Image Compression

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Image Compression

• Mohamed N. Ahmed, Ph.D.

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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.

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Image Compression

• We want to remove redundancy from the data
• Mathematically

Transformation
Statistically Uncorrelated data
2D array Of pixels
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Day4 Image Compression

• Outline
• Fundamentals
• Coding Redundancy
• Interpixel Redundancy
• Psychovisual Redundancy
• Fidelity Criteria
• Error-Free Compression
• Variable-length Coding
• LZW Coding
• Predictive Coding
• Lossy Compression
• Transform Coding
• Wavelet Coding
• Image Compression Standards

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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
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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

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Data Redundancy

• If n1 n2 , CR1 and RD0 no redundancy
• If n1 gtgt n2 , CR and RD high
  redundancy
• If n1 ltlt n2 , CR and RD undesirable
• A compression ration of 10 (101) 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

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Coding Redundancy
Variable-Length Coding
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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.
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Run-Length Encoding
Example of Inter-pixel Redundancy removal
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Psycho-visual Redundancy
The human visual system is more sensitive to
edges Middle Picture Uniform quantization from
256 to 16 gray levels CR 2 Right picture
Improved gray level quantization (IGS) CR 2
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Fidelity Criteria
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
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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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Fidelity Criteria
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Compression Model
The source encoder is responsible for removing
redundancy (coding, inter-pixel,
psycho-visual) The channel encoder ensures
robustness against channel noise.
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Compression Types
Compression
Error-Free Compression (Loss-less)
Lossy Compression
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Error-Free Compression

• Some applications require no error in compression
  (medical, business documents, etc..)
• CR2 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.

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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

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Huffman Codes
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Huffman Codes
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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 ?

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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

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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.

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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

Initial Dictionary










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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 symbol39

39-39
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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 ?

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LZW Coding
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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.

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Bit-Plane Decomposition
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Bit-Plane Encoding
Constant Area Coding One Dimensional Run Length
coding Two Dimensional Run Length coding
1b 2w 1b 3w
4b 1w
12 w
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Loss-less Predictive Encoding
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Loss-less Predictive Encoding
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Lossy Compression
Quantizer
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Lossy Compression
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DPCM
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DPCM
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Transform Coding

• 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

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Transform Coding
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2D Transforms

• Energy packing
• 2D transforms pack most of the energy
• into small number of coefficients located
• at the upper left corner of the 2D array

Energy Packing
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2D Transforms

• 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

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2D Transforms

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

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Discrete Cosine Transform

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

for u0 for u1, 2, , N-1
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Discrete Cosine Transform
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2D Transforms
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Effect of Window Size
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Quantization
Quantizer
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Quantization

• Each transformed coefficient is quantized

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Quantization
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Bit allocation and Zig Zag Ordering
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DCT and Quantization
Right Column
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Wavelet Coding
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Wavelet Transform
1
2
3
4
Put a pixel in each quadrant-? No size change
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Wavelet Transform
a
b
c
d

• Now let
• a (x1x2x3x4)/4
• b (x1x2-x3-x4)/4
• c (x1x3-x2-x4)/4
• d (x1x4-x2-x3)/4

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Wavelet Transform
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Wavelet Transform
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Wavelet Transform
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Wavelet Coding

• High Frequency coefficients tend to be very small
  ---? 0
• They can be quantized very effectively without
  distorting the results

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Wavelet Transform
DCT
Wavelet
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Wavelet Transform
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Image Compression Standards

• Binary Compression Standards
• CCITT G3 -gt 1D Run Length Encoding
• CCITT G4 -gt 2D Run Length encoding
• JBIG1 -gt Lossless adaptive binary compression
• JBIG2 -gt Lossy/Lossless adaptive binary
  compression

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JBIG/JBIG2
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Image Compression Standards

• Continuous Tone Still Image Compression Standards
• JPEG
• JPEG 2000
• Mixed Raster Content (MRC)

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MRC
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Video Compression