Lossy Compression

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Chapter 21

• Lossy Compression

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Introduction

• Some applications can tolerate losses of original
  data (e.g. voice, video, image)
• Compression algorithms offer tradeoffs between
  fidelity and compression ratio
• typically greater compression lower fidelity
• Chapter Overview
• Discrete Cosine Transform (DCT)
• Wavelet compression
• JPEG compression
• MPEG compression

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Discrete Cosine Transform (DCT)

• Provides one-to-one mapping of an array of pixel
  values into an array in the spatial frequency
  domain
• DCT does not provide any compression, but puts
  image information in a form more suitable for
  compression
• One-dimensional DCT of p(x) (0 ? x ? N)

C(f)
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One-Dimensional DCT - Example
0 white, 255 black
grayscale function is shifted to obtain symmetry
around 0, p(x)
one-dimensional DCT of p(x)
inverse DCT of S(f)
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Discrete Cosine Basic Functions
cos(0)
cos(2x1)4?/16
cos(2x1)?/16
cos(2x1)5?/16
cos(2x1)6?/16
cos(2x1)2?/16
cos(2x1)3?/16
cos(2x1)7?/16
General form cos(2x1)?f/2N
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Properties of One-Dimensional DCT Basic Functions

• Complete a weighted sum of N cosine functions
  can be found for any N pixel sample values
• Minimal none of the N waveforms can be
  represented by any weighted combination of the
  other N-1 waveforms, and all N are required for
  completeness
• Unique no other set of cosine waveforms, other
  than scaled versions of the original N waveforms,
  can be used to represent all possible sequences
  of N pixel values

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Two-Dimensional DCT - Example
Similarly, a matrix of N x N pixels can be
uniquely represented by a weighted sum of N x N
cosine functions.
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Wavelet Compression

• Wavelet
• a function that is non-zero only over a short
  region, and zero elsewhere
• a wavelet function is represented by summing
  wavelets expansions and contractions of a single
  wavelet, knowas the mother wavelet
• Simplest form of wavelet used in wavelet analysis
  is the Haar wavelet

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Haar Wavelets
Mother wavelet
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Compression Using Haar Wavelets
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Matrix Formed from Haar Mother Wavelet
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JPEG Image Compression

• Sequential DCT-based
• Single Left to Right, Top to Bottom scan
• uses quantization to reduce matrix before
  compression
• Progressive DCT-based
• multiple scans, increasing clarity (resolution)
• precision of coefficients increased in each scan
• Lossless
• uses predictor matrix, followed by Huffman or
  arithmetic coding of ordered matrix
• Hierarchical
• encodes a set of progressively lower resolution
  images
• compress each image, form low to high resolution
  using one of above methods

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JPEG Modes of Operation
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JPEG Sequential DCT
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Quantization Effects
Quantization Matrix Q (u, v)
Quantized DCT Matrix K (u, v)
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Zig-Zag Scan
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Image Reconstruction
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JPEG Scheme
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Progressive DCT Compression
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Progressive DCT Compression (continued)
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JPEG Lossless Mode
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JPEG 2000
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MPEG Structure
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Block Motion Compensation
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Temporal Picture Structure -Example