software research image compression mohamed n. ahmed, ph.d

Software ResearchSoftware Research

Digital Image ProcessingDigital Image Processing

Image Compression

Mohamed N. Ahmed, Ph.D.



Image Compression

• Everyday an enormous amount of information is stored, processed, and transmitted

• Financial data

• Reports

• Inventory

• Cable TV

• Online Ordering and tracking

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Image Compression

• Because much of this information is graphical or pictorial in nature, the storage and communications requirements are immense.

• Image compression addresses the problem of reducing the amount of data requirements to represent a digital image.

• Image Compression is becoming an enabling technology: HDTV.

• Also it plays an important role in Video Conferencing, remote sensing, satellite TV, FAX, document and medical imaging.



Image Compression

• We want to remove redundancy from the data

• Mathematically

Transformation2D array Of pixels

Statistically Uncorrelated data



Day4 : Image Compression

Outline:1. Fundamentals

Coding RedundancyInterpixel RedundancyPsychovisual RedundancyFidelity Criteria

2. Error-Free CompressionVariable-length CodingLZW CodingPredictive Coding

3. Lossy CompressionTransform CodingWavelet Coding

4. Image Compression Standards



Fundamentals

• The term data compression refers to the process of reducing the amount of data required to represent a given quantity of information

• Data Information

• Various amount of data can be used to represent the same information

• Data might contain elements that provide no relevant information : data redundancy

• Data redundancy is a central issue in image compression. It is not an abstract concept but mathematically quantifiable entity

Some Images are adopted from R. C. Gonzalez & R. E. Woods



Data Redundancy

• Let n1 and n2 denote the number of information carrying units in two data sets that represent the same information

• The relative redundancy RD is define as :

where CR, commonly called the compression ratio, is

2

1

11

n

nC

CR

R

RD



Data Redundancy

• If n1 = n2 , CR=1 and RD=0 no redundancy

• If n1 >> n2 , CR and RD high redundancy

• If n1 << n2 , CR and RD undesirable

• A compression ration of 10 (10:1) means that the first data set has 10 information carrying units (say, bits) for every 1 unit in the second (compressed) data set.

• In Image compression , 3 basic redundancy can be identified» Coding Redundancy

» Interpixel Redundancy

» Psychovisual Redundancy

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Coding Redundancy

• Recall from the histogram calculations

where p(rk) is the probability of a pixel to have a certain value rk

If the number of bits used to represent rk is l(rk), then

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Coding Redundancy

• Example:



Coding Redundancy

Variable-LengthCoding



Inter-pixel Redundancy

Here the two pictures have Approximately the same Histogram.

We must exploit Pixel Dependencies.

Each pixel can be estimated From its neighbors.



Run-Length Encoding

Example of Inter-pixel Redundancy removal



Psycho-visual Redundancy

The human visual system is more sensitive to edges

Middle Picture: Uniform quantization from 256 to 16 gray levelsCR= 2

Right picture: Improved gray level quantization(IGS)CR= 2



Fidelity Criteria

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The error between two functions is given by:

So, the total error between the two images is

The root-mean-square error averaged over the whole image is



Fidelity Criteria

• A closely related objective fidelity criterion is the mean square signal to noise ratio of the compressed-decompressed image

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SNR M

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Fidelity Criteria



Compression Model

The source encoder is responsible for removing redundancy(coding, inter-pixel, psycho-visual)

The channel encoder ensures robustness against channel noise.



Compression Types

Compression

Error-Free Compression(Loss-less)

Lossy Compression



Error-Free Compression

• Some applications require no error in compression (medical, business documents, etc..)

• CR=2 to 10 can be expected.

• Make use of coding redundancy and inter-pixel redundancy.

• Ex: Huffman codes, LZW, Arithmetic coding, 1D and 2D run-length encoding, Loss-less Predictive Coding, and Bit-Plane Coding.



Huffman Coding

• The most popular technique for removing coding redundancy is due to Huffman (1952)

• Huffman Coding yields the smallest number of code symbols per source symbol

• The resulting code is optimal



Huffman Codes



Workshop

Obtain the Huffman codes for the following sequence:

– 5 5 5 5 8 8 4 2 7 7 7 2 2 2 2 4 4 7 7 7 7 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4

– What is the average code length with and without compression ?



Fixed Length: LZW Coding

• Error Free Compression Technique

• Remove Inter-pixel redundancy

• Requires no priori knowledge of probability distribution of pixels

• Assigns fixed length code words to variable length sequences

• Patented Algorithm US 4,558,302

• Included in GIF and TIFF and PDF file formats



LZW Coding

• Coding Technique– A codebook or a dictionary has to be constructed

– For an 8-bit monochrome image, the first 256 entries are assigned to the gray levels 0,1,2,..,255.

– As the encoder examines image pixels, gray level sequences that are not in the dictionary are assigned to a new entry.

– For instance sequence 255-255 can be assigned to entry 256, the address following the locations reserved for gray levels 0 to 255.



LZW Coding

• Example

Consider the following 4 x 4 8 bit image

39 39 126 126

39 39 126 126

39 39 126 126

39 39 126 126

Dictionary Location Entry

0 01 1. .255 255256 -

511 -

Initial Dictionary



LZW Coding

39 39 126 126

39 39 126 126

39 39 126 126

39 39 126 126

•Is 39 in the dictionary……..Yes•What about 39-39………….No•Then add 39-39 in entry 256•And output the last recognized symbol…39

Dictionary Location Entry

0 01 1. .255 255256 -

511 -

39-39



Workshop

• Code the following image using LZW codes

39 39 126 126

39 39 126 126

39 39 126 126

39 39 126 126

* How can we decode the compressed sequence to obtain the original image ?



LZW Coding



Bit-Plane Coding

• An effective technique to reduce inter pixel redundancy is to process each bit plane individually

• The image is decomposed into a series of binary images.

• Each binary image is compressed using one of well known binary compression techniques.



Bit-Plane Decomposition



Bit-Plane Encoding

Constant Area Coding

One Dimensional Run Length coding

Two Dimensional Run Length coding

1b 2w 1b 3w 4b 1w

12 w



Loss-less Predictive Encoding



Lossy Compression

Quantizer



Lossy Compression



DPCM



• A reversible linear transform (such as Fourier Transform) is used to map the image into a set of transform coefficients

• These coefficients are then quantized and coded.• The goal of transform coding is to decorrelate

pixels and pack as much information into small number of transform coefficients.

• Compression is achieved during quantization not during the transform step

Transform Coding



Transform Coding



• Energy packing– 2D transforms pack most of the energy

into small number of coefficients located

at the upper left corner of the 2D array

2D Transforms

EnergyPacking



• Consider an image f(x,y) of size N x N

Forward transform

g(x,y,u,v) is the forward transformation kernel or basis functions

2D Transforms

.1,...,2,1,0,

),,,(),(),(1

0

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0

Nvu

vuyxgyxfvuTN

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N

y



2D Transforms

• Inverse transform

h(x,y,u,v) is the inverse transformation kernel or basis functions

.1,...,2,1,0,

),,,(),(),(1

0

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0

Nyx

vuyxhvuTyxfN

u

N

v



Discrete Cosine Transform

• One of the most frequently used transformations for image compression is the DCT.

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uxvu

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)(

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)12(cos

2

)12(cos)()(

),,,(),,,(

for u=0

for u=1, 2, …, N-1



Discrete Cosine Transform



2D Transforms



Effect of Window Size



Quantization

Quantizer



Quantization

• Each transformed coefficient is quantized

),(

),(),(ˆ

vuZ

vuTroundvuT



Quantization



Bit allocation and Zig Zag Ordering



DCT and Quantization

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vuZ

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Right Column



Wavelet Coding



Wavelet Transform

Put a pixel in each quadrant- No size change

1 23 4



• Now let» a = (x1+x2+x3+x4)/4

» b =(x1+x2-x3-x4)/4

» c =(x1+x3-x2-x4)/4

» d =(x1+x4-x2-x3)/4

Wavelet Transforma bc d

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Wavelet Transform



Wavelet Coding

• High Frequency coefficients tend to be very small --- 0

• They can be quantized very effectively without distorting the results



Wavelet Transform

Wavelet DCT



Wavelet Transform



Image Compression Standards

• Binary Compression Standards• CCITT G3 -> 1D Run Length Encoding

• CCITT G4 -> 2D Run Length encoding

• JBIG1 -> Lossless adaptive binary compression

• JBIG2 -> Lossy/Lossless adaptive binary compression



JBIG/JBIG2



• Continuous Tone Still Image Compression Standards

• JPEG

• JPEG 2000

• Mixed Raster Content (MRC)

Image Compression Standards



MRC



Video Compression

software research image compression mohamed n. ahmed, ph.d

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