1. Transform Coding Heejune AHN Embedded Communications Laboratory
Seoul National Univ. of Technology
Fall 2013
Last updated

2. Agenda Transform Coding Concept Transform Theory Review
DCT (Discrete Cosine Transform)
DCT in Video coding
DCT Implementation & Fast Algorithms
Appendix: KL Transform

3. 1. Transform Coding X1= lum(2n), X2= lum(2n+1), neighbor pixels Y1, Y2
X1 ~ U(0, 255), X2~ U(0,255)
Quantization of X1 and X2 => 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)
255
-255
255

4. Which ones are easier to encode (quantize)?
f(X1)
f(X2)
255
255
f(Y1)
f(Y2)
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

6. 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 [X11+X21, X21+X22]
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

7. x = T-1 y = TT y T-1 =TT => ||y||2 = yTy = xTTT Tx = ||x||2
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
Parseval’s Theorem
Signal Power/Energy conserves between Transform Domain v1 v2 v3 vN
T-1 =TT =>
x = T-1 y = TT y
||y||2 = yTy = xTTT Tx = ||x||2

8. Example of Orthonormal transform

9. 2D Transform y = T x x = T-1 y Data Forward Transform
2D pixel value matrix, 2D transform coefs matrix
2D matrix => 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

10. 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 )

11. Hadamard transform
Core Matrix
1차원
N 차원
2차원
Transform

12. DCT Transform 1D Forward DCT (pixel domain to frequency domain)
1D Inverse DCT (frequency domain to pixel domain)

13. 2D DCT 2D DCT basis Functions Coef. Distribution
DC ~ Uniform dist., AC ~ Laplacian dist.

14. Properties DCT performance DCT complexity Orthonormal transform
Separable transform
Real valued coefficients
DCT performance
very resembles KLT for image input
Image input model (1 order Markov chain)
xn+1 = rho * xn+1 + 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

15. 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 N=8

16. Energy concentration Performance
measured for typical natural images, block size 1x32
KLT is optimum
DCT performs only slightly worse than KLT

17. 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 N
column-wise N-transform
row-wise
N-transform

18. 4. Transform in Image Coding
Transform coding Procedure
Transform T(x) usually invertible
Quantization not invertible, introduces distortion
Combination of encoder and decoder lossless

19. DCT in Image Coding Original 8x8 block Transformed 8x8 blockQ
Run-level coding
Original 8x8 block
Transformed 8x8 block
Zig-zag scan
Transmission
Reconstructed 8x8 block
Scaling and inverse DCT
Inverse zig-zag scan

20. DCT in Image Coding Uniform deadzone quantizer Entrphy coding
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
Baseline JPEG does not use deadzone

21. DCT Examples
Note that only a few coefficients has sizable value.

22. quantizer stepsize for AC coefficients: 25 quantizer stepsize
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

23. 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)

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

25. Chen’s FDCT
See Code at

26. How the fast algorithm works?
Exploiting the symmetry of cosine function.
STEP 1
STEP 2

27. HW Implementation 2D DCT using 1D DCT Function Block 1-D DCT 8x8 RAM
Input sample
1-D DCT
Output coef
MUX
8x8 RAM
Row order input
Column order output

28. 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

29. IDCT Mismatch DCT x IDCT = I ? DCT mismatch in MC-DCT
DCT is defined: in “floating point” and “direct form.”
Integer Implementation induces ‘error’ after Inverse DCT.
different FDCT has different ‘error’s.
DCT mismatch in MC-DCT
different reference image at encoder and decoder
very small error but it accumulates.
orgE
DCT Q
VLC
VLD IQ IQ
IDCTE
IDCTD
Should Equal but Mismatch !
recE
recD

30. 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 F[63], the last Coef.
Decoder check and correct the values
H.264
(modified) Integer DCT is used
adding random error cancelation

31. KL Transform, The Optimal Transform
Appendix
KL Transform, The Optimal Transform

32. Optimal Transform Optimality K-L (Karhunen-Loeve) transform
(No) Redundancy in input signal => (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 (Y=T X) such that RY,Y = T RX,X TT meets diagonal matrix (i.e., completely uncorrelated Y)

33. Optimal Transform Solution Issue in KLT
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.