15-211 Fundamental Structures of Computer Science

15-211Fundamental Structuresof Computer Science

February 18, 2003

Ananda Guna

Lossy Compression

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Lossy Compression

Lecture outline The concept The formats Techniques

Discrete Cosine Transform Wavelets etc..

An Example Lossy Compression using Singular Value

Decomposition (SVD) Why SVD works and its drawbacks

Data Compression

We have studied two important data compression algorithms Huffman Code Lemple-Ziv Dictionary Method.

They provide good introduction to lossless compression.

What if we can compress the image by degrading the image a bit?

What techniques are used in jpeg and gif compression algorithms?

Lossy compression methods use mathematical theories

The Concept

A Data compression technique where some amount of information is lost (mainly redundant or unnecessary information)

MPEG (Moving Picture Experts Group) Store only the changes from one frame to another

(rather than the frame itself) Video information is encoded using DCT MPEG Formats

MPEG-1 – 30 fps – VHS quality MPEG-2 – 60 fps – higher resolution and CD quality audio MPEG-4 – based on wavelet technology – smaller files

Techniques

Discrete Cosine Transform (DCT) Used in JPEG algorithm

Wavelet based image compression Used in MPEG-4

JPEG

JPEG

Joint Photographic Expert Group Voted as international standard in 1992 Works well for both color and grayscale

imagesMany steps in the algorithm

Some requiring sophistication in mathematics

We’ll skip many parts and focus on just the main elements of JPEG

JPEG in a nutshell

BG

RY I

QRGB to YIQ(optional)

for each plane(scan)

for each 8x8 blockDCTQuantZig-zag

DPCM

RLE

Huffman 11010001…

JPEG in a nutshell

BG

RY I

QRGB to YIQ(optional)

for each plane(scan)

for each 8x8 blockDCTQuantZig-zag

DPCM

RLE

Huffman 11010001…

Linear transform coding

For video, audio, or images, one key first step of the compression will be to encode values over regions of time or space

The basic strategy is to select a set of linear basis functions i that span the space sin, cos, wavelets, … defined at discrete points

Linear transform coding

Coefficients:

In matrix notation:

Where A is an nxn matrix, and each row defines a basis function

Cosine transform

Discrete Cosine Transform

DCT separates the image into spectral sub-bands of differing importance

With input image A, the output coefficients B are given by the following equation:

N1 and N2 give the image’s height and width

Basis functions

JPEG in a nutshell

BG

RY I

QRGB to YIQ(optional)

for each plane(scan)

for each 8x8 blockDCTQuantZig-zag

DPCM

RLE

Huffman 11010001…

Quantization

The purpose of quantization is to encode an entire region of values into a single value For example, can simply delete low-order bits:

101101 could be encoded as 1011 or 101

When dividing by power-of-two, this amounts to deleting whole bits

Other division constants give finer control over bit loss

JPEG uses a standard quantization table

JPEG quantization table

q =

Each B(k1,k2) is divided by q(k1,k2).Eye is most sensitive to low frequencies (upper-left).

JPEG in a nutshell

BG

RY I

QRGB to YIQ(optional)

for each plane(scan)

for each 8x8 blockDCTQuantZig-zag

DPCM

RLE

Huffman 11010001…

Zig-zag scan

Purpose is to convert 8x8 block into a 1x64 vector, with low-frequency coefficients at the front

JPEG in a nutshell

BG

RY I

QRGB to YIQ(optional)

for each plane(scan)

for each 8x8 blockDCTQuantZig-zag

DPCM

RLE

Huffman 11010001…

Final stages

The DPCM (differential pulse code modulation) and RLE (run length encoding) steps take advantage of a common characteristic of many images: An 8x8 block is often not too different

than the previous one Within a block, there are often long

sequences of zeros

Data Compression with SVD

Singular Value Decomposition(SVD)

Suppose A is an mxn matrix We can find a decomposition of the matrix

A such that A = U S VT, where U and V are orthonormal matrices (I.e. UUT = I

and V VT = I, where I-identity matrix S is a diagonal matrix such that S = diag(s1, s2,

s3, … sk, 0,0,…0), where si ‘s are called the singular values of A and k is the rank of A. It is possible to choose U and V such that s1> s1> …. > sk

More on SVD in your linear algebra course..

Here is another way of expressing A

A = s1 U1V1T + s2 U2V2

T + ….+ sK UKVKT

where Ui and Vi are ith column of U and V respectively

Bit of a knowledge about block matrix multiplication will convince you that this sum is indeed equal to A.

So How does this applies to image compression? It is very very interesting

Any image is really an mxn matrix of pixels and in a bitmap color image each pixel is represented by 3-bytes (R,G,B)

Here is a look at it

Consider this color image

This is part of a famous image(Do you know who? Hint: Splay)The image is a 16x16 bitmap image enlarged

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Here is the Red part of the image

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Green Part

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Blue Part

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The Red matrix representation of the image (16x16 matrix)

173 165 165 165 148 132 123 132 140 156 173 181 181 181 189 173 198 189 189 189 181 165 148 165 165 173 181 198 206 198 181 165 206 206 206 206 198 189 181 181 198 206 206 222 231 214 181 165 231 222 206 198 189 181 181 181 206 222 222 222 231 222 198 181 231 214 189 173 165 165 173 181 181 189 198 222 239 231 206 214 206 189 173 148 148 148 148 165 156 148 165 198 222 231 214 239 181 165 140 123 123 115 115 123 140 148 140 148 165 206 239 247 165 82 66 82 90 82 90 107 123 123 115 132 140 165 198 231 123 198 74 49 57 82 82 99 107 115 115 123 132 132 148 214 239 239 107 82 82 74 90 107 123 115 115 123 115 115 123 198 255 90 74 74 99 74 115 123 132 123 123 115 115 140 165 189 247 99 99 82 90 107 123 123 123 123 123 132 140 156 181 198 247 239 165 132 107 148 140 132 132 123 132 148 140 140 156 214 198 231 165 156 132 156 156 140 140 140 148 148 132 140 156 222 247 239 222 181 181 140 156 140 148 148 148 140 132 156 206 222 214 198 181 181 181 181 173 148 156 148 140 140 165 198 222 239

So the idea is to apply SVD to this matrix and get a close enough approximation using as fewer columns of U and V as possible.

So for example, if the above matrix has one large dominant singular value, we might be able to get a pretty good approximation using just a single vector each from U and V

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173 165 165 165 148 132 123 132 140 156 173 181 181 181 189 173 198 189 189 189 181 165 148 165 165 173 181 198 206 198 181 165 206 206 206 206 198 189 181 181 198 206 206 222 231 214 181 165 231 222 206 198 189 181 181 181 206 222 222 222 231 222 198 181 231 214 189 173 165 165 173 181 181 189 198 222 239 231 206 214 206 189 173 148 148 148 148 165 156 148 165 198 222 231 214 239 181 165 140 123 123 115 115 123 140 148 140 148 165 206 239 247 165 82 66 82 90 82 90 107 123 123 115 132 140 165 198 231 123 198 74 49 57 82 82 99 107 115 115 123 132 132 148 214 239 239 107 82 82 74 90 107 123 115 115 123 115 115 123 198 255 90 74 74 99 74 115 123 132 123 123 115 115 140 165 189 247 99 99 82 90 107 123 123 123 123 123 132 140 156 181 198 247 239 165 132 107 148 140 132 132 123 132 148 140 140 156 214 198 231 165 156 132 156 156 140 140 140 148 148 132 140 156 222 247 239 222 181 181 140 156 140 148 148 148 140 132 156 206 222 214 198 181 181 181 181 173 148 156 148 140 140 165 198 222 239

The red image, again

Byte values (0…255) indicate intensity of the color at each pixel

Matrix decomposition

Suppose A is an mn matrix, e.g.:

We can decompose A into three matrices, U, S, and V, such that

A = 120 100 120 100 10 10 10 10 60 60 70 80 150 120 150 150

A = USVT

Decomposition example

A = 120 100 120 100 10 10 10 10 60 60 70 80 150 120 150 150

U = 0.5709 -0.6772 -0.4532 0.1009 0.0516 -0.0005 -0.1539 -0.9867 0.3500 0.7121 -0.5984 0.1113 0.7409 0.1854 0.6425 -0.0615

S = 386.154 0 0 0 0 20.6541 0 0 0 0 7.5842 0 0 0 0 0.9919

V = 0.5209 -0.5194 0.6004 -0.3137 0.4338 -0.1330 -0.7461 -0.4873 0.5300 -0.1746 -0.1886 0.8081 0.5095 0.8259 0.2176 -0.1049

Orthonormal:UUT = I

Orthonormal:VVT = I

Diagonal, with decreasing singular values

Singular value decomposition

Such a factoring of a matrix, or decomposition is a called an SVD.

Exactly how to find U, V, and S is beyond the scope of this course. But you’ll find out in your matrix/linear algebra

course… Note: Very important also for

graphics/animation algorithms

So what about compression?

Let: si be the ith eigen value in S

Ui be the ith column in U

Vi be the ith column in V

Then, another formula for matrix A is

A = s1 U1V1T + s2 U2V2

T + ….+ sK UKVKT

s1

U1

V1

A = 120 100 120 100 10 10 10 10 60 60 70 80 150 120 150 150

U = 0.5709 -0.6772 -0.4532 0.1009 0.0516 -0.0005 -0.1539 -0.9867 0.3500 0.7121 -0.5984 0.1113 0.7409 0.1854 0.6425 -0.0615

S = 386.154 0 0 0 0 20.6541 0 0 0 0 7.5842 0 0 0 0 0.9919

V = 0.5209 -0.5194 0.6004 -0.3137 0.4338 -0.1330 -0.7461 -0.4873 0.5300 -0.1746 -0.1886 0.8081 0.5095 0.8259 0.2176 -0.1049

SVD example

A1 = s1U1V1T

= 115 96 117 112

10 9 11 10

70 59 72 69

149 124 152 146This is called the “rank-1 approximation

Apply SVD to a simple matrix

Lets take a look at applying the SVD to a smaller matrix This example will allow us to understand what is going on here

A = 120 100 120 100 10 10 10 10 60 60 70 80 150 120 150 150

U = 0.5709 -0.6772 -0.4532 0.1009 0.0516 -0.0005 -0.1539 -0.9867 0.3500 0.7121 -0.5984 0.1113 0.7409 0.1854 0.6425 -0.0615

S = 386.1540 0 0 0 0 20.6541 0 0 0 0 7.5842 0 0 0 0 0.9919

V = 0.5209 -0.5194 0.6004 -0.3137 0.4338 -0.1330 -0.7461 -0.4873 0.5300 -0.1746 -0.1886 0.8081 0.5095 0.8259 0.2176 -0.1049

Note that first eigen value is substantially larger than the others.

Form a rank-1 sum

A1 = s1 U1 V1T

A1 = 115 96 117 112

10 9 11 10

70 59 72 69

149 124 152 146

Error Matrix |A - A1| is 5 4 3 12 0 1 1 0 10 1 2 11 1 4 2 4 Error is relatively small with a rank-1 approximation.

What have we learnt here?

Perhaps in this case we only need to know one column vector, one row vector, one singular value to get a pretty good approximation to the original image.

So instead of 4x4 = 16 bytes, we can store = 4 + 4 + 1 bytes to get an image fairly “close” to original image (almost 50% savings)

What if we do a rank-2 approximation?

A2 = s1 U1 V1T + s2 U2 V2

T

A2 = 122 98 119 100

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62 57 69 81

147 123 151 149

Error Matrix |A - A2|

2 2 1 0 0 1 1 0 2 3 1 1 3 3 1 1

Even smaller error matrix

Analysis

To get an idea of how close the approximation to the original matrix is, we can calculate:

Mean of Rank1 error matrix = 3.8125

Mean of Rank2 error matrix = 1.3750

Where mean is the average of the all entries

We really don’t gain much by calculating the rank-2 approximation (why?)

Why is that?

If you look at the first singular value it is fairly large compared to others.

Therefore, contribution from the rank-1 sum is very significant compared to the sum of all other rank approximations.

So even if you leave out all other rank sums, you still get a pretty good approximation with just two vectors. So we are up to something here

It is time to look at some samples

Some Samples (128x128)

Original mage 49K Rank 1 approx 825 bytes

Samples ctd…

Rank 16 approx 13K Rank 8 approx 7K

Some size observations

Note that theoretically the sizes of the compressed images should be Rank 1 = 54 + (128 + 128 + 1)*3

Rank 8 = 54 + (128+128+1)*3*8 = 6K Rank 16 = 54 + (128 + 128 + 1)*3*16 = 12K Rank 32 = 54 + (128 + 128 + 1)*3*32 = 24K Rank 64 = 48K (pretty close to the original)

Bmp Header U1 V1 + S1 bytes/pixel

Implementation (compression)

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SVD

SVD

SVD

001100100

000110000

1100000100

COMPRESSION STEP

Compressed file stores U, V and S for the Rank selected for each of the colors R , G and B and header bytes

Header

bytes

Implementation (decompression)

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001100100

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DECOMPRESSION

Compressed file stores U, V and S for the Rank selected for each of the colors R , G and B and the bmp header

Header

bytes

Form rank sum

Form rank sum

Form rank

sum

Matlab Code for SVD

Matlab is a computer algebra system (www.mathworks.com)

Here is Matlab code that can perform SVD on an image. A=imread('c:\temp\rhino64','bmp'); N = size(A)[1]; R = A(:,:,1); // extract Red matrix G = A(:,:,2); // extract Green Matrix B = A(:,:,3); // extract blue matrix Apply SVD to each of the matrices

[ur,sr,vr]=svd(double(R));[ug,sg,vg]=svd(double(G));[ub,sb,vb]=svd(double(B));

Complete Matlab Code for SVD ctd..

A=imread('c:\temp\rosemary','bmp');s=size(A)%imagesc(A)R = A(:,:,1); G = A(:,:,2); B = A(:,:,3);[ur,sr,vr]=svd(double(R));[ug,sg,vg]=svd(double(G));[ub,sb,vb]=svd(double(B));%initialize matrices to zero matricesRk=zeros(s(1),s(2));Gk=zeros(s(1),s(2));Bk=zeros(s(1),s(2));k = 8; % k is the desired rank% form the rank sumsfor i=1:k, Rk=Rk + sr(i,i)*ur(:,i)*transpose(vr(:,i)); endfor i=1:k, Gk=Gk + sg(i,i)*ug(:,i)*transpose(vg(:,i)); endfor i=1:k, Bk=Bk + sb(i,i)*ub(:,i)*transpose(vb(:,i)); end% Now form the rank-k approximation of AAk = A;Ak(:,:,1)=Rk; Ak(:,:,2)=Gk; Ak(:,:,3)=Bk;% now plot the rank-k approximation of imageimagesc(Ak)

Matlab outputs

Rank-4

Rank-4 Rank-8

Rank-1

Rank-1

Rank-8

Original images were approximately 128x128

original

original

A Java Implementation-Compressor Interface

/** * This interface describes an object that can

compress and decompress a stream of bits. */public interface Compressor {public void compress (BitReader reader, BitWriter

writer) throws IOException;

public void decompress (BitReader reader, BitWriter writer) throws IOException;

}

A Java Implementation

Class Lossy implements Compressor { // Helper methods

public void compress (BitReader reader, BitWriter writer) throws IOException

{ ……}

public void decompress (BitReader reader, BitWriter writer) throws IOException

{…….}

}

Adaptive Rank Methods

All popular image compression programs apply compression algorithm to sub-blocks of the image

This exploit the uneven characteristics of the original image

If parts of the image are less complex than the others, then a smaller number of singular values are needed to obtain a "close" approximation

Adaptive Rank Methods ctd..

So instead of picking same fixed rank for each sub-block, we decide how many singular values to pick from each sub-block by looking at the following:

Percent of r values = s1 + s2 + ….+ sr

------------------ s1 + s2 + ….+ sk

Where k is the max number of non zero singular values of A.

Results of applying the Adaptive Ranking Method

We applied the adaptive ranking method to Prof. Danny Sleator. Here are the results.

80% of singular

values

26K

Original

49K

50% of singular

values

15K

10% of singular

values

14K

SVD Complexity & Resources

The best known algorithm for SVD is O(m2n + n3) in theory (Source: Golub & Van Horn – Matrix Computations)

If only the singular values are needed, then we can do O(m n2)

There is a Java package called JAMA (Java Matrix Package) available at http://math.nist.gov/javanumerics/jama/

import Jama.*;Use the class:

SingularValueDecomposition

Why does SVD works?

SVD is a dimension reduction method. Applications in Data mining Signal Processing Compression

The idea is to isolate what is important and discard or ignore the others.

Remove redundant information or noise in data.

Conclusion

As with any other compression technique there are drawbacks to SVD. For eg: we need to store the matrices U and V. So if we have to store more than n2 /(2n+1) singular

values, this is not good. Instead of selecting a fixed block size for each block, is it

possible to achieve a better compression using dynamic block sizes? Can this be done by sub-dividing the original image until an optimal block size is found?

Optimize the storage scheme for U and V. Pack as many bits as possible to obtain a greater compression.

We encourage you to explore the idea further.