Orthogonal Transforms

1. Orthogonal Transforms Fourier

2. Review • Introduce the concepts of base functions: • For Reed-Muller, FPRM • For Walsh • Linearly independent matrix • Non-Singular matrix • Examples • Butterflies, Kronecker Products, Matrices • Using matrices to calculate the vector of spectral coefficients from the data vector Our goal is to discuss the best approximation of a function using orthogonal functions

3. Orthogonal Functions

4. Orthogonal Functions

5. Note that these are arbitrary functions, we do not assume sinusoids

6. Illustrate it for Walsh and RM

10. Mean Square Error

11. Mean Square Error

14. Important result

15. We want to minimize this kinds of errors. • Other error measures are also used.

16. Unitary Transforms

17. Unitary Transforms • Unitary Transformation for 1-Dim. Sequence • Series representation of • Basis vectors : • Energy conservation : Here is the proof

18. Unitary Transformation for 2-Dim. Sequence • Definition : • Basis images : • Orthonormality and completeness properties • Orthonormality : • Completeness :

19. Unitary Transformation for 2-Dim. Sequence • Separable Unitary Transforms • separable transform reduces the number of multiplications and additions from to • Energy conservation

20. Properties of Unitary Transform transform Covariance matrix

21. Example of arbitrary basis functions being rectangular waves

22. This determining first function determines next functions

23. 0 1

25. Small error with just 3 coefficients

26. This slide shows four base functions multiplied by their respective coefficients

27. This slide shows that using only four base functions the approximation is quite good End of example

28. Orthogonality and separability

29. Orthogonal and separable Image Transforms

30. Extending general transforms to 2-dimensions

31. Forward transform inverse transform separable

32. Fourier Transforms in new notations We emphasize generality Matrices

33. Fourier Transform separable

34. Extension of Fourier Transform to two dimensions

38. Discrete Fourier Transform (DFT) New notation

40. Fast Algorithms for Fourier Transform Task for students: Draw the butterfly for these matrices, similarly as we have done it for Walsh and Reed-Muller Transforms 2 Pay attention to regularity of kernels and order of columns corresponding to factorized matrices

41. Fast Factorization Algorithms are general and there is many of them

42. 1-dim. DFT (cont.) • Calculation of DFT : Fast Fourier Transform Algorithm (FFT) • Decimation-in-time algorithm Derivation of decimation in time

43. Decimation in Time versus Decismation in Frequency

44. Butterfly for Derivation of decimation in time • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-time algorithm (cont.) Please note recursion

45. 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.) • Derivation of Decimation-in-frequency algorithm

46. Decimation in frequency butterfly shows recursion • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.)

47. Conjugate Symmetry of DFT • For a real sequence, the DFT is conjugate symmetry

48. Use of Fourier Transforms for fast convolution

49. Calculations for circular matrix

50. By multiplying