Image Processing Fundamentals

1
Digital Image Processing
Chapter 13 Image Compression
Prepared by Eng. Mohamed Hassan Supervised by
Dr. Ashraf Aboshosha http//www.icgst.com/A_Abosh
osha.html editor_at_icgst.com Tel.
0020-122-1804952 Fax. [phone]
2
Goal of Image Compression

• Digital images require huge amounts of space for
  storage and large bandwidths for transmission.
• A 640 x 480 color image requires close to 1MB of
  space.
• The goal of image compression is to reduce the
  amount of data required to represent a digital
  image.
• Reduce storage requirements and increase
  transmission rates.

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Approaches

• Lossless
• Information preserving
• Low compression ratios
• Lossy
• Not information preserving
• High compression ratios
• Trade-off image quality vs compression ratio

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Data ? Information

• Data and information are not synonymous terms!
• Data is the means by which information is
  conveyed.
• Data compression aims to reduce the amount of
  data required to represent a given quantity of
  information while preserving as much information
  as possible.

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Data vs Information

• The same amount of information can be represented
  by various amount of data, e.g.

Your wife, Helen, will meet you at Logan Airport
in Boston at 5 minutes past 600 pm tomorrow
night Your wife will meet you at Logan Airport
at 5 minutes past 600 pm tomorrow night Helen
will meet you at Logan at 600 pm tomorrow night
Ex1
Ex2
Ex3
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Data Redundancy
compression
Compression ratio
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Data Redundancy

• Relative data redundancy

Example
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Types of Data Redundancy

• Coding
• Interpixel
• Psychovisual
• Compression attempts to reduce one or more of
  these redundancy types.

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Coding Redundancy

• Code a list of symbols (letters, numbers, bits
  etc.)
• Code word a sequence of symbols used to
  represent a piece of information or an event
  (e.g., gray levels).
• Code word length number of symbols in each code
  word

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Coding Redundancy
N x M image rk k-th gray level P(rk)
probability of rk l(rk) of bits for rk
Expected value
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Coding Redundancy

• l(rk) constant length

Example
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Coding Redundancy

• l(rk) variable length
• Consider the probability of the gray levels

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Interpixel redundancy

• Interpixel redundancy implies that any pixel
  value can be reasonably predicted by its
  neighbors (i.e., correlated).

autocorrelation f(x)g(x)
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Interpixel redundancy

• To reduce interpixel redundnacy, the data must be
  transformed in another format (i.e., through a
  transformation)
• e.g., thresholding, differences between adjacent
  pixels, DFT

(110) bits/pair
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Psychovisual redundancy

• The human eye does not respond with equal
  sensitivity to all visual information.
• It is more sensitive to the lower frequencies
  than to the higher frequencies in the visual
  spectrum.
• Idea discard data that is perceptually
  insignificant!

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Psychovisual redundancy
Example quantization
256 gray levels
16 gray levels
16 gray levels
i.e., add to each pixel a small pseudo-random
number prior to quantization
C8/4 21
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How do we measure information?

• What is the information content of a
  message/image?
• What is the minimum amount of data that is
  sufficient to describe completely an image
  without loss of information?

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Modeling Information

• Information generation is assumed to be a
  probabilistic process.
• Idea associate information with probability!

A random event E with probability P(E) contains
Note I(E)0 when P(E)1
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How much information does a pixel contain?

• Suppose that gray level values are generated by a
  random variable, then rk contains

units of information!
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How much information does an image contain?

• Average information content of an image

using
Entropy
units/pixel
(assumes statistically independent random events)
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Redundancy (revisited)

• Redundancy

where
Note of Lavg H, the R0 (no redundancy)
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Entropy Estimation

• It is not easy to estimate H reliably!

image
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Entropy Estimation

• First order estimate of H

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Estimating Entropy

• Second order estimate of H
• Use relative frequencies of pixel blocks

image
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Estimating Entropy

• The first-order estimate provides only a
  lower-bound on the compression that can be
  achieved.
• Differences between higher-order estimates of
  entropy and the first-order estimate indicate the
  presence of interpixel redundancy!

Need to apply transformations!
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Estimating Entropy

• For example, consider differences

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Estimating Entropy

• Entropy of difference image

• Better than before (i.e., H1.81 for original
  image)

• However, a better transformation could be found
  since

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Image Compression Model
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Image Compression Model

• Mapper transforms input data in a way that
  facilitates reduction of interpixel redundancies.

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Image Compression Model

• Quantizer reduces the accuracy of the mappers
  output in accordance with some pre-established
  fidelity criteria.

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Image Compression Model

• Symbol encoder assigns the shortest code to the
  most frequently occurring output values.

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Image Compression Models

• Inverse operations are performed.
• But quantization is irreversible in general.

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Fidelity Criteria

• How close is to ?
• Criteria
• Subjective based on human observers
• Objective mathematically defined criteria

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Subjective Fidelity Criteria
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Objective Fidelity Criteria

• Root mean square error (RMS)
• Mean-square signal-to-noise ratio (SNR)

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Objective Fidelity Criteria (contd)
RMSE 5.17
RMSE 15.67
RMSE 14.17
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Lossless Compression
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Lossless Methods Taxonomy
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Huffman Coding (coding redundancy)

• A variable-length coding technique.
• Optimal code (i.e., minimizes the number of code
  symbols per source symbol).
• Assumption symbols are encoded one at a time!

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Huffman Coding

• Forward Pass
• 1. Sort probabilities per symbol
• 2. Combine the lowest two probabilities
• 3. Repeat Step2 until only two probabilities
  remain.

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Huffman Coding

• Backward Pass
• Assign code symbols going backwards

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Huffman Coding

• Lavg using Huffman coding
• Lavg assuming binary codes

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Huffman Coding/Decoding

• After the code has been created, coding/decoding
  can be implemented using a look-up table.
• Note that decoding is done unambiguously.

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Arithmetic (or Range) Coding (coding redundancy)

• No assumption on encode source symbols one at a
  time.
• Sequences of source symbols are encoded together.
• There is no one-to-one correspondence between
  source symbols and code words.
• Slower than Huffman coding but typically achieves
  better compression.

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Arithmetic Coding (contd)

• A sequence of source symbols is assigned a single
  arithmetic code word which corresponds to a
  sub-interval in 0,1.
• As the number of symbols in the message
  increases, the interval used to represent it
  becomes smaller.
• Smaller intervals require more information units
  (i.e., bits) to be represented.

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Arithmetic Coding (contd)
Encode message a1 a2 a3 a3 a4
1) Assume message occupies 0, 1)
2) Subdivide 0, 1) based on the probability of
ai
3) Update interval by processing source symbols
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Example
Encode
a1 a2 a3 a3 a4
0.06752, 0.0688) or, 0.068
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Example

• The message a1 a2 a3 a3 a4 is encoded using 3
  decimal digits or 3/5 0.6 decimal digits per
  source symbol.
• The entropy of this message is
• Note finite precision arithmetic might cause
  problems due to truncations!

-(3 x 0.2log10(0.2)0.4log10(0.4))0.5786
digits/symbol
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Arithmetic Decoding
1.0
0.8
0.72
0.592
0.5728
a4
0.8
0.72
0.688
0.5856
0.57152
Decode 0.572
a3
0.4
0.56
0.624
0.5728
056896
a2
a3 a3 a1 a2 a4
0.2
0.48
0.592
0.5664
0.56768
a1
0.0
0.4
0.56
0.56
0.5664
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LZW Coding (interpixel redundancy)

• Requires no priori knowledge of pixel probability
  distribution values.
• 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

• A codebook (or dictionary) needs to be
  constructed.
• Initially, the first 256 entries of the
  dictionary are assigned to the gray levels
  0,1,2,..,255 (i.e., assuming 8 bits/pixel)

Initial Dictionary
Consider a 4x4, 8 bit image 39 39 126
126 39 39 126 126 39 39
126 126 39 39 126 126
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LZW Coding (contd)

• 39 39 126 126
• 39 39 126 126
• 39 39 126 126
• 39 39 126 126

As the encoder examines image pixels, gray level
sequences (i.e., blocks) that are not in the
dictionary are assigned to a new entry.
- Is 39 in the dictionary..Yes - What
about 39-39.No - Then add 39-39 in entry
256
39-39
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Example
Concatenated Sequence CS CR P
39 39 126 126 39 39 126 126 39 39
126 126 39 39 126 126
(CR)
(P)
CR empty
If CS is found (1) No Output (2) CRCS
else (1) Output D(CR) (2) Add CS to D (3)
CRP
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Decoding LZW

• The dictionary which was used for encoding need
  not be sent with the image.
• Can be built on the fly by the decoder as it
  reads the received code words.

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Differential Pulse Code Modulation (DPCM) Coding
(interpixel redundancy)

• A predictive coding approach.
• Each pixel value (except at the boundaries) is
  predicted based on its neighbors (e.g., linear
  combination) to get a predicted image.
• The difference between the original and predicted
  images yields a differential or residual image.
• i.e., has much less dynamic range of pixel
  values.
• The differential image is encoded using Huffman
  coding.

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Run-length coding (RLC) (interpixel redundancy)

• Used to reduce the size of a repeating string of
  characters (i.e., runs)
• 1 1 1 1 1 0 0 0 0 0 0 1 ? (1,5) (0,
  6) (1, 1)
• a a a b b b b b b c c ? (a,3) (b, 6) (c,
  2)
• Encodes a run of symbols into two bytes (symbol,
  count)
• Can compress any type of data but cannot achieve
  high compression ratios compared to other
  compression methods.

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Bit-plane coding (interpixel redundancy)

• An effective technique to reduce inter pixel
  redundancy is to process each bit plane
  individually.
• (1) Decompose an image into a series of binary
  images.
• (2) Compress each binary image (e.g., using
  run-length coding)

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Combining Huffman Coding with Run-length Coding

• Assuming that a message has been encoded using
  Huffman coding, additional compression can be
  achieved using run-length coding.

e.g., (0,1)(1,1)(0,1)(1,0)(0,2)(1,4)(0,2)
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Lossy Compression

• Transform the image into a domain where
  compression can be performed more efficiently
  (i.e., reduce interpixel redundancies).

(N/n)2 subimages
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Example Fourier Transform
The magnitude of the FT decreases, as u, v
increase!
K ltlt N
K-1
K-1
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Transform Selection

• T(u,v) can be computed using various
  transformations, for example
• DFT
• DCT (Discrete Cosine Transform)
• KLT (Karhunen-Loeve Transformation)

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DCT

• forward

• inverse

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DCT (contd)

• Basis set of functions for a 4x4 image
  (i.e.,cosines of different frequencies).

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DCT (contd)
DFT
WHT
DCT
8 x 8 subimages 64 coefficients per
subimage 50 of the coefficients truncated
RMS error 2.32 1.78
1.13
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DCT (contd)

• Subimage size selection

4 x 4 subimages
8 x 8 subimages
2 x 2 subimages
original
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JPEG Compression

• JPEG is an image compression standard which was
  accepted as an international standard in 1992.
• Developed by the Joint Photographic Expert Group
  of the ISO/IEC for coding and compression of
  color/gray scale images.
• Yields acceptable compression in the 101 range.
• A scheme for video compression based on JPEG
  called Motion JPEG (MJPEG) exists

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JPEG Compression (contd)

• JPEG uses DCT for handling interpixel redundancy.
• Modes of operation
• (1) Sequential DCT-based encoding
• (2) Progressive DCT-based encoding
• (3) Lossless encoding
• (4) Hierarchical encoding

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JPEG Compression (Sequential DCT-based encoding)
Entropy decoder
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JPEG Steps

• 1.Divide the image into 8x8 subimages
• For each subimage do
• 2. Shift the gray-levels in the range -128,
  127
• - DCT requires range be centered
  around 0
• 3. Apply DCT (i.e., 64 coefficients)
• 1 DC coefficient F(0,0)
• 63 AC coefficients F(u,v)

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JPEG Steps

• 4. Quantize the coefficients (i.e., reduce the
  amplitude of coefficients that do not contribute
  a lot).

Q(u,v) quantization table
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Example

• Quantization Table Qij

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Example (contd)
Quantization
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JPEG Steps (contd)

• 5. Order the coefficients using zig-zag ordering
• - Place non-zero coefficients first
• - Create long runs of zeros (i.e., good for
  run-length encoding)

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Example
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JPEG Steps (contd)

• 6. Form intermediate symbol sequence and encode
  coefficients
• 6.2 AC coefficients variable length coding

6.1 DC coefficients predictive encoding
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Intermediate Coding
DC
symbol_1 (SIZE)
symbol_2 (AMPLITUDE)
end of block
DC (6)
(61) AC (0,2)
(-3)
symbol_1 (RUN-LENGTH, SIZE) symbol_2
(AMPLITUDE)
SIZE bits for encoding amplitude RUN-LENGTH
run of zeros
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DC/AC Symbol Encoding

• DC encoding
• AC encoding

symbol_1 symbol_2 (SIZE)
(AMPLITUDE)
predictive coding
-2048, 2047
-211, 211-1 1 SIZE11
0 0 0 0 0 0 476 (6,9)(476)
-210, 210-1 1 SIZE10
If RUN-LENGTH gt 15, use symbol (15,0) , i.e.,
RUN-LENGTH16
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Effect of Quality
90 (58k bytes)
50 (21k bytes)
10 (8k bytes)
worst quality, highest compression
best quality, lowest compression
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Example 1 homogeneous 8 x 8 block
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Example 1 (contd)
Quantized
De-quantized
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Example 1 (contd)
Reconstructed
Error
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Example 2 less homogeneous 8 x 8 block
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Example 2 (contd)
Quantized
De-quantized
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Example 2 (contd)
Error
Reconstructed spatial
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JPEG for Color Images

• Could apply JPEG on R/G/B components .
• It is more efficient to describe a color in terms
  of its luminance and chrominance content
  separately, to enable more efficient processing.
• YUV
• Chrominance can be subsampled due to human vision
  insensitivity

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JPEG for Color Images

• Luminance Received brightness of the light
  (proportional to the total energy in the visible
  band).
• Chrominance Describe the perceived color tone of
  the light (depends on the wavelength composition
  of light
• Hue Specify the color tone (i.e., depends on the
  peak wavelength of the light).
• Saturation Describe how pure the color is (i.e.,
  depends on the spread or bandwidth of the light
  spectrum).

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JPEG for Color Images
Encoder
Decoder
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JPEG Modes

• JPEG supports several different modes
• Sequential Mode
• Progressive Mode
• Hierarchical Mode
• Lossless Mode
• Sequential is the default mode
• Each image component is encoded in a single
  left-to-right, top-to-bottom scan.
• This is the mode we have been describing.

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Progressive JPEG

• The image is encoded in multiple scans, in order
  to produce a quick, rough decoded image when
  transmission time is long.

Sequential
Progressive
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Progressive JPEG (contd)

• Send DCT coefficients in multiple scans
• (1) Progressive spectral selection algorithm
• (2) Progressive successive approximation
  algorithm
• (3) Hybrid progressive algorithm

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Progressive JPEG (contd)

• (1) Progressive spectral selection algorithm
• Group DCT coefficients into several spectral
  bands
• Send low-frequency DCT coefficients first
• Send higher-frequency DCT coefficients next

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Progressive JPEG (contd)

• (2) Progressive successive approximation
  algorithm
• Send all DCT coefficients but with lower
  precision.
• Refine DCT coefficients in later scans.

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Progressive JPEG (contd)

• (3) Hybrid progressive algorithm
• Combines spectral selection and successive
  approximation.

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Hierarchical JPEG

• Hierarchical mode encodes the image at several
  different resolutions.
• Image is transmitted in multiple passes with
  increased resolution at each pass.

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Hierarchical JPEG (contd)
f4
N/4 x N/4
f2
N/2 x N/2
f
N x N
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Hierarchical JPEG (contd)
 
 
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Hierarchical JPEG (contd)
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Lossless Differential Pulse Code Modulation
(DPCM) Coding

• Each pixel value (except at the boundaries) is
  predicted based on certain neighbors (e.g.,
  linear combination) to get a predicted image.
• The difference between the original and predicted
  images yields a differential or residual image.
• Encode differential image using Huffman coding.

xm
dm
Entropy Encoder
pm
predictor
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Lossy Differential Pulse Code Modulation (DPCM)
Coding

• Similar to lossless DPCM except that (i) it uses
  quantization and (ii) the pixels are predicted
  from the reconstructed values of certain
  neighbors.

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Block Truncation Coding

• Divide image in non-overlapping blocks of pixels.
• Derive a bitmap (0/1) for each block using
  thresholding.
• e.g., use mean pixel value in each block as
  threshold.
• For each group of 1s and 0s, determine
  reconstruction value
• e.g., average of corresponding pixel values in
  original block.

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Subband Coding

• Analyze image to produce components containing
  frequencies in well defined bands (i.e.,
  subbands)
• e.g., use wavelet transform.
• Optimize quantization/coding in each subband.

102
Vector Quantization

• Develop a dictionary of fixed-size vectors (i.e.,
  code vectors), usually blocks of pixel values.
• Partition image in non-overlapping blocks (i.e.,
  image vectors).
• Encode each image vector by the index of its
  closest code vector.

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Fractal Coding

• What is a fractal?
• A rough or fragmented geometric shape that can be
  split into parts, each of which is (at least
  approximately) a reduced-size copy of the whole.

Idea store images as collections of
transformations!
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Fractal Coding (contd)
Generated by 4 affine transformations!
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Fractal Coding (contd)

• Decompose image into segments (i.e., using
  standard segmentations techniques based on edges,
  color, texture, etc.) and look them up in a
  library of IFS codes.
• Best suited for textures and natural images.

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Fingerprint Compression

• An image coding standard for digitized
  fingerprints, developed and maintained by
• FBI
• Los Alamos National Lab (LANL)
• National Institute for Standards and Technology
  (NIST).
• The standard employs a discrete wavelet
  transform-based algorithm (Wavelet/Scalar
  Quantization or WSQ).

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Memory Requirements

• FBI is digitizing fingerprints at 500 dots per
  inch with 8 bits of grayscale resolution.
• A single fingerprint card turns into about 10 MB
  of data!

A sample fingerprint image 768 x 768 pixels
589,824 bytes
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Preserving Fingerprint Details
The "white" spots in the middle of the black
ridges are sweat pores Theyre admissible points
of identification in court, as are the little
black flesh islands in the grooves between
the ridges
These details are just a couple pixels wide!
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What compression scheme should be used?

• Better use a lossless method to preserve every
  pixel perfectly.
• Unfortunately, in practice lossless methods
  havent done better than 21 on fingerprints!
• Would JPEG work well for fingerprint compression?

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Varying compression ratio

• FBIs target bit rate is around 0.75 bits per
  pixel (bpp)
• i.e., corresponds to a target compression ratio
  of 10.7 (assuming 8-bit images)
• This target bit rate is set via a knob on the
  WSQ algorithm.
• i.e., similar to the "quality" parameter in many
  JPEG implementations.
• Fingerprints coded with WSQ at a target of 0.75
  bpp will actually come in around 151

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End
Thank you