Image Compression

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Image Compression
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Topics Covered

• Introduction
• Static Image Compression
• JPEG

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Introduction
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Introduction

• We all know what Image Compression is

File Format
Original Image
Reproduction
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Introduction Lossless Compression

• Compression often restricted, but reproduction is
  identical to original

Original Image
Reproduction
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Introduction Lossy Compression

• Higher Compression, but some image information is
  lost

gt
Original Image
Reproduction
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Introduction The Need for Standards

• Growth of communications networks (WWW, Mobile
  Phones, etc)
• Prevent the proliferation of different types of
  file formats, keep everything manageable
• Transmission of images of particular kinds
  requires that some minimum level of service can
  be expected.

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Static Image Compression
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Compression of Static Images.

• In any image there is redundant information.
• Compression seeks to remove the redundant
  information so that only essential information
  remains.
• We use mathematical notions of what constitutes
  information.

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Information

• Information Uncertainty (Claude Shannon, 1948)

Take 1 pixel
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Two Approaches to Compression

• Lossless most efficient encoding so as to
  reduce redundancy in representation.
• Lossy most efficient coding, and reducing
  perceptual redundancy.

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Lossless Encoding - Huffman

• Huffman Encoding variable length encoding
  system, to reduce redundancy in a data set.
• Eg. Take an image with n greylevels.
• For the image calculate the probability of each
  grey level

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Huffman Encoding
0
1
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Huffman Encoding

• The two grey-levels (symbols) with the smallest
  probabilities are combined to make a new compound
  symbol, whose probability is the summed
  individual probabilities.
• The symbols are then reordered in terms of their
  probability, and the symbols with the two
  smallest probabilities are combined until we
  arrive at two symbols.

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Huffman Coding

• The easiest way of encoding two symbols is with 0
  and 1.
• The symbol with the highest probability therefore
  obtains the code 0, and the other symbol is a 1.
• We then decompose the compound symbol into its
  two components, and add a 0 and 1 to its
  components.
• We continue to decompose until we end up with a
  complete set of code symbols.

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Huffman Encoding
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Lossy Encoding Frequency Analysis

• Spatial Frequency transforms reorder the image
  such that the intensity of the power at each
  frequency, represents the number of pixels at
  that frequency.
• Most power in images is usually at the lower
  frequencies.

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Low Pass Filtering

• Usually higher Frequencies have lower power.
• This correlates to large areas of redundant
  information.
• LP Filtering in the frequency domain is a good
  compression strategy.

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What Transform?

• Several Means of transforming images into
  Frequency domain
• Fourier
• Wavelet
• Cosine
• Walsh, Hadamard, Hartley, etc.

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Problems with Transforms

• Often based on difficult numbers.
• Fourier Transform is rendered unsuitable, because
  it relies on complex numbers less tractable in
  compression algorithms.
• Other techniques more suitable because they rely
  on real numbers, usually converted to integers
  for storage using rounding.

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Spatial Decorrelation

• Frequency transforms decorrelate the spatial
  information, placing it in increasing order of
  frequency.
• Inverse transformation is required in order to
  reconstruct the spatial image at the inverse
  stage.

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Low Pass Filtering

• Attenuate (set to zero) all information above a
  certain distance from the origin.

0.9,0.8,0.75,0.7,0.4, 0.1,0.12,0.115,0.115,0.1
Power
0.9,0.8,0.75,0.7,0.4, 0.1,0,0,0,0
w
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Low Pass Filtering
Original Image
FFT
LPF in Fourier Domain
IFFT
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Low Pass Filtering
Wavelet Transform
Original Image
Wavelet LPF
Restored Image
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The JPEG Compression Standard

• Highly configurable standard, used extensively
  in
• Graphic Arts
• Desktop Publishing
• Faxes
• Medical Images
• Etc.

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JPEG

• Supports both lossless, and lossy compression, of
  any size and bit depth.
• The technique is also parameterisable so users
  can trade off degree of compression, against
  quality.
• Offers four modes of operation
• Sequential encoding in the order the image was
  scanned.
• Progressive mulitipass encoding so that a
  coarse image is transmitted rapidly, followed by
  repeated passes at higher resolutions
• Lossless lossless compression
• Hierarchical, the image is encoded at multiple
  resolutions.

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JPEG

• Basic lossy compression is based on the Direct
  Cosine Transform (DCT).
• Image is subdivided into 8x8 blocks, to which the
  DCT is applied.
• The coefficients are quantised according to a
  quantisation table supplied as part of the
  application.
• The quantisation table becomes part of the
  compressed stream.

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JPEG Quantisation

• DCT version of the 8x8 image is divided by a
  quantisation table.
• Where Iq is rounded.
• The quantisation table is itself an 8x8 matrix of
  integers, where large values imply coarser
  quantisation.
• Normally the range of values in a quantisation
  table are between 0 and 255

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JPEG

• Each image can have its own quantisation table,
  which can be submitted as part of the format.
• If using a bespoke quantisation table, ensure
  that pixels with high degrees of variation obtain
  finer quantisation, than those pixels with lower
  quantisation.

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JPEG

• The transformed image is then stored as a series
  of lists of coefficients, using zig-zag ordering.

This creates a 1D array of coefficient values,
so that the information rich parts of the
frequency spectrum are at the beginning, and the
zeros are at the end.
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JPEG

• An End of Block marker (EOB) identifies the point
  at which all coefficients become zero.
• This facilitates the the next step which is
  statistical coding (typically Huffman encoding).

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JPEG

• High resolution colour images, JPEG can compress
  to a ratio of 101 with results being more than
  adequate in many cases.
• JPEG forms the basis of the MPEG system for
  encoding images, as will be discussed in due
  course.