Transform Coding

1
Transform Coding

• Heejune AHN
• Embedded Communications Laboratory
• Seoul National Univ. of Technology
• Fall 2013
• Last updated 2013. 9. 30

2
Agenda

• Transform Coding Concept
• Transform Theory Review
• DCT (Discrete Cosine Transform)
• DCT in Video coding
• DCT Implementation Fast Algorithms
• Appendix KL Transform

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1. Transform Coding

• X1 lum(2n), X2 lum(2n1), neighbor pixels
• X1 U(0, 255), X2 U(0,255)
• Quantization of X1 and X2 gt same data
• Cross-Correlation of X1 and X2
• Y1, Y2
• 45 degree rotation
• Y1 (X1 X2) /2
• Average or DC value
• Y2 (X2 X1) /2
• Difference or AC value
• Y1 F(0, 255), Y2 F(-255,255)

0
0
-255
255
255
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• Which ones are easier to encode (quantize)?

f(X1)
f(X2)
0
0
255
255
f(Y1)
f(Y2)
0
0
-255
255
255
5

• Origins of Transform Coding Benefits
• Signal Theory
• Make the representation easier to manipulate
• energy concentration
• Image and HVS Properties
• HVS is more sensitive to Low frequency
• More dense quantizer to Low frequency

Vilfredo Pareto Economist 1848-1923
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2. Transform Theory Review

• Definition of Transform
• N to M mapping, Y1, Y2, . . ., YN F
  X1,X2, . . ., XM
• Linear Transform (cf. Non-Linear Transform)
• if Y11, Y12 F X11,X12 and Y21, Y22
  F X21,X22
• Y11 Y21, Y12 Y22 F X11X21, X21X22
• Matrix representation of Linear Transform
• Forward
• Inverse

y T x
N transform coefficients, arranged as a vector
Transform matrix of size NxN
Input signal block of size N, arranged as a vector
x T-1 y
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• Basis Vectors
• Orthogonal
• Vl Vm 0 for basis Vector V1, V2, . . ., VN
• Each vectors are disjointed, separated.
• Orthonormal
• Vl 1 for basis Vector V1, V2, . . ., VN
• Parsevals Theorem
• Signal Power/Energy conserves between Transform
  Domain

v1
v2
v3
vN
x T-1 y TT y
T-1 TT gt
y2 yTy xTTT Tx x2
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• Example of Orthonormal transform

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2D Transform

• Data
• 2D pixel value matrix, 2D transform coefs matrix
• 2D matrix gt 1D vector
• Forward Transform
• Inverse transform

y T x
NxN transform coefficients, arranged as a vector
Transform matrix of size N2xN2
Input signal block of size NxN, arranged as a
vector
x T-1 y
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3. Transforms

• Various transforms in image compression
• DFT (Discrete Fourier Transform)
• DCT (Discrete cosine Transform)
• DST (Discrete sine Transform)
• Hadamard Transfrom
• Discrete Wavelet Transform
• and more (HAAR etc )

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Hadamard transform

• Core Matrix
• 1??
• N ??
• 2??
• Transform

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DCT Transform

• 1D Forward DCT (pixel domain to frequency
  domain)
• 1D Inverse DCT (frequency domain to pixel domain)

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2D DCT

• 2D DCT basis Functions
• Coef. Distribution
• DC Uniform dist., AC Laplacian dist.

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• Properties
• Orthonormal transform
• Separable transform
• Real valued coefficients
• DCT performance
• very resembles KLT for image input
• Image input model (1 order Markov chain)
• xn1 rho xn1 e(n)
• DCT complexity
• 2D DCT 1D DCT for vertical 1D DCT for
  horizontal
• Not for 3D (for delay and memory size)
• DCT size (4x4, 8x8, 16x16, 32x32 )
• Larger better performance, but blocking artifact
  (?) and HW complexity

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Coding Performance of DCT
Karhunen Loève transform 1948/1960
Haar transform 1910
Walsh-Hadamard transform 1923
Slant transform Enomoto, Shibata, 1971
Discrete CosineTransform (DCT) Ahmet,
Natarajan, Rao, 1974
Comparison of 1-d basis functions for block size
N8
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• Energy concentration Performance
• measured for typical natural images, block size
  1x32
• KLT is optimum
• DCT performs only slightly worse than KLT

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Complexity Performance of DCT

• Separation of 2D DCT
• Cascading 1-D DCT
• Reduction of the complexity (multiplication) from
  O(N4) to O(N3)
• 8x8 DCT
• For 64 each Coefs, 64 multiplications
• 2 times 64 Coefs x 8
• Can you derive this ?

NxN block of transform coefficients
NxN block of pixels
column-wise N-transform
row-wise N-transform
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4. Transform in Image Coding

• Transform coding Procedure
• Transform T(x) usually invertible
• Quantization not invertible, introduces
  distortion
• Combination of encoder and decoder lossless

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DCT in Image Coding
DCT
Q
Run-level coding
Transformed 8x8 block
Original 8x8 block
Zig-zag scan

Transmission
Reconstructed 8x8 block
Scaling and inverse DCT
Inverse zig-zag scan
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DCT in Image Coding

• Uniform deadzone quantizer
• transform coefficients that fall below a
  threshold are discarded.
• Entrphy coding
• Positions of non-zero transform coefficients are
  transmitted in addition to their amplitude
  values.
• Efficient encoding of the position of non-zero
  transform coefficients zig-zag-scan
  run-level-coding

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DCT Examples

• Note that only a few coefficients has sizable
  value.

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DCT coding with increasingly coarse
quantization, block size 8x8
quantizer stepsize for AC coefficients 25
quantizer stepsize for AC coefficients 100
quantizer stepsize for AC coefficients 200
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4. Implementation

• Implementation issue
• HW or SW
• Computational Cost, Speed, Implementation Size
• Performance Cost
• Implementation complexity
• SW Implementation decision factors
• Computational cost of multiplication
• Whether Fixed or Float point operation (esp.
  multiplication)
• Special Coprocessor and Instruction set (e.g. MMX)

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Fast DCT Algorithm

• Original DCT/IDCT
• Computation load
• 64 Add 64 Mult.
• 8 (7) Addition 8 multiplication / one coeff.
  (from eqn.)
• Scaling
• input range 0, 255 gt output range -2024,
  2024
• Fast DCT
• Similar to Fast DFT
• Share same computation between nodes.
• O(NxN) gt O (N log2N)
• N Width (num of coeff.)
• log2N Steps of algorithm
• Several version Chen, Lee, Arai etc

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Chens FDCT
See Code at http//www.cmlab.csie.ntu.edu.tw/chen
hsiu/tech/fastdct.cpp
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• How the fast algorithm works?
• Exploiting the symmetry of cosine function.
• STEP 1
• STEP 2

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HW Implementation

• 2D DCT using 1D DCT Function Block

Input sample
1-D DCT
Output coef
MUX
8x8 RAM
Row order input
Column order output
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• Distributed Arithmetic DCT
• Multiplier-less architecture
• Lookup, Shift, accumulators only

4 bits from u input
Shift(2-1)
LUT (ROM)
accumulator
Output coef Fx
Add or subtract
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IDCT Mismatch

• DCT x IDCT I ?
• DCT is defined in floating point and direct
  form.
• Integer Implementation induces error after
  Inverse DCT.
• different FDCT has different errors.
• DCT mismatch in MC-DCT
• different reference image at encoder and decoder
• very small error but it accumulates.

orgE
DCT
Q
VLC
VLD
IDCTD
IDCTE
Should Equal but Mismatch !
recD
recE
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• IDCT Mismatch control
• Minimum accuracy of DCT algorithm is defined in
  SPEC.
• H.261/3,MPEG-1/2 Restrict the sum of coefficients
  values
• Oddification rule of sum of all DCT coefficients,
• Make LSB of F63, the last Coef.
• Decoder check and correct the values
• H.264
• (modified) Integer DCT is used

adding random error cancelation
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Appendix

• KL Transform, The Optimal Transform

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Optimal Transform

• Optimality
• (No) Redundancy in input signal gt (No) Redundant
  Quantization Result
• No cross-correlation between different components
  (coefs)
• K-L (Karhunen-Loeve) transform
• Assumption
• Input Covariance is given
• Problem Definition
• find a transform (YT X) such that RY,Y T
  RX,X TT meets diagonal matrix (i.e., completely
  uncorrelated Y)

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Optimal Transform

• Solution
• Build T with eigenvectors of RX,X as basis vector
  
• Then, by the definition of Eigen-vectors values
  (of RX,X)
• So.
• Issue in KLT
• RX,X is varying for image to image Need to
  calculate new T, transmit it to decoder
• Not Separable (vertical, horizontal)
• But, good for benchmarking performance of other
  transform.