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Image Compression. Chapter 8. 1 Introduction and background. The problem: Reducing the amount of data required to represent a digital image. Compression is achieved by removing the data redundancies: Coding redundancy Interpixel redundancy Psychovisual redundancy.

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Image Compression

Chapter 8


1 Introduction and background

  • The problem: Reducing the amount of data required to represent a digital image.

  • Compression is achieved by removing the data redundancies:

    • Coding redundancy

    • Interpixel redundancy

    • Psychovisual redundancy


1 Introduction and background (cont.)

  • Structral blocks of image compression system.

    • Encoder

    • Decoder

  • The compression ratio

    where n1 and n2denote the number of information carrying units (usually bits) in the original and encoded images respectively.


1 Introduction and background (cont.)


1 Introduction and background (cont.)

function cr = imratio(f1,f2)

%IMRATIO Computes the ratio of the bytes in two images / variables.

%CR = IMRATIO( F1 , F2 ) returns the ratio of the number of bytes in

%variables / files F1 and F2. IfF1 and F2 are an original and compressed

%image, respectively, CR is the compression ratio.

error(nargchk(2,2,nargin)); %check input arguments

cr = bytes(f1) / bytes(f2); %compute the ratio

The Function that finds the compression ratio between two images:


1 Introduction and background (cont.)

(cont.)

% return the number of bytes in input f. If f is a string, assume that it

% is an image filename; if not, it is an image variable.

function b = bytes(f)

if ischar(f)

info = dir(f); b = info.bytes;

elseif isstruct(f)

b = 0;

fields = fieldnames(f);

for k = 1:length(fields)

b = b + bytes(f.(fields{k}));

end

else

info = whos('f'); b = info.bytes;

end


1 Introduction and background (cont.)

>> r =imratio( (imread('bubbles25.jpg')), bubbles25.jpg')

r =

0.9988

>> f = imread('bubbles.tif');

>> imwrite(f,'bubbles.jpg','jpg')

>> r = imratio( (imread( 'bubbles.tif' ) ) , 'bubbles.jpg')

r =

14.8578


1 Introduction and background (cont.)

  • Let denote the reconstructed image.

  • Two types of compression

    • Loseless compression: if

    • Loseless compression: if


1 Introduction and background (cont.)

  • In lossy compression the error between is defined by root mean square which is given by


1 Introduction and background (cont.)

The M- Function that computes e(rms) and displays both e(x,y) and its histogram

%COMPARE Computes and displays the error between two matrices. RMSE =COMPARE (F1 , F2, SCALE) returns the root-mean-square error between inputsF1 and F2, displays a histogram of the difference, and displays a scaleddifference image. When SCALE ,s omitted, a scale factor of 1 is used

function rmse = compare(f1 , f2 , scale)

error(nargchk(2,3,nargin));

if nargin < 3

scale = 1;

end

%compute the root mean square error

e = double(f1) - double(f2);

[m,n] = size(e);

rmse = sqrt (sum(e(:) .^ 2 ) / (m*n));


1 Introduction and background (cont.)

(cont.)

%output error image & histogram if an error

if rmse

%form error histogram.

emax= max(abs(e(:)));

[h,x]=hist(e(:),emax);

if length(h) >= 1

figure; bar(x,h,'k');

%scale teh error image symmetrically and display

emax = emax / scale;

e = mat2gray(e,[-emax,emax]);

figure; imshow(e);

end

end


1 Introduction and background (cont.)

>> r1 = imread('bubbles.tif');

>> r2 = imread('bubbles.jpg');

>> compare(r1, r2,1)

>> In E:\matlab\toolbox\images\images\truesize.m (Resize1) at line 302

In E:\matlab\toolbox\images\images\truesize.m at line 40

In E:\matlab\toolbox\images\images\imshow.m at line 168

In E:\matlab\work\matlab_code\compare.m at line 32

ans =

1.5660


1 Introduction and background (cont.)

Error histogram


1 Introduction and background (cont.)

Error image


2 Coding redundancy

  • Let nk denotethe number of times that the kth gray level appears in the image and n denote the total number of pixels in the image. The associated probability for the kth gray level can be expressed as


2 Coding redundancy (cont.)

  • If the number of bits used to represent each value of rk isl(rk), then the average number of bits required to represent each pixel is

  • Thus the total number of bits required to code an M×N image is MNLavg


2 Coding redundancy (cont.)

  • When fixed variable length that is l(rk) =m then


2 Coding redundancy (cont.)

  • The average number of bits requred by Code 2 is


2 Coding redundancy (cont.)

How few bits actually are needed to represent the gray levels of an image?

Is there a minimum amount of data that is sufficient to describe completely an image without loss information?

Information theory provides the mathematical framework to answer these questions.


2 Coding redundancy (cont.)

Formulation of generated information

Note that if P(E)=1 that is if the event always occurs then I(E)=0 and no information is attributed to it. No uncertainity associated with the event


2 Coding redundancy (cont.)

Given a source of random events from the discrete set of possible events with associated probabilities

The average information per source output, called the entropy, is


2 Coding redundancy (cont.)

If we assume that the histogram based on gray levels is an estimate of the true probability distribution, the estimate of H can be expressed by


1 Introduction and background (cont.)

function h = entropy(x,n)

%ENTROPY computes a first-order estimate of the entropy of a matrix.

%H = ENTROPY(X,N) returns the first-order estimate of matrix X with N

%symbols in bits / symbol. The estimate assumes a statistically independent

%source characterized by the relative frequency of occurence of the

%elements in X

error(nargchk(1,2,nargin));

if nargin < 2

n=256;

end


1 Introduction and background (cont.)

(cont)

x = double(x);

xh = hist(x(:),n);

xh = xh/sum(xh(:));

% make mask to eliminate 0's since log2(0) = -inf

i = find(xh);

h = -sum(xh(i) .* log2(xh(i))); %compute entropy


1 Introduction and background (cont.)

f = [119 123 168 119;123 119 168 168] ;

f = [f; 119 119 107 119 ; 107 107 119 119] ;

f =

119 123 168 119

123 119 168 168

119 119 107 119

107 107 119 119

p=hist(f(:),8)

p =

3 8 2 0 0 0 0 3


1 Introduction and background (cont.)

p =p/sum(p)

p =

Columns 1 through 7

0.1875 0.5000 0.1250 0 0 0 0 0.1875

H = entropy(f)

h =

1.7806


Huffman codes

  • Huffman codes are widely used and very effective technique for compressing data.

  • We consider the data to be a sequence of charecters.


Huffman codes (cont.)

Consider a binary charecter code wherein each charecter is represented by a unique binary string.

Fixed-length code:

a = 000, b = 001, c = 010, d = 011, e = 100, f = 101

variable-length code:

a = 0, b = 101, c = 100, d = 111, e = 1101, f = 1100


Huffman codes (cont.)

100

100

1

0

1

0

55

a:45

a:45

14

86

0

1

0

0

1

58

28

25

30

14

0

0

1

0

1

0

1

0

1

1

a:45

b:13

d:16

d:16

e:9

f:5

c:12

b:13

14

d:16

c:12

0

1

f:5

e:9

Fixed-length code

Variable-length code


Huffman codes (cont.)

Prefix code:

Codes in which no codeword is also a prefix of some other codeword.

Encoding for binary code:

Example:Variable-length prefix code.

a b c

Decoding for binary code:

Example:Variable-length prefix code.


Constructing Huffman codes


Constructing Huffman codes

  • Huffman’s algorithm assumes that Q is implemented as a binary min-heap.

  • Running time:

  • Line 2 : O(n) (uses BUILD-MIN-HEAP)

  • Line 3-8: O(n lg n) (the for loop is executed exactly n-1 times and each heap operation requires time O(lg n) )


Constructing Huffman codes: Example

f:5

e:9

c:12

b:13

d:16

a:45

c:12

b:13

d:16

a:45

14

1

0

0

f:5

e:9

14

d:16

25

a:45

0

1

0

0

1

f:5

e:9

c:12

b:13


Constructing Huffman codes: Example

a:45

25

30

0

1

0

1

c:12

b:13

d:16

14

0

1

f:5

e:9


Constructing Huffman codes: Example

55

a:45

1

0

30

25

0

0

1

1

d:16

14

c:12

b:13

0

1

f:5

e:9


Constructing Huffman codes: Example

100

1

0

55

a:45

1

0

30

25

0

1

0

1

d:16

14

c:12

b:13

1

0

f:5

e:9


Huffman Codes

function CODE = huffman(p)

%check the input arguments for reasonableness

error(nargchk(1,1,nargin));

if (ndims(p) ~= 2) | (min(size(p))>1)| ~isreal(p)|~isnumeric(p)

error('P must be a real numeric vector');

end

global CODE

CODE = cell(length(p),1);

if length (p) > 1

p=p /sum(p);

s=reduce(p);

makecode(s,[]);

else

CODE ={'1'};

end;


Huffman Codes(cont.)

%Create a Huffman source reduction tree in a MATLAB cell structure byperforming %source symbol reductions until there are only two reducedsymbols remaining

function s = reduce(p);

s= cell(length(p),1)

for i=1:length(p)

s{i}=i;

end

while numel(s) > 2

[p,i] = sort(p);

p(2) = p(1) + p(2);

p(1) = [];

s = s(i);

s{2} = {s{1},s{2}};

s(1) = [];

end


Huffman Codes(cont.)

%Scan the nodes of a Huffman source reduction tree recursively to generate

%the indicated variable length code words.

function makecode(sc,codeword)

global CODE

if isa(sc,'cell')

makecode(sc{1},[codeword 0]);

makecode(sc{2},[codeword 1]);

else

CODE{sc} = char('0'+codeword);

end


Huffman Codes(cont.)

>> p = [0.1875 0.5 0.125 0.1875];

>> c = huffman(p)

c =

'011'

'1'

'010'

'00'


Huffman Encoding

>> f2 = uint8 ([2 3 4 2; 3 2 4 4; 2 2 1 2; 1 1 2 2])

f2 =

2 3 4 2

3 2 4 4

2 2 1 2

1 1 2 2

>> whos('f2')

Name Size Bytes Class

f2 4x4 16 uint8 array

Grand total is 16 elements using 16 bytes


Huffman Encoding(cont.)

>> c = huffman(hist(double(f2(:)),4))

c =

'011'

'1'

'010'

'00'

>> h1f2 = c(f2(:))'

h1f2 =

Columns 1 through 9

'1' '010' '1' '011' '010' '1' '1' '011' '00'

Columns 10 through 16

'00' '011' '1' '1' '00' '1' '1'


Huffman Encoding(cont.)

>> whos('h1f2')

Name Size Bytes Class

h1f2 1x16 1018 cell array

Grand total is 45 elements using 1018 bytes

>> h2f2 = char (h1f2)'

h2f2 =

1010011000011011

1 11 1001 0

0 10 1 1

>> whos('h2f2')

Name Size Bytes Class

h2f2 3x16 96 char array

Grand total is 48 elements using 96 bytes


Huffman Encoding(cont.)

>> h2f2 = h2f2(:);

>> h2f2(h2f2 == ' ') = [];

>> whos('h2f2')

Name Size Bytes Class

h2f2 29x1 58 char array

Grand total is 29 elements using 58 bytes


Huffman Encoding(cont.)

>> h3f2 = mat2huff(f2)

h3f2 =

size: [4 4]

min: 32769

hist: [3 8 2 3]

code: [43867 1944]

>> whos('h3f2')

Name Size Bytes Class

h3f2 1x1 518 struct array

Grand total is 13 elements using 518 bytes


Huffman Encoding(cont.)

>> hcode = h3f2.code;

>> whos('hcode')

Name Size Bytes Class

hcode 1x2 4 uint16 array

Grand total is 2 elements using 4 bytes

>> dec2bin(double(hcode))

ans =

1010101101011011

0000011110011000


Huffman Encoding(cont.)

>> f = imread('Tracy.tif');

>> c=mat2huff(f);

>> cr1 = imratio(f,c)

cr1 =

1.2191

>> save SqueezeTracy c;

>> cr2 = imratio ('Tracy.tif','SqueezeTracy.mat')

cr2 =

1.2627


Huffman Decoding

function x = huff2mat(y)

% HUFF2MAT decodes a Huffman encoded matrix

if ~isstruct(y) | ~isfield(y,'min') | ~isfield(y,'size') | ~isfield(y,'hist') | ~isfield(y,'code')

error('The input must be a structure as returned by MAT2HUFF');

end

sz = double(y.size); m=sz(1); n= sz(2);

xmin = double(y.min) - 32768;

map = huffman(double(y.hist));

code = cellstr(char('','0','1'));

link = [2; 0; 0]; left = [2 3];

found = 0; tofind = length(map);


Huffman Encoding(cont.)

(cont.)

while length(left) & (found < tofind)

look = find(strcmp(map, code {left(1)}));

if look

link(left(1)) = -look;

left = left (2:end);

found = found + 1;

else

len = length (code);

link(left(1)) = len + 1;

link = [link; 0; 0];

code{end + 1} = strcat(code{left(1)},'0');

code{end + 1} = strcat(code{left(1)},'1');

left = left(2:end);

left = [left len + 1 len + 2];

end

end

x = unravel (y.code',link, m*n); %Decode using C 'unravel'

x = x + xmin - 1;

x = reshape(x,m,n);


Huffman Encoding(cont.)

(cont.)

#include mex.h

void unravel (unsigned short *hx, double *link, double *x, double xsz, int hxsz)

{

int i = 15, j=0, k=0, n=0;

while (xsz - k)

{

if (*(link + n) > 0)

{

if((*(hx + j) >> i) & 0x0001)

n = *(link + n);

else n = *(link + n) - 1;

if (i) i--;

else {j++; i=15;}

if( j > hxsz )

mexErrMsgTxt("OutOf code bits??");

}


Huffman Encoding(cont.)

(cont.)

else {

*(x + k++) = -*(link + n);

n = 0;}

}

if ( k == xsz - 1 )

*(x + k++) = -*(link + n);

}

void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[])

{

double *link, *x, xsz;

unsigned short *hx;

int hxsz;

if(nrhs != 3)

mexErrMsgTxt("Three inputs required.");


Huffman Decoding(cont.)

(cont.)

else if (nlhs > 1)

mexErrMsgTxt("Too many output arguments");

if( !mxIsDouble(prhs[2]) || mxIsComplex(prhs[2]) || mxGetN(prhs[2]) * mxGetM(prhs[2]) != 1 )

mexErrMsgTxt("Input XSIZE must be a scalar");

hx = mxGetPr(prhs[0]);

link = mxGetPr(prhs[1]);

xsz = mxGetScalar(prhs[2]);

hxsz = mxGetM(prhs[0]);

plhs[0] = mxCreateDoubleMatrix( xsz, 1, mxREAL);

x = mxGetPr(plhs[0]);

unravel(hx, link, x, xsz, hxsz);

}


Interpixel redundancy

  • Consider these two images:

  • They have virtually identical histograms

  • The histograms are three modal indicating the presence of three dominant grey levels


Interpixel redundancy (cont.)

  • Because the gray levels of the image are not equally probable, variable-length coding can be used to reduce the coding redundancy.

  • Note thatthe entropy estimates and the compression ratios of the two images are about the same.

  • The variable length coding does not take advantage of the obvious structral relationship between the aligned matches


Interpixel redundancy (cont.)

  • In order to reduce pixel redundancies, the 2-D pixel array normally used for human viewing and interpretation must be transformed into a more efficient format.

  • For example the difference between the adjacent pixels can be used to represent an image.

  • Transformations of this type are referred to as mapping.

  • If the original image can be reconstructed from such transformed set is referred to as reversible mappings.


Interpixel redundancy (cont.)

  • The following figure showsa lossless predictive coding.


Interpixel redundancy (cont.)

  • The following figure showsa lossless predictive coding.

The decoder reconstructs the prediction error from the received variable-length code words and performs the inverse operation


Interpixel redundancy (cont.)

  • Various methods can be used to generate .

  • In most cases the prediction is formed by a linearcombination of m previous pixels.

wherem is the order of the predictor and

are prediction coefficients. For 1-D linear predictive coding this equation can be rewritten


Interpixel redundancy (cont.)

Example: Consider encoding the Figure 8.7(c) using the simple

linear predictor


Psychovisual Redundancy

  • Psychovisual Redundancy is associated with real or quantifiable visual information.

  • Its elimination is desirable because the information itself is not essential for normal visual processing.


JPEG compression (cont.)

In transform coding, a reversible, linear transform like DFT or DCT

is used to map an image into a set of transform coefficients which are then quantized and coded.


JPEG compression (cont.)

  • One of the most popular and comprehensive compression standard is the JPEG standard.

  • In the JPEG baseline coding system, which is based on the discrete cosine transform and is adequate for most compression applications.

  • It is performed in four sequential steps:


JPEG compression (cont.)


JPEG compression (cont.)


JPEG compression (cont.)

  • After computing the DCT coefficients, they are normalized in accordance with

where , u,v=0,1, ...,7 are the resulting normalized and quantized coefficients, T(u,v) DCT of an 8x8 block of image f(x,y) and Z(u,v) is a transform normalization array like that of Figure 8.12(a)


JPEG compression (cont.)

  • After each block’s DCT coefficientsare quantized, the elements of

    are recorded in accordance with zigzag pattern of Figure 8.12(b).

  • Since the resulting on-dimensionally arrayqualitatively arranged according to increasing spatial frequency,the encoder in Figure 8.11(a) is designed to take advantage of long runs of zero that normally result from the reordering.


JPEG 2000

  • JPEG 2000is based on the idea that the coefficients of a transform that decorrelates the pixels of an image can be coded more efficiently than the original pixels themselves. If the transform basis functions-wawelets in the JPEG 2000 case- pack most of the important visual information into a small number of coefficients, the remaining coefficients can be quantized coarsely or truncated to zero with little image distortion.


JPEG 2000


Fractal Image Compression


Resim İçindeki Benzer Parçalar

  • Range : Küçük bloklar

  • Domain: Büyük bloklar


Görüntünün bloklara bölünmesi

Orjinal 8x8

Domain parçaları

4x4’e indirgenmiş halleri

4x4 Range

Parçaları

Orjinal

görüntü


Parçaların Benzerliği


Parçaların Benzerliği-3

  • Benzerini arayacağımız Range üsttedir.

  • Domainler’in orijinal halleri ilk kolonda gösterilmiştir.

  • Regresyon modeline göre düzeltilmiş domainler ise ikinci sütunda gösterilmiştir.

  • Yandaki rakamlar residual mean square(rms) değerlerini göstermektedir.

  • Rms değeri en küçük olan domain eldeki range ile eşleştirilir.


Rangeleri Domainler Cinsinden İfade Etmek

  • Benzer Range-Domainler bulunduktan sonra bunlar bir dosyada saklanarak resim bilgisini oluştururlar.

  • Örnek resmimiz 320x200x8 = 512,000 bit (64,000 byte) içeriyor.

  • Domain numaraları 0-999 arasında olduğu için 10 bitle ifade edilebilir.

  • Parlaklık ve konrast tamsayıya çevrilip, 0-15 ve 0-31 aralığında kuantize edilirse, 4 ve 5 bitle ifade edilebilir.

  • Dolayısıyla, 4000 Range’e karşılık gelen Domain bilgilerini dosyaya yazarsak :4000*(Domain Numarası+Parlaklık+Kontrast)= 4000*(10+4+5) = 76,000 bit (9,500 byte) olmaktadır.

  • 512/76 = 6.73:1, (512-76)/512 = %85 sıkıştırmak demektir, üstelik %85 sıkıştıran diğer algoritmalardan çok daha iyi bir kalitede !


Görüntü Kalitesi

  • Decompression, siyah bir ekrandan başlanarak 8-10 iterasyonla gerçekleştirilir.

  • Resmin kalitesi sonucun orjinale ne kadar yakın olduğuna yani herbir Range-Domain çiftinin ne kadar benzer olduk-larına bağlıdır.


Decompression: Iteration-1


Decompression: Iteration-2


Decompression: Iteration-3


Decompression: Iteration-10


Aynı Resimden Daha Fazla Domain Elde Etmek : Domain Transformasyonları

  • Range’e uygun Domaini ararken Domainleri olduğu gibi bırakmayıp transformasyon işlemlerine sokabiliriz.

  • Orjinal, 90o, 180o ve 270o derece döndürülür.

  • X eksenine göre yansımaları alınır.

  • Elde edilen 8 ayrı Domainin negatifi alınarak, bir Domain-den hareketle 16 farklı Domain elde edilir.


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