric space of compact sets in Rn• W has a

unique fixed point in H, say A. Then A is a compact subset of Rn, and is called the

attractor of W.

The Collage Theorem states that if an IPS gives a collage of B which is avery

good approximation of B (i.e. Band WeB) are very close in Hausdorff distance), then the

attractor will be very close to (looks like) the original set B. For compression purposes,

the requirement is such that the number of transforms should be as small as possible

with decoding error below a specified quality level.

3.3 Methodology

The fractal compression procedure consists of encoding and decoding modules.

3.3.1 Encoding

The fractal image compression program first partitions the original image into

nonoverlapping domain regions of any size or shape. Then a collection of possible range

regions is defined. The range regions can overlap and need not cover the entire image,

but must be larger than the domain regions, For each domain region the algorithm then

searches for a suitable range region that, when applied with an appropriate affine

transformation, very closely resembles the domain region. Afterward, a FIF (Fractal

Image Format) file is generated for the image. This file contains information on the

choice of domain regions, and the list of affine coefficients (i.e. the entries of the

transformation matrix) of all associated affine transformations. So all the pixels' data in a

given region are compressed into a small set of entries of the transform matrix, with each

entry, corresponding to an integer between 0 and 255, taking up one byte.

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This process is independent of the resolution of the original image. The output

graphic will look like the original at any resolution, since the compressor has found an

IPS whose attractor replicates the original one (i.e. a set of equations describing the

original image). The process does a lot of work, during the search for the suitable range

regions. But once the compression is done, the FIF file can be decompressed very

quickly. Thus, fractal image compression is asymmetrical. The practical implementations

of a fractal compressor offer different levels of compression. The lower levels have more

relaxed search criteria to cut down processing time, but with the loss of more detail. The

higher levels give very good detail, but takes a long time to process each image

3.3.2 Decoding

To decompress an image, the compressor first allocates two memory buffers of

equal size, with arbitrary initial content. The iterations then begin, with buffer 1

containing the range image and buffer 2 containing the domain image. The domain

image is partitioned into domain regions as specified in the FIF file. For each domain

region, its associated range region is located in the range image. Then the corresponding

affine map is applied to the content of the range region, pulling the content towards the

map's attractor. Since each of the affine map is contractive, the range region is contracted

by the transformation. This is the reason why the range regions are required to be larger

than the domain regions during compression.

For the next iteration, the roles of the domain image and range image are

switched. The process of mapping the range regions (now in buffer 2) to their respective

domain regions (in buffer 1) is repeated, using the prescribed affine transformations.

Then the entire step is repeated again and again, with the content of buffer 1 mapped to

buffer 2, then vice versa. At every step, the content is pulled ever closer to the attractor of

the IPS which forms a collage of the original image. Eventually the differences between

the two images become very small, and the content of the first buffer is the output

decompressed image (Anson, 1993).

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3.3.3 Computational Complexity

The main computations in fractal coding are trying every potential domain block

for each range, finding the best transformations for it and the computation of the

matching of the result with the range block. If the size of the image is 256 x 256 with

ranges 4 pixels wide and domains 8 pixels wide, altogether there are 64 x 64 = 4096

ranges and 249 x 249 = 62001 domains, if the domain separation is one pixel. Thus

253956096 domain-range comparisons are required. Each comparison involves an error

analysis to find the optimal affine map. Such lengthy computations can be reduced by

increasing the domain separation. The maximum separation possible in this case is eight,

which is the size of the domain. Though the computational complexity gets minimised,

this reduces the quality of the reconstructed image.

Another method of reducing the computational complexity is by domain and

range classification. By classifying domains and ranges, the domain-range comparisons

that possess the probability of a domain-range match is avoided

3.3.4 Comparison with DCT

Fractal-transform image compression overcomes many drawbacks of DCT.

Decompression speed, resolution independence and the compression ratios distinguish

fractal image compression from DCT. DCT breaks an image into 8 by 8 pixel blocks

and then uses some mathematical computations to decide the image information that can

be thrown away without damaging the appearance of the image too much. DCT

transforms the image data in the 8 by 8 block mathematically from x,y space into

frequency space. Instead of viewing the data as an array of 64 values arranged in an 8 by

8 grid, DCT views it as a varying signal that can be apprOXimated by a collection of 64

cosine functions with appropriate amplitudes. Each cosine that DCT uses as a basis

function is associated with a value called its DCT coefficient, which determines each

cosine function's amplitude. Most of the important visual information for typical

continuous-tone images is concentrated in the cosine functions with lower frequencies.

So, by giving less weight to higher frequency cosines and approximating small DCT

42

coefficients to zero, compression can be achieved without too much image degradation.

Storage space can be reduced if the remaining OCT coefficients are quantised to a

predefined set of values. The OCT method is effective at low compression ratios. Also,

the assumption that higher frequencies are unimportant does not hold if there are sharp

edges in the picture. Attenuating the higher frequency OCT basis functions result in

artifacts that look like ripples spreading out from the edges. This effect, known as Gibb's

phenomenon is an unavoidable aspect of DCT. Also, the compressed images are

resolution dependent. Display of the decompressed image with higher resolution than the

original will result in blockiness due to pixel replication. The fractal transform processes

use much larger and more complex regions when it deals with high resolution images. So

the size of a compressed FIF file does not have to increase in proportion to the number of

pixels in the image. Instead of suppressing the higher-frequency data associated with

sharp edges, fractal compression predicts edges at higher resolutions from the fractal

model determined during compression. The fractal-transform process is inherently

asymmetric. So more computation is required for compression than for decompression.

Fractal transform is relatively slow but decompression is fast. Also, compression ratios

can be improved by taking more time during compression without any increase in

decompression time .Many application developers prefer fractal image compression for

multimedia applications where quick access to high quality images is essential. Fractal

image compression is used by Microsoft in its Encarta multimedia encyclopedia. ( Anson,

1993).

3.4 Applications of fractal compression

Barnsley's fractal transform was the first practical algorithm used to

mathematically describe a real-world bitmap image in terms of its fractal properties.

Two tremendous benefits of converting conventional bitmap images to fractal data are

(i) The ability to scale any fractal image up or down in size without the introduction of

image artifacts or a loss in detail that occurs in bitmap images. This process of "fractal

43

zooming" is independent of the resolution of the original bitmap image, and the zooming

is limited only by the amount of available memory in the computer.

(ii)The size of the physical data used to store fractal codes is much smaller than the size

of the original bitmap data. Hence fractal images are more than 100 times smaller than

their bitmap sources. It is this aspect of fractal technology, called fractal compression,

that has promoted the greatest interest within the computer imaging industry. Fractal

compression is an asymmetrical process, taking longer time to compress an image than

to decompress it. This characteristic limits the usefulness of fractally compressed data to

applications where image data is constantly decompressed but never recompressed.

Fractal compression is therefore highly suited for use in image databases for searching

and retrieval purposes and CD-ROM applications. The content and resolution of the

source bitmap can greatly affect fractal compression. Images with a high fractal content

(e.g., faces, landscapes, and intricate textures) result in much higher compression ratios

than images with a low fractal content (e.g., charts, diagrams, text, and flat textures).

High-resolution images may be compressed to achieve higher compression ratios and will

still retain a high image quality. To retain a high quality for lower resolution images, the

resulting compression ratio will be much lower. Images with a greater bit depth (such as

24-bit truecolor images) will also compress more efficiently than images with fewer bits

per pixel (such as 8-bit gray-scale images) (Fisher, 1995 ; McGregor et al., 1996).

3.5 Variation of compression with image characteristics

The properties of fractal compression can be better understood by comparing

certain image parameters which are obtained before and after compression. The

parameters chosen for our study are fractal dimension and spectral flatness measure. The

parameter fractal dimension is increasingly becoming important in processing of images

in various disciplines. High-resolution images of solar granulation contain many

structures and a large amount of noise. Compressing them with common losses

algorithms such as gzip or compress leads to a compression with data reduction of only

15 to 20%. The noise in these images is responsible for the bad compression rate, since it

44

is incompressible. Therefore, it is necessary to make use of a lossy algorithm if high

compression ratios are to be achieved (Muhlmann and Hanslmeier, 1996).

Image compression is required for preview functionality in large image databases

such as Hubble Space Telescope archive with interactive sky atlases, linking image and

catalog information (e.g. Aladin, Strasbourg Observatory) and for image data

transmission, where more global views are communicated to the user followed by more

detail, if discussed. Much work has been progressing in the area of fractal analysis of

astronomical and astrophysical structures. Wu et al. (1997) investigated the application

of multifractal formalism to characterize the galactic structure by the distribution of Hii

regions. The physical processes leading to fractal structure are fragmentation and

clustering. The galaxy distribution has been described to be formulated by fractals. Here

fractals with both upper and lower scale limits are considered. The upper limit is fixed

at least by the relic background temperature fluctuation limits and the lower limit is given

by galaxy sizes. The main property of a fractal, namely self-similarity is demonstrated

by the observed scaling of mean diameters of large voids by sample size. The first

number characterizing a fractal is its dimension. The fractal dimension for the

distribution of galaxies can be computed using the idea of capacity dimension

(Mandelbrot, 19~). Dividing the volume under consideration into separate cubic cells of

size 1, the number of cells containing at least one galaxy is N(l). The fractal dimension

using capacity dimension is (Fairwall, 1988 ; Saar, 1988)

D = -d (log N) / d (log 1) ............ (3.3)

This can be extended for larger scales which make galaxy counts fluctuate across

the sky. Peebles(1989) in his studies on fractal galaxy distribution concludes that the

galaxy distribution changes from a pure fractal with dimension

3-1 = 1.23

45

where r < ro ............ (3.4)

Feitzinger and Galinski (1987) studied the morphological structure of the distribution of

the star forming sites in galaxies and found it to be fractal in nature. Hence, fractal

dimension, being an important parameter for galaxy studies is chosen in this study.

Various indices can be defined for an image. The indices tried out in this work are

image activity measure obtained from edge information denoted by ledge and another

index Igradient obtained from the gradient of the image and finally the spectral flatness

measure (sfm) (Saha and Vemuri, 2000). The image activity measure obtained from

edge information is defined as

M*N

ledge = (11 ( M * N) L B (Ii) ) * 100

i=l

...........(3.5)

The image activity measure obtained from gradient information is defined as

M-l N M N-l

Igradient=1IM*N [L L I (I (i,j)-I(i+l,j» I121 + L L ICICi,j)-ICi,j+1»1I21]

i=l j=l i=l j=l

............ (3.6)

The spectral flatness measure is an important index of the image. This parameter

defines the level of activity of an image and is defined as the ratio of the arithmetic mean

to the geometric mean of the Fourier coefficients (Revathy et al., 2000). It is expressed

as

MN MN

sfm ={ [(1/MN)*LLIF(i,j)1 2]/[I1I1] }*[IF(i,j)1 2]IIMN •••••••(3.7)

i=lj=l i=lj=l

where F(i, j) is the (i, j)th Fourier coefficient of the image.

These indices are evaluated for a set of galaxy images. To make a comparison of

the three indices, they are plotted as in Figure 3.1. A denotes Igradienh B denotes sfm, and

C denotes ledge. It could be observed that spectral flatness measure varies the most, for

46

different images and hence it is the most active measure . So it has been chosen as

another important parameter in addition to fractal dimension in the present study.

50

40

A 3)II

m:::3

Sl 20

~"0.!:

10

0

..

A

B

/- ........._. ""- . - - C

/" . .../ .. /"

C') LO co f::: 0> co

~0> eo eo

~ ~ 80 ;1j r:: ~I"- eo

~LO

~ C')~ 0 0~ ~

,... ;C\J C\J C\J C\J C') C') C') ~ "'" ~

Figure 3.1: The variation of activity indices for a set of images

3.6 Image Catalogue

The availability of images, variety of structures in these images and well defined

classification schemes make the images of galaxies suitable for our present study which

involves compression, analysis and classification of images.

A data set of 113 galaxies developed by Frei et aI. (1996) at AT & T Bell

Laboratories. The galaxies were chosen to span the Hubble classification classes. All the

galaxies in the set are nearby, well resolved and bright with the faintest having the total

magnitude Br of 12.90. The sample was chosen to be suitable to test automatic galaxy

classification techniques with the idea that automatic methods of classifying galaxies are

necessary to handle the huge amount of data that will soon be available from large survey

47

projects, such as the Sloan Digital Sky Survey. All the images of the set were recorded

with charge coupled devices (CCDs) at the Palomar Observatory with the 1.5 meter

telescope and at the Lowell Observatory with the 1.1 meter telescope. The images were

stored in FITS (Flexible Image Transport System) format with important data on these

galaxies published in the Third Reference Catalog of Bright Galaxies (de Vaucouleurs et

al., 1991). Table 3.1 shows the number of spiral and elliptical galaxies observed at both

of the observatories along with the broad band pass wavelengths of the filters through

which they were observed. The images were processed to a point where the flat field and

bias corrections were made and stars were removed from them (Thankee, 1999).

Observatory Spiral Galaxies Elliptical Galaxies Bands (nanometer)Palomar 31 0 500, 650 and 820Lowell 58 14 450 and 650

Table 3.1: Details of data set

3.7 Algorithms

The algorithm for fractal compression implemented in this study consists of

compression module and decompression module. The algorithms for compression and

decompression are given below.

3.7.1 Algorithm for compression module

I. Main module

a. Read the input file name

b. Read the size of the image(x_size,y_size), overlap (to compute the domain

separation) and max_error.

c. Open the input file in read mode

d. Find the output filename by changing the extension to FIF

e. Open the output file in write mode

f. Call the comp_img() submodule

g. Stop

48

II. Algorithm for comp_img()

a. Call initialize( ) function

b. Call initdomain() function

c. for s = domsizemin to domsizemax,

call domclassify( ) function

d. Write horizontal and vertical image dimensions(x_size and y-size) and

the overlap parameter to the output file

e. Compute maxerror =(max_error)2

f. Call scanimage( ) function

g. Stop

III Algorithm for initialize() !initializes image data and cumulative datal

a. for y =0 to y_size

for x =0 to x_size

read pixel values from the input image and store it in range (y,x) array

b. for y =0 to y_size/2

for x = 0 to x_size/2

compute the domain from the range image.

c. for x = 0 to x_size/2

set cumulative data to zero

d. for y =0 to y_size/2

for x =0 to x_size/2

find the cumulative sum of domain and range

e. Stop

IV Algorithm for initdomain() /initializes the domain information!

a. for n =domsizemin to domsizemax

initialize domain information

b. Stop

V Algorithm for domnclassify( ) /classifies all domains of a particular size.

it is inserted in a linked list according to its class and size!

a. for class =0 to classmax

49

initialize all domain lists to zero

b. for y = 0 to (y_size - domsize)

for x = 0 to (x_size - domsize)

classify all domains of this size by calling the getclass( ) function

c. Stop

VI Algorithm for getclass () /classifies a domain or range by ordering the image

Brightness in the four quadrants of the domain or range/

a. Initialize array delta (3) = {6,2 I}

b. Get the cumulative values of each quadrant, sum(i)

c. for i =0 to 2

for j = 0 to 2

if sum( i) < sumO) then class = class + delta (i)

d. return class

e. Stop

VII Algorithm for scanimage ( ) /splits the image recursively/

a. Compute the size of the range using log...base2(s_size=1«log...base2)

b. If (log...base2 > domsizemax) call scanimage ( ) function recursively

four times . else call compcrng( ) function

c. If (x_size>s_size), call scanimage() recursively

d. If (y_size>s_size), call scanimage() recursively

e. Stop

VIII Algorithm for compr_rng( ) /compresses a range by searching a map

with all domains of the same class/

a. Get range size as csize = 1 «log...base2

b. Call getclass( ) function

c. For each domain domnhead(class,log...base2),call getrnap() function

d. If (bestdom = NULL) or (maperror2 < bestmaperror2) the~

bestmap = map, bestdom = dom

e. If( log...base2=domsizemin) then write' l' to the output file

Write bestmap30ntrast and bestmap_brightness to the output file

If(bestmap_contrast = 0) then return

50

Write domain number to the output file

End

Else

Begin

Write '0' to the output file , call compcrng( ) four times recursively.

End

f. Stop

IX Algorithm for getmap( ) lfinds out the best possible mapping from a domain

to rangel

a. Find csize as r_size=1«range_Io~base2

b. For ry = rangep_y to (rangep-y + r_size), compute rd

c. Compute csum = 0.25 x rd

d. Compute contrast and quantize it by calling the quantize( ) function

e. Compute brightness and quantize it by calling the quantize( ) function

f. Compute the sum of squared errors.

g. Return affine map

h. Stop

X. Algorithm for quantize() Iquantizes a value in the range 0.0 to max to

the range 0 to imaxl

a. Get the values for max and imax

b. ivaI = (integer)/(value/max)x(imax +1)

c. if (ivaI < 0) return 0, else if (ival >imax) return imax

d. return ivaI

e. Stop

3.7.2 Algorithm for decompression module

1. Main module( )

a. Read the filename of the compressed image

b. Read the number of iterations

c. Open the input file in read mode

d. Find the output filename by removing the extension FIF

e. Open the output file in write mode

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f. Call the expandfile( ) function

g. Stop

II Algorithm for expandfile ( )

a. Read x_size, y_size and overlap parameters from the input file

b. Call initdomain( ) function

c. For iter = 1 to no. of iterations, call iterimage( ) function

d. Call filterboundary( ) function

e. For y =0 to y_size( )

For x=O to x_size( )

Write range (y,x) to the output file

e. Stop

III Algorithm for initdomain( ) /initializes the domain information!

a. for n = domsizemin to domsizemax

initialize domain information

b. Stop

IV Algorithm for scanimage( ) /splits the squares recursively/

a. Compute the size of the range using lo~base2(s_size=I« lo~base2)

b. If (lo~base2 > domsizemax), call scanimage( ) function recursively four

times

c. If(x_size > s_size), call scanimage( ) recursively

d. If(y_size> s_size) call scanimage() recursively

e. Stop

V Algorithm for decmp_mg( ) /reads the affine map for a range or

split the range if the compressor did it in comp_mg( )/

a. if(lo~base2 * domsizemin) read check value from input file

b. if(lo~base2 = domsizemin) and (check = 1)

begin

read affinemap from input file

compute contrast and brightness

if(mapcontrast * 0)

begin

52

calculate the mapcontrast using contrast information

read domain number from input file

calculate x and y coordinates of the domain

end

else x and y coordinates of the domain are set to zero

end

else

call decmp_mg( ) functions four times recursively

c. Stop

VI Algorithm for iterimage( ) /refines the image by applying the

affine maps on the image!

a. For map =maphead to map =NULL

Apply affine map to each range

b. Stop

vn Algorithm for filterboundary( ) /smoothens the transition between adjacent

ranges by going through all the ranges

a. for map =maphead to map =NULL

begin

if (mapsize =1 «domsizemin) continue since the range is too small

if (map_x>1)

smoothen the left boundary of the range and the right boundary of the

adjacent range to the left

if (map-y>1)

smoothen the top boundary of the range and the bottom boundary of the range

above.

End

b. Stop

3.8 Performance Measures

A number of objective and subjective distortion measures for the performance

analysis is available to measure the amount of degradation of the reconstructed image.

53

The choice of a measure depends on factors such as easily computable, analytically

tractable and adapt to the human visual system characteristics.

A widely used measure of reconstructed image fidelity is the mean square error

MSE. MSE for an M x N image is defined as

1 N MMSE = -LL(X(i,j)- Y(i,j))2

NM i=l j=1

..................... (3.8)

Where X(iJ) and Y(IJ) represent original and reconstructed images respectively.

Though MSE is widely used, it completely ignores the psycho visual properties of the

human vision system. The Peak Signal to Noise Ratio (PSNR) defined as

P SNR =10 log 10 (Peak to peak value of the original image datar..(3.9). MSE

has been evaluated for various compression levels and for different images with different

fractal dimension and spectral flatness dimension.

3.9 Results

The fractal dimension (obtained by box counting) and spectral flatness measure

of 113 galaxies are given in Table 3.2. The features of fractal compression can be studied

by comparing compression ratio with different image parameters obtained before and

after compression. The parameters chosen in this study are fractal dimension and spectral

flatness measure (sfm).

54

GALAXYID FD SFM GALAXYID FD SFMNGC2403 1.518 728.464 NGC4406 1.557 987.378NGC2541 1.251 74.2313 NGC4414 1.741 866.526NGC2683 1.662 999.084 NGC4429 1.639 834.189NGC2715 1.506 729.234 NGC4442 1.454 708.556NGC2768 1.599 1392.81 NGC4449 1.670 1018.94NGC2775 1.603 380.999 NGC4450 1.635 850.514NGC2903 1.680 1148.44 NGC4472 1.694 4810.07NGC2976 1.613 933.724 NGC4477 1.419 1071.41NGC2985 1.456 335.554 NGC4486 1.721 5412.93NGC 3031 1.794 9119.12 NGC4487 1.163 347.647NGC3077 1.746 4506.09 NGC4498 1.448 140.890NGC3079 1.629 915.749 NGC4501 1.639 1388.92NGC 3147 1.258 295.127 NGC4526 1.686 1214.05NGC 3166 1.434 386.193 NGC4527 1.434 617.043NGC 3184 0.744 1096.79 NGC4535 1.292 816.900NGC 3198 1.46 593.099 NGC4548 1.405 1328.99NGC 3319 0.988 117.165 NGC4559 1.442 687.504NGC3344 1.478 1358.48 NGC4564 1.051 822.200NGC 3351 1.671 2984.17 NGC4569 1.579 1755.23NGC 3368 1.731 1816.34 NGC4571 1.464 258.584NGC3377 1.266 422.871 NGC4579 1.550 1821.95NGC3379 1.394 674.677 NGC4593 1.435 542.130NGC3486 1.560 458.099 NGC4594 1.598 411.025NGC3556 1.657 831.945 NGC4621 1.287 773.816NGC3596 1.576 536.911 NGC4636 1.604 1802.09NGC3623 1.625 2181.93 NGC 4651 1.619 471.965NGC 3631 1.226 867.067 NGC4654 1.503 538.753NGC3672 1.603 941.778 NGC4689 1.323 361.266NGC3675 1.562 942.208 NGC4710 1.618 1053.97NGC3726 0.800 771.595 NGC4725 1.552 1652.78NGC3810 1.530 1238.40 NGC4731 1.347 250.773NGC3877 1.606 1113.24 NGC4754 1.428 538.923NGC 3893 1.547 1031.08 NGC4826 1.637 1974.87NGC 3938 1.580 2137.23 NGC4861 1.550 178.940NGC3953 1.604 1203.43 NGC4866 1.048 1312.81NGC4013 1.245 115.970 NGC5005 1.501 1239.40NGC4030 1.660 739.456 NGC5033 1.376 532.915NGC4088 1.689 889.796 NGC5055 1.584 1392.80NGC4123 0.910 378.211 NGC5204 1.541 308.636NGC 4125 1.173 483.078 NGC5248 1.501 558.875NGC4136 1.458 378.950 NGC5322 1.270 510.201NGC4144 1.479 813.850 NGC 5334 1.307 57.4779NGC 4157 1.122 751.298 NGC5364 1.609 1159.68NGC 4178 1.360 318.254 NGC 5371 1.239 633.234NGC4189 1.423 268.493 NGC5377 1.292 319.319NGC 4192 1.534 1263.53 NGC 5585 1.379 723.914NGC4216 1.459 1108.64 NGC5669 1.244 262.189NGC4242 1.490 2059.26 NGC 5701 1.565 707.354NGC4254 1.543 690.790 NGC5746 1.623 1349.82

NGC4258 1.739 2059.26 NGC 5792 1.413 575.952

NGC4303 1.665 578.501 NGC 5813 1.355 664.867

NGC 4321 1.546 983.399 NGC 5850 1.280 341.473

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NGC4340 1.120 919.043 NGC 5985 1.657 1320.31NGC4365 1.608 537.914 NGC 6015 1.605 666.885NGC4374 1.510 688.845 NGC 6118 1.225 860.594NGC4394 1.533 508.440 NGC 6384 1.617 476.84

NGC 6503 1.548 794.48

Table 3.2: Fractal dimension and sfm of 113 galaxies

3.9.1 Variation of fractal dimension with compression

The galaxy images with different fractal dimension and spectral flatness measure

were compressed and reconstructed based on fractal techniques. The fractal dimension

of reconstructed images are higher compared to that of the original images. Typical

examples of fractal compression are given is Figure 3.2.

(a) (b) (c) (d)

(e) (0 (g) (h)

Figure 3.2 Images of NGC 2403, NGC 2541, NGC 3079, NGC 4144(a) - (d) : original images; (e)-(h) reconstructed images

3.9.2 Variation of spectral flatness measure with compression

It could be observed that spectral flatness measure of reconstructed images are more,

compared to that of the original images. This is depicted in Table 3.3.

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Images from FD after Sfm after Compression PsnrNGC compression compression ratio

2403 1.81 1225.42 91 21.482541 1.64 248.66 93 27.612768 1.58 1533.24 94 26.563079 1.69 1308.76 92 24.684123 1.9 886.98 84 26.964125 1.29 1031.68 80 24.324136 1.54 1027.87 88 26.374178 1.45 883.67 92 25.274498 1.56 883.69 92 23.924501 1.66 1575.29 91 29.02

Table 3.3 : Variation of fd and sfm with compression

3.9.3 Variation of fractal dimension with compression ratio

The variation of fractal dimension with compression ratio has been studied for

images from the near by galaxy catalogue. Based on this result, .a linear relationship

between fractal dimension with compression ratio is found out. The relation has come

to be

y = 80.35 + 5.79 X •.•••••••(3.10)

where X denotes fractal dimension and Y denotes the compression ratio. From this

relation, given an image, its fractal dimension can determine the range of compression

one can obtain by fractal compression even before actually compressing it.

3.9.4 Variation of spectral flatness measure with compression ratio

The variation of spectral flatness measure with compression ratio has been

studied. A linear relationship between spectral flatness measure with compression

ratio is found out. The relation is

y = 88.64 + 0.0009 X

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••••••••. (3.11)

where X denotes spectral flatness measure and Y denotes the compression ratio. This

relation is an important result as, given an image, its spectral flatness measure can

determine the range of compression one can obtain by fractal compression even before

actually compressing it.

3.9.5 Relationship of sfm with psnr

In order to obtain a relation between psnr and sfm, compression ratios of a few

galaxies were found out and a linear fit was made between these two parameters. The

psnr - sfm equation is found to be

y = 25.77 - 0.0001 X ......... (3.12)

where X is the spectral flatness measure and Y is the psnr. So, given an image, the

spectral flatness measure can be computed from which the quality of an image can be

known.

3.9.6 Relation of fractal dimension with psnr

, Working on the similar lines as above, a relation between psnr- fractal dimension

was found out. It is

Y = 24.47 + 0.70 X ......... (3.13)

where X is the fractal dimension and Y is the psnr. Given an image, the fractal dimension

can be computed from which the ratio of signal to noise can be computed even without

compressing it.

3.10 Conclusion

In this chapter, fractal based compression and its dependence on fractal dimension

for a set of images have been studied. The features fractal dimension and spectral flatness

58

measure of these images are evaluated. The compression levels, psnr (peak signal-to

noise-ratio) values and their variation with spectral flatness measure and fractal

dimension are also evaluated. Relations between the psnr and fractal dimension as well as

spectral flatness measure has been obtained. These can determine beforehand the amount

and range of compression one can get for a given image, even before actually

compressing it.

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