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Orthogonal Transforms

Orthogonal Transforms. Fourier Walsh Hadamard. Review. Introduce the concepts of base functions: For Reed-Muller, FPRM For Walsh Linearly independent matrix Non-Singular matrix Examples Butterflies, Kronecker Products, Matrices

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Orthogonal Transforms

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  1. Orthogonal Transforms Fourier Walsh Hadamard

  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

  3. Orthogonal Functions

  4. Illustrate it for Walsh and RM

  5. Mean Square Error

  6. Important result

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

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

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

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

  11. Properties of Unitary Transform transform Covariance matrix

  12. Example of arbitrary basis functions being rectangular waves

  13. This determining first function determines next functions

  14. 0 1

  15. Small error with just 3 coefficients

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

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

  18. Orthogonal and separable Image Transforms

  19. Extending general transforms to 2-dimensions

  20. Forward transform inverse transform separable

  21. Fourier Transform separable

  22. Extension of Fourier Transform to two dimensions

  23. Discrete Fourier Transform (DFT) New notation

  24. 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

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

  26. 1-dim. DFT (cont.) • Calculation of DFT : Fast Fourier Transform Algorithm (FFT) • Decimation-in-time algorithm Derivation of decimation in time

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

  28. 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.) • Derivation of Decimation-in-frequency algorithm

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

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

  31. Use of Fourier Transforms for fast convolution

  32. Calculations for circular matrix

  33. By multiplying

  34. W *= Cw* In matrix form next slide

  35. w *= Cw*

  36. Here is the formula for linear convolution, we already discussed for 1D and 2D data, images

  37. Linear convolution can be presented in matrix form as follows:

  38. As we see, circular convolution can be also represented in matrix form

  39. Important result

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