Image Transforms: Robust Communication with Lossless and Lossy Compression, Slides of Electrical Engineering

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Image Transforms for Robust

Coding

Introduction

• “The fundamental problem of communication is that

of reproducing at one point either exactly or

approximately a message selected at another point.”

(Claude Shannon, 1948)

• What information should be transmitted?

• How should it be transmitted?

Source Coding

• Goals

– Reduction of redundancy

– Removal of irrelevancy (irreversible)

• Stages

– Transformation

– Quantization (lossy)

– Entropy coding

Image samples

Transform

coefficients

Symbol stream

Bit stream

Lossless and Lossy

• Lossless (2:1 to 8:1)

– Limited by the entropy of the message (Claude E.

Shannon)

• Lossy (100:1)

Entropy is a measure of information content:

the more probable the message, the lower its

information content, the lower its entropy

Transformations for Image Coding

• Block transforms (blocks in the spatial domain)

– Fixed size

– Quadtrees

• Sub-band decompositions (blocks in the frequency

domain)

– Uniform

– Logarithm

• Pyramids

• Wavelet

• Wavelet packets

Quantization

Two-bit resolution Three-bit resolution

Four-level digital

representation

Eight-level digital

representation

Can be used to exploit features of the human visual system

Entropy Coding

• Methods based on repeated characters: run-length encoding

– The repeated character is replaced by the number of occurrences and by the

character itself

• e.g. AAAABBBCCCCC (12 symbols) is coded as 4A3B5C (6 symbols) [2:1]

• Methods based on probability of occurrence: Huffman coding,

arithmetic coding

– Symbols that occur more often are assigned shorter codes

• e.g. natural languages: ‘a’, ‘is’, ‘the’ (shorter words) are those with higher

probability

• Dictionary-based methods: Lempel-Zip coding

– Words and phrases within a text stream are likely to be repeated

• e.g. acronyms: The Lempel-Zip encoding methods (LZ) are string-matching

techniques. LZ were invented in 1977 and 1978

Fourier Analysis

Fourier Transform

Short-time Fourier Analysis

Wavelet Transform Bases

Time vs. Frequency

Wavelet Transform

2-D Wavelet Transform