1. Image Coding and Compression
Introduction to Digital Image Processing with MATLAB® Asia Edition
McAndrew‧Wang‧Tseng
Chapter 14:
Image Coding and Compression
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2. 14.1 Lossless and Lossy Compression
It is thus important for both reasons of storage and file transfer to make these file sizes smaller, if possible
It will be necessary to distinguish between two different classes of compression methods:
Lossless compression, where all the information is retained
Lossy compression, where some information is lost
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14.2 Huffman Coding
The average number of bits per pixel can be calculated easily as the expected value (in a probabilistic sense):
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14.2 Huffman Coding
Determine the probabilities of each gray value in the image
Form a binary tree by adding probabilities two at a time, always taking the two lowest available values
Now assign 0 and 1 arbitrarily to each branch of the tree from its apex
Read the codes from the top down
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FIGURE 14.1
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14.2 Huffman Coding
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14.2 Huffman Coding
We can evaluate the average number of bits per pixel as an expected value
Huffman codes are uniquely decodable
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14.3 Run-length Encoding
Run-length encoding (RLE) is based on a simple idea: to encode strings of 0s and 1s by the number of repetitions in each string
Encode each line separately starting with the number of 0s (binary image)
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14.3 Run-length Encoding
Encode each row as a list of pairs of numbers, the first number in each pair giving the starting position of a run of 1s and the second number its length
Grayscale images can be encoded by breaking them up into their bit planes
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14.3 Run-length Encoding
Each plane can then be encoded separately using our chosen implementation of RLE
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14.3 Run-length Encoding
However, small changes of gray value may cause significant changes in bits
To overcome this difficulty, we may encode the gray values with their binary Gray codes
A Gray code is an ordering of all binary strings of a given length so that there is only one bit change between one string and the next
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14.3 Run-length Encoding
4-bit gray codes
Binary bit plane
Gray codes
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14.3 Run-length Encoding
Run-length Encoding in MATLAB
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FIGURE 14.3
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14.3 Run-length Encoding
We can reduce the size of the output by storing it using the data type uint16
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14.3 Run-length Encoding
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14.4 The JPEG Algorithm
Lossy compression trades some acceptable data loss for greater rates of compression
The algorithm developed by the Joint Photographic Experts Group (JPEG) has become one of the most popular
It uses transform coding, where the coding is done not on the pixel values themselves, but on a transform
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14.4 The JPEG Algorithm
discrete cosine transform (DCT)
In the JPEG algorithm it is applied only to 8 × 8 blocks
If f(j, k) is one such block, then the forward (2-D) DCT is defined as
The corresponding inverse DCT as
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14.4 The JPEG Algorithm
where C(w) is defined as
The DCT has a number of properties that make it particularly suitable for compression:
It is real-valued, so there is no need to manipulate complex numbers
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14.4 The JPEG Algorithm
It has a high information-packing ability because it packs large amounts of information into a small number of coefficients
It can be implemented very efficiently in hardware
Like the FFT, there is a “fast” version of the transform that maximizes efficiency
The basis values are independent of the data [7]
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14.4 The JPEG Algorithm
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FIGURE 14.4
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14.4 The JPEG Algorithm
The JPEG baseline compression scheme is applied as follows:
The image is divided into 8 × 8 blocks, with each block transformed and compressed separately
For a given block, the values are shifted by subtracting 128 from each value
The DCT is applied to this shifted block
The DCT values are normalized by dividing by a normalization Matrix Q. It is this normalization that provides the compression by making most of the elements of the block zero
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14.4 The JPEG Algorithm
This matrix is formed into a vector by reading off all nonzero values from the top left in a zigzag fashion:
The first coefficients of each vector are encoded by listing the difference between each value and the values from the previous block. This helps keep all values (except for the first) small
These values are then compressed using RLE
All other values (known as the AC coefficients) are compressed using a Huffman coding
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14.4 The JPEG Algorithm
To decompress, the steps above are applied in reverse
The Huffman encoding and RLE can be decoded with no loss of information
The vector is read back into an 8 × 8 matrix
The matrix is multiplied by the normalization matrix
The inverse DCT is applied to the result
The result is shifted back by 128 to obtain the original image block
The normalization matrix
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14.4 The JPEG Algorithm
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14.4 The JPEG Algorithm
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14.4 The JPEG Algorithm
The normalization matrix
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14.4 The JPEG Algorithm
Where EOB signifies the end of the block. By this stage we have reduced an 8 × 8 block to a vector of length 21, containing only small values
To uncompress
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14.4 The JPEG Algorithm
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14.4 The JPEG Algorithm
It can be seen that these values are very close to the values in the original block. The differences between original and reconstructed values are
The algorithm works best on regions of low frequency; in such cases the original block can be reconstructed with only very small errors
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FIGURE 14.5
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FIGURE 14.6
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FIGURE 14.7
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FIGURE 14.8
Suppose we take our extra parameter n to be 2. This has the effect of doubling each value in the normalization matrix, thus should set more of the DCT values to 0:
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14.4 The JPEG Algorithm
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FIGURE 14.9
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FIGURE 14.10
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FIGURE 14.11~14.13
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14.4 The JPEG Algorithm
We have seen how changing the compression rate may affect the output. The JPEG algorithm, however, is particularly designed for storage
The output vector is further encoded using Huffman coding
DC (first value)
k contains all elements x whose
absolute value satisfies
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14.4 The JPEG Algorithm
e.g.
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