2. An Example of 1D Transform with Two Variablesy2 y1
(1,1)
(1.414,0) x1
Transform matrix

3. Generalization into N Variables
forward transform
basis vectors (column vectors of transform matrix)

4. Decorrelating Property of Transformx2 y2 y1 x1
x1 and x2 are highly correlated
y1 and y2 are less correlated
p(x1x2)  p(x1)p(x2)
p(y1y2)  p(y1)p(y2)

5. Transform=Change of Coordinates
Intuitively speaking, transform plays the role of facilitating the source modeling
Due to the decorrelating property of transform, it is easier to model transform coefficients (Y) instead of pixel values (X)
An appropriate choice of transform (transform matrix A) depends on the source statistics P(X)
We will only consider the class of transforms corresponding to unitary matrices

6. Unitary Matrix and 1D Unitary Transform
Definition
conjugate
transpose
A matrix A is called unitary if A-1=A*T
When the transform matrix A is unitary, the defined
transform is called unitary transform
Example
For a real matrix A, it is unitary if A-1=AT

7. Inverse of Unitary Transform
For a unitary transform, its inverse is defined by
Inverse Transform
basis vectors corresponding to inverse transform

8. Properties of Unitary Transform
Energy compaction: only few transform coefficients have large magnitude
Such property is related to the decorrelating role of unitary transform
Energy conservation: unitary transform preserves the 2-norm of input vectors
Such property essentially comes from the fact that rotating coordinates does not affect Euclidean distance

9. Energy Compaction Example
Hadamard matrix
significant
insignificant

10. Energy Conservation*
A is unitary
Proof

11. Numerical Example
Check:

12. 2D Transform=Two Sequential 1D Transforms
column transform
(left matrix multiplication first)
row transform
row transform
(right matrix multiplication first)
column transform
Conclusion:
 2D separable transform can be decomposed into two sequential
 The ordering of 1D transforms does not matter

13. Energy Compaction Property of 2D Unitary Transform
 Example
A coefficient is called significant if its magnitude
is above a pre-selected threshold th
insignificant coefficients (th=64)

14. Energy Conservation Property of 2D Unitary Transform
2-norm of a matrix X
A unitary
Example:
You are asked to prove such property in your homework

15. Important 2D Unitary Transforms
Discrete Fourier Transform
Widely used in non-coding applications (frequency-domain approaches)
Discrete Cosine Transform
Used in JPEG standard
Hadamard Transform
All entries are 1
N=2: Haar Transform (simplest wavelet transform for multi-resolution analysis)

16. Discrete Cosine Transform

43. 1D DCT  Real and orthogonal  DCT is NOT the real part of DFT
forward transform
inverse transform
Properties
 Real and orthogonal
excellent energy compaction property
 DCT is NOT the real part of DFT
Fact
The real and imaginary parts of DFT are
generally not orthogonal matrices
 fast implementation available: O(Nlog2N)

46. (2451 significant coefficients, th=64)
2D DCT
Its DCT coefficients Y
(2451 significant coefficients, th=64)
Original cameraman image X
Notice the excellent energy compaction property of DCT