1. CS654: Digital Image Analysis Lecture 15: Image Transforms with Real Basis Functions

2. Recap of Lecture 14 Discrete Fourier Transform Orthogonal sinusoidal waveform Computational complexity is high Involves complex multiplication

3. Outline of Lecture 15 Basis function with real (Integer) values Hadamard Transform Haar Transform KL Transform

4. Hadamard Transform Core matrix

5. Generation of transformation matrix Using Kronecker product recursion Example

6. Unitary Hadamard Transform General unitary transformation equation Using Hadamard transform Forward transformation Inverse transformation Forward transformation Inverse transformation What happens in case of images?

7. Summation expression Forward transformation Inverse transformation LSB, MSB ?

8. Properties of Hadamard Transformation Sequency 0 7 3 4 1 6 2 5

9. Natural Ordering vs. Sequency Ordering Natural Order (h) Sequency (s) 0000 0 10011117 20100113 3 1004 4 0011 51011106 6 0102 71111015 000 100 010 110 001 101 011 111 Natural order of the Hadamard transform coefficients = bit reversed gray code representation of its sequency

10. Haar Transform

11. Haar Function

12. Haar Basis Function Computation Determine the order of N Calculate the Haar function

13. Haar Basis Function Computation

14. Haar basis for N=2

15. KL Transform Exploits the statistical properties of an image Basis functions are orthogonal Eigen vectors of the covariance matrix Optimally de-correlates the input data Energy compaction Input dependent, and high computational complexity

16. Eigen analysis Inverse Transform is defined as:

17. Thank you Next Lecture: Convolution and Correlation