Image Compression Coding Schemes
Short Description
CS 804B, M3_2, Lecture Notes...
Description
Resmi N.G. Reference: Digital Image Processing 2nd Edition Rafael C. Gonzalez Richard E. Woods
Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Error-Free Compression Devise an alternative representation of the image in which
its interpixel redundancies are reduced. Code
the representation redundancies.
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to
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eliminate
the
coding
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Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Variable-Length Coding Reduces only coding redundancy. Assign the shortest possible codewords to the most
probable graylevels.
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Huffman Coding Yields the smallest possible number of code symbols per
source symbol. Creates a series of source reductions by ordering the probabilities of the symbols. Combines the lowest probability symbols into a single symbol that replaces them in the next source reduction. Codes each reduced source, starting with the smallest source and working back to the original source. This operation is repeated for each reduced source until the original source is reached.
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Huffman’s procedure creates optimal code which is an
instantaneous uniquely decodable block code. Block code – each source symbol is mapped into a fixed
sequence of code symbols. Instantaneous – each codeword can be decoded without referencing succeeding symbols. Uniquely decodable – any string of code symbols can be coded in only one way.
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Arithmetic Coding Generates non-block codes. An entire sequence of source symbols is assigned a single
arithmetic codeword. The codeword defines an interval of real numbers between 0 and 1. As the number of symbols in the message increases, the interval used to represent it becomes smaller and the number of information units required to represent the interval becomes larger. 3/24/2012
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Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Lempel-Ziv-Welch (LZW) Coding Assigns fixed length codewords to variable length
sequences of source symbols. Requires no apriori knowledge of the probability of occurrence of the symbols to be encoded. A codebook or dictionary containing the source symbols to be coded is constructed while the data are being encoded. An LZW decoder builds an identical decompression dictionary during decoding. 3/24/2012
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Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Bit-Plane Coding Reduces interpixel redundancies. Decomposes a multilevel image into a series of binary
images. Compresses each image using one of the binary compression methods.
m bit gray scaleimage am1 2m1 am2 2m2 .... a1 21 a0 20
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Constant Area Coding Use special codewords to identify large areas of
contiguous 1’s or 0’s. Image is divided into blocks of size pxq pixels, which are
classified as all white, all black or mixed intensity. Most probable or frequently occurring category is
assigned 1-bit codeword 0, the other two are assigned 2bit codes 10 and 11. 3/24/2012
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One-dimensional run-length coding Represents each row of an image or bit plane by a
sequence of lengths that describe successive runs of black and white pixels. Code each contiguous group of 0’s or 1’s encountered in a
left to right scan of a row by its length and specify the value of the first run of each row. Black and white run lengths may be coded separately
using variable-length codes. 3/24/2012
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Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Lossless Predictive Coding
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Based on eliminating the interpixel redundancies of
closely spaced pixels by extracting and coding only the new information in each pixel. The new information of a pixel is defined as the difference
between the actual and predicted value of that pixel. System consists of an encoder and a decoder each
containing an identical predictor. The prediction error is coded using variable-length code.
en f n f n 3/24/2012
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The decoder reconstructs the error from the received
variable-length codewords and performs the inverse operation.
f n en f n
Prediction is usually formed by a linear combination of m
previous pixels.
f n round i f ni i 1
m
In 1-D linear predictive coding,
m f n ( x, y) round i f ( x, y i ) i 1
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Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Error-Free Compression Variable-Length Coding
Huffman Coding Other Near Optimal Variable Length Codes Arithmetic Coding
LZW Coding Bit-Plane Coding Bit-Plane Decomposition Constant Area Coding One-Dimensional Run-Length Coding Two-Dimensional Run-Length Coding Lossless Predictive Coding
Lossy Compression Lossy Predictive Coding 3/24/2012
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Lossy Predictive Coding
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Lossy predictive coding – Quantizer absorbs the nearest-integer function of the error-
free encoder and is inserted between the symbol encoder and the point at which the prediction error is formed. It maps prediction error into a limited range of outputs and
establishes the amount of compression and distortion associated with lossy predictive coding. The lossy encoder’s predictor is placed within a feedback .
loop, where its input f n , is generated as a function of past predictions and the corresponding quantized errors.
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.
.
f n en f n m f n round i f n i i 1
.
.
f n en f n
Output of the decoder is also given by
Optimum predictors minimize the mean square prediction
error,
2 2 E{en } E f n f n , subject tothe constraints .
.
m
f n en f n en f n f n and f n i f ni i 1
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Optimum
quantizers minimize the mean square 2 quantization error, E s ti .
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Transform Coding
Transform Selection Subimage Size Selection Bit Allocation Zonal Coding Implementation Threshold Coding Implementation Wavelet Coding Wavelet Selection Decomposition Level Selection Quantizer Design Image Compression Standards Binary Image Compression Standards One Dimensional Compression Two Dimensional Compression
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Transform Coding
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Predictive coding – directly operates on the pixels of an
image (spatial domain method). Transform coding – based on modifying the transform of
an image. A reversible linear transform (eg; Fourier transform) is
used to map the image into a set of transform coefficients. These coefficients are then quantized and coded.
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Transform Coding Subimage decomposition – NxN image is decomposed
into subimages of size nxn. Transformation – This process decorrelates the pixels of
each subimage or packs as much information as possible into the smallest number of coefficients. Quantization – Selectively eliminates or more coarsely
quantizes the coefficients that carry the least information. Encoding – Codes the quantized coefficients. 3/24/2012
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Transform Selection Transform is selected based on the amount of reconstruction error that can be tolerated and the available computational resources. The forward discrete transform T(u,v) of an image f(x,y) of
size NxN can be expressed as, N 1 N 1
T (u , v) f ( x, y ) g ( x, y, u , v) x 0 y 0
for u , v 0,1,..., N 1
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The inverse discrete transform f(x,y) can be obtained as, N 1 N 1
f ( x, y ) T (u , v)h( x, y, u , v) u 0 v 0
for x, y 0,1,..., N 1 g(x,y,u,v) and h(x,y,u,v) are called the forward and inverse
transformation kernels respectively and are also called as basis functions or basis images. T(u,v)
for u,v=0,1,…N-1 coefficients.
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are
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called
the
transform
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The forward kernel g(x,y,u,v) is said to be separable if
g(x,y,u,v) = g1(x,u)g2(y,v) The kernel is symmetric if g1 is functionally equal to g2. The same applies for inverse kernel. A 2-D transform with a separable kernel can be computed
using row-column or column-row corresponding 1-D transform.
passes
of
the
The forward and inverse kernels determine the type of
transform that is computed and the overall computational complexity and reconstruction error of the transform coding system. 3/24/2012
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Most well-known transform-pair (DFT)
1 g ( x, y , u , v ) 2 e N h ( x, y , u , v ) e
j 2 ( ux vy ) N
j 2 ( ux vy ) N
Walsh-Hadamard transform-pair m 1
bi ( x ) pi (u ) bi ( y ) pi ( v )
g ( x, y, u, v) h( x, y, u , v) ( 1) i 0 where N 2 . m
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The summation in the exponent is performed in modulo 2
arithmetic and bk(z) is the kth bit(from right to left) in the binary representation of z.
If m 3, z 6 (110 inbinary ), b2 ( z ) 1, b1 ( z ) 1, b0 ( z ) 0 p0 (u ) bm1 (u ) p1 (u ) bm1 (u ) bm2 (u ) p2 (u ) bm2 (u ) bm3 (u ) : pm1 (u ) b1 (u ) b0 (u ) 3/24/2012
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Discrete cosine transform
g ( x, y , u , v ) h ( x, y , u , v ) (2 x 1)u (2 y 1)v (u ) (v) cos cos 2 N 2 N 1 N where (u ) 2 N
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for u 0 for u 1, 2,...N 1
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Consider a sub image of size nxn n 1 n 1
f ( x, y ) T (u , v)h( x, y, u , v) u 0 v 0
for x, y 0,1,..., n 1 n 1 n 1
F T (u , v)H uv u 0 v 0
h(0,1, u , v) ... h(0, n 1, u , v) h(0,0, u , v) h(1,0, u , v) h(1,1, u , v) ... h(1, n 1, u , v ) H uv : : : : h(n 1,0, u , v) h(n 1,1, u , v) ... h(n 1, n 1, u , v) 3/24/2012
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The matrix F with pixels of input sub-image is a linear
combination of n2 matrices of size nxn, Huv for u,v = 0,1,…n-1. These n2 matrices of size nxn are the basis images and the
associated T(u,v) are the expansion coefficients. F can be compressed by truncating the transform
coefficients (setting transform coefficients to zero based on a specified truncation criterion). For this purpose, we define a transform coefficient
masking function. 3/24/2012
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0 if T (u , v) staisfies a specified truncation (u , v) otherwise 1 Approximation of F fromthetruncated expansion
n 1 n 1
F (u , v)T (u , v) H uv u 0 v 0
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The mean square error between subimage F and
approximation F is erms
2 E F F n 1 n 1 n1 n1 E T (u , v)H uv (u , v)T (u , v) H uv u 0 v 0 u 0 v0 2 n1 n1 E T (u , v)H uv 1 (u , v) u 0 v0
2
n 1 n 1
T2 (u ,v ) 1 (u , v) u 0 v 0
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The total mean-square approximation error is thus the sum
of the variances of the discarded transform coefficients (coefficients for which γ(u,v) = 0 so that 1- γ(u,v) = 1). The mean-square error of the N/n2 subimages of an NxN
image are identical. Thus, the mean-square error of the NxN image equals that of a single subimage.
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Subimage size selection Subdivide the image so that the correlation or redundancy
between adjacent subimages is reduced to a specified acceptable level. Bit Allocation The retained coefficients are mostly selected on the basis
of maximum variance (zonal coding) or on the basis of maximum magnitude (threshold coding). The overall process of truncating, quantizing and coding
the coefficients of a transformed subimage is called bit allocation. 3/24/2012
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Zonal Coding Implementation The transform coefficients of maximum variance carry the
most image information and must be retained in the coding process. sampling – Multiplying each transformed coefficient T(u,v) with the corresponding element in a zonal mask (1 for locations with maximum variance and 0 for all other locations).
Zonal
Coefficients of maximum variance are usually located
around the origin of an image transform. 3/24/2012
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The coefficients retained during zonal coding process
must be quantized and coded. Mostly, the coefficients are allocated the same number of
bits (coefficients are generally normalized by their standard deviations and are uniformly quantized) or a fixed number of bits is distributed among them unequally (a quantizer is designed for each coefficient). The retained coefficients, selected on the basis of
maximum variance, are allocated bits proportional to the logarithm of the coefficient variances. 3/24/2012
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Threshold Coding Implementation Zonal coding – uses single fixed mask for all subimages.
Threshold coding Adaptive transform coding approach. Based on the concept that for any subimage, the transform coefficients of the largest magnitude make the most significant contribution to reconstructed subimage quality. The location of transform coefficients retained for each subimage therefore varies from one subimage to another. The elements of γ(u,v)T(u,v) are reoredered to form a 1-D run length coded sequence.
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Three ways to threshold a transformed subimage: A single global threshold to all subimages – Level of compression varies from image to image depending on the number of coefficients that exceed the threshold. A different threshold for each subimage – (called N-largest
coding) Same number of coefficients are discarded for each subimage. The code rate is therefore constant and known in advance. Threshold as a function of the location of each coefficient
within the subimage – Results in variable code rate.
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In the third approach, thresholding and quantization can
be combined by replacing γ(u,v)T(u,v) in n 1 n 1
F (u, v)T (u , v)H uv u 0 v 0
with
T (u, v) T (u, v) round Z (u, v)
where
is a thresholded and quantized approximation of T(u,v) and Z(u,v) is an element of the transform normalization array Z.
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T (u, v)
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A normalized subimage transform, T (u, v)
must be denormalized by multipying with Z(u,v) before it is inverse transformed to obtain the approximation of subimage f(x,y). .
T (u, v) T (u, v)Z (u, v) 3/24/2012
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Transform Coding
Transform Selection Subimage Size Selection Bit Allocation Zonal Coding Implementation Threshold Coding Implementation Wavelet Coding Wavelet Selection Decomposition Level Selection Quantizer Design Image Compression Standards Binary Image Compression Standards One Dimensional Compression Two Dimensional Compression
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Wavelet Coding
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Based on the idea that the coefficients of a transform that
decorrelates the pixels of an image can be coded more efficiently than the original pixels themselves. If the transform’s basis functions(here, wavelets) can pack
most of the information into a small number of coefficients, the remaining coefficients can be truncated to zero. Discrete wavelet transform of the image is computed
which converts a large portion of the original image to vertical, horizontal and diagonal decomposition coefficients. 3/24/2012
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Many of the computed coefficients carry little information
and hence can be quantized and coded to minimize the intercoefficient and coding redundancy. Lossless
coding methods (Run-length, Huffman, Arithmetic, and Bit-plane coding) can be used for the final symbol coding process.
Main difference between wavelet coding and transform
coding – Wavelet coding does not require subdivision of original image. Less reconstruction error. 3/24/2012
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Wavelet selection Most widely used are Daubechies wavelets and biorthogonal
wavelets. Decomposition level selection The number of operations in the computation of the forward and inverse transforms increases with the number of decomposition levels. Quantizer design An enlarged quantization interval around zero Adapting the size of quantization interval from scale to scale. The selected intervals are transmitted to decoder with the encoded image bit stream. 3/24/2012
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Transform Coding
Transform Selection Subimage Size Selection Bit Allocation Zonal Coding Implementation Threshold Coding Implementation Wavelet Coding Wavelet Selection Decomposition Level Selection Quantizer Design Image Compression Standards Binary Image Compression Standards One Dimensional Compression Two Dimensional Compression
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International Standardization Organization (ISO) Consultative Committee of the International Telephone
and Telegraph (CCITT) Binary and Continuous-tone image compression Still-frame and Video(Sequential frame) applications
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Transform Coding
Transform Selection Subimage Size Selection Bit Allocation Zonal Coding Implementation Threshold Coding Implementation Wavelet Coding Wavelet Selection Decomposition Level Selection Quantizer Design Image Compression Standards Binary Image Compression Standards One Dimensional Compression Two Dimensional Compression
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Continuous Tone Still Image Compression Standards
JPEG Lossy Baseline Coding System Extended Coding System Lossless Independent Coding System JPEG 2000 Video Compression Standards
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Continuous Tone Still Image Compression Standards Based principally on lossy transform coding techniques. Original DCT-based JPEG standard Wavelet-based JPEG 2000 standard JPEG-LS standard
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JPEG Three different coding systems: Lossy baseline coding system – based on DCT Extended coding system – for higher precision Lossless independent coding system for compression
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Baseline coding system Input and output data precision is limited to 8 bits. Quantized DCT values are restricted to 11 bits. Compression is performed in 3 steps: DCT computation Quantization Variable-length code assignment
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The image is first sub-divided into pixel blocks of size
8x8, which are processed from left to right, top to bottom. The pixels in each block are level shifted by subtracting
the quantity 2n-1, where 2n is the maximum number of gray levels. 2D-DCT of the block is then computed, quantized and
reordered to coefficients.
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form
a
1D-sequence
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quantized
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The original 8x8 sub-image
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Level shifted subimage (subtract 27 from each pixel)
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Apply forward DCT
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Normalization array Z
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T (u, v) T (u , v) round ; Z (u , v)
T (0,0) 415 T (0,0) round round 26 16 Z (0,0)
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The coefficients are then reordered in a zig-zag pattern, resulting in a 1D coefficient sequence.
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The default JPEG code is then constructed for this sequence.
First, the difference between current DC coefficient and that of the previously encoded subimage is computed.
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For a DC difference category K, an additional K bits are
needed and are computed as either the K LSBs of the positive difference or the K LSBs of the negative difference minus 1. Example: DC difference = -9 DC Difference category = 4 (ie; K = 4) DC Code = 101 (requires additional 4 bits) 9 : 1001 -9: 0110 + 0001 = 0111 0111 – 1 = 0110 (K LSBs of 0110) Therefore, the complete code is 1010110. 3/24/2012
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Decoder regenerates the normalized transform coefficients.
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.
Denormalization : T (u, v) T (u, v) Z (u, v) .
T (0,0) T (0,0) Z (0,0) (26)(16) 416
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Inverse DCT of denormalized reconstructed subimage.
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array
gives
completely
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Level shift each inverse transformed pixel by adding 27.
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Difference between original and reconstructed images.
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JPEG 2000 Extends the initial JPEG standard to provide increased
flexibility. Portions of JPEG 2000 compressed image can be
extracted for retransmission, storage, display and editing. Based on wavelet coding. Quantized coefficients are arithmetically coded on bit-
plane basis. 3/24/2012
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Wavelet Theory – Some Concepts Wavelet transform – based on small waves of varying
frequency and limited duration called wavelets.
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1. First step of the encoding process is to DC level shift the
samples of the n-bit image to be coded by subtracting 2n-1. If image has more than one component(R,G,B), each component is individually shifted. 2. If there are exactly 3 components, they can be decorrelated using a reversible or irreversible linear combination of the components. 3. eg; irreversible component transform Y0(x,y) = 0.299I0(x,y) + 0.587I1(x,y) + 0.114I2(x,y) Y1(x,y) = - 0.169I0(x,y) – 0.331I1(x,y) + 0.5I2(x,y) Y2(x,y) = 0.5I0(x,y) – 0.419I1(x,y) - 0.081I2(x,y) 3/24/2012
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I0 , I1, I2 are the level-shifted input components. Y0, Y1, Y2 are the corresponding decorrelated components. 3. After the image has been level-shifted and decorrelated, its components are divided into tiles, which are rectangular arrays of pixels containing same relative proportion of all components. These components can be extracted and reconstructed independently. 4. The 1D – Discrete Wavelet Transform(DWT) of rows and columns of each tile component is then computed. A lifting-based approach involving 6 lifting and scaling operations is used. 3/24/2012
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The resultant even-indexed values of Y are equivalent to fast wavelet transform lowpass filtered output; oddindexed values of Y correspond to highpass filtered output.
This produces 4 components: a low-resolution approximation of the tile component, and its horizontal, vertical and diagonal frequency characteristics.
When each of the tile components has been processed, the total number of transform coefficients is equal to the number of samples in the original image.
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5.
Quantization
ab (u, v) qb (u, v) sign[ab (u, v)]. floor b b is the quantization step size. 6.
Final step: coefficient bit modeling, arithmetic coding, bitstream layering and packetizing. Coefficients of each transformed tile component are arranged into rectangular blocks called code blocks, which are individually coded one bit plane at a time.
Starting from the most significant bit plane with nonzero element, each bit plane is processed in 3 passes. 3/24/2012
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Each bit is coded in one of the 3 passes:
propagation – coefficients that are insignificant but has atleast one significant coefficient in neighbourhood are coded. Magnitude refinement – current bits of significant coefficients are coded. Cleanup - remaining insignificant coefficients are coded. Significance
The outputs are arithmetically coded and grouped with similar
passes from other code blocks to form layers. These layers are partitioned into packets, which are the
fundamental units of the encoded stream. 3/24/2012
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JPEG 2000 decoders simply invert these operations. Decodes the bit-modeled, arithmetically coded, layered and
packetized code-stream. A user-selected number of the original image’s tile component
subbands are reconstructed. Coefficients are dequantized. Dequantized coefficients are inverse transformed by column
and by row using inverse fast wavelet transform or inverse lifting operations. Final steps are the assembly of tile components, inverse
component transformation and DC level shifting. 3/24/2012
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For Self-Study Binary Image Compression Standards Video Compression Standards
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Thank You
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