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Image Transform

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  1. EE535 Digital Image Processing (Spring 2000’) Image Transform Fundamentals of digital image processing Anil K. Jain, chap. 5

  2. Introduction • Topics • Unitary transform • Discrete Fourier Transform (DFT) • Discrete Cosine Transform (DCT) • Discrete Sine Transform (DST) • Discrete Walsh Transform (DWT) • Discrete Hadamard Transform (DHT) • Haar transform • Slant transform • Karhunen-Loeve(KL) transform EE535 Digital Image Processing (Spring 2000’)

  3. Unitary Transforms • Unitary Transformation for 1-Dim. Sequence • Series representation of • Basis vectors : • Energy conservation : EE535 Digital Image Processing (Spring 2000’)

  4. Unitary Transforms (cont.) • Unitary Transformation for 2-Dim. Sequence • Definition : • Basis images : • Orthonormality and completeness properties • Orthonormality : • Completeness : EE535 Digital Image Processing (Spring 2000’)

  5. Unitary Transforms (cont.) • Unitary Transformation for 2-Dim. Sequence • Separable Unitary Transforms • separable transform reduces the number of multiplications and additions from to • Energy conservation EE535 Digital Image Processing (Spring 2000’)

  6. Discrete Fourier Transform (DFT) • 1-dim. DFT • Definition • Inverse DFT • Forward DFT EE535 Digital Image Processing (Spring 2000’)

  7. DFT (cont.) • 1-dim. DFT (cont.) • DFS and DFT • Discrete Fourier Series for periodic signals (DFS) • DFT for one period of periodic signals • DTFT and DFT for one period of periodic signals EE535 Digital Image Processing (Spring 2000’)

  8. S(f) Ga(f) ... ... f f -2fs -fs fs 2fs 0 -B B reconstruction filter GS(f) ... ... f -2fs -fs B fs 2fs 3fs 4fs -B DFT (cont.) • 1-dim. DFT (cont.) • Periodic Sampling , CTFT and DTFT EE535 Digital Image Processing (Spring 2000’)

  9. DFT (cont.) • 1-dim. DFT (cont.) • Periodic Sampling , CTFT and DTFT (cont.) EE535 Digital Image Processing (Spring 2000’)

  10. DFT (cont.) • 1-dim. DFT (cont.) • Calculation of DFT : Fast Fourier Transform Algorithm (FFT) • Decimation-in-time algorithm EE535 Digital Image Processing (Spring 2000’)

  11. DFT (cont.) • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-time algorithm (cont.) EE535 Digital Image Processing (Spring 2000’)

  12. DFT (cont.) • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.) EE535 Digital Image Processing (Spring 2000’)

  13. DFT (cont.) • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.) EE535 Digital Image Processing (Spring 2000’)

  14. DFT (cont.) • 2-Dim. DFT • Definition • Inverse DFT • Forward DFT EE535 Digital Image Processing (Spring 2000’)

  15. (a) Original Image (b) Magnitude (c) Phase DFT (cont.) • 2-Dim. DFT (cont.) • example EE535 Digital Image Processing (Spring 2000’)

  16. DFT (cont.) • 2-Dim. DFT (cont.) • Properties of 2D DFT • Separability EE535 Digital Image Processing (Spring 2000’)

  17. DFT (cont.) • 2-Dim. DFT (cont.) • Properties of 2D DFT • Translation • Conjugate symmetry For real EE535 Digital Image Processing (Spring 2000’)

  18. (a) a sample image (b) its spectrum (c) rotated image (d) resulting spectrum DFT (cont.) • 2-Dim. DFT (cont.) • Properties of 2D DFT (cont.) • Rotation EE535 Digital Image Processing (Spring 2000’)

  19. DFT (cont.) • 2-Dim. DFT (cont.) • Properties of 2D DFT • Circular convolution and DFT • Correlation EE535 Digital Image Processing (Spring 2000’)

  20. DFT (cont.) • 2-Dim. DFT (cont.) • Calculation of 2-dim. DFT • Direct calculation • Complex multiplications & additions : • Using separability • Complex multiplications & additions : • Using 1-dim FFT • Complex multiplications & additions : ??? EE535 Digital Image Processing (Spring 2000’)

  21. Discrete Cosine Transform (DCT) • 2-dim. DCT • Definition • Inverse DCT • Forward DCT EE535 Digital Image Processing (Spring 2000’)

  22. DCT (cont.) • 2-dim. DCT (cont.) • Basis Functions for 1-dim. DCT (N=16) EE535 Digital Image Processing (Spring 2000’)

  23. DCT (cont.) • Fast algorithm of 1-dim. DCT EE535 Digital Image Processing (Spring 2000’)

  24. Discrete Sine Transform (DST) • 1-dim. DST • Definition • Inverse DST • Forward DST • Reference for fast algorithm of DST • P.Yip and K. R. Rao, “A Fast Computational Algorithm for the Discrete Sine Transform,” IEEE Trans. On Communicatins, Vol. COM-28, No. 2, Feb., 1980 EE535 Digital Image Processing (Spring 2000’)

  25. DST (cont.) • 1-dim. DST (cont.) • Basis Functions for 1-dim. DST (N=16) EE535 Digital Image Processing (Spring 2000’)

  26. x DST (cont.) • 1-dim. DST (cont.) • Fast algorithm for 1-D DST g(x) (2N+2)-FFT x x N-1 N 2N+1 EE535 Digital Image Processing (Spring 2000’)

  27. x 5 6 7 k 0 1 2 3 4 0 + + + + + + + + + + 1 + - - + - - + 2 + - - - - + + + - - - 3 + - + + - - - + + + 4 + - - - - + 5 - + + + - 6 + - - - + + + - - - + + + + 7 - Walsh Transform (DWT) • 1-dim. DWT • Definition • Inverse DWT • Forward DWT • Basis • : k-th bit of z EE535 Digital Image Processing (Spring 2000’)

  28. Walsh Transform (DWT) • 1-dim. DWT • Fast algorithm for Walsh transformation EE535 Digital Image Processing (Spring 2000’)

  29. G(0) g(0) G(1) g(2) N/2-point DWT G(2) g(4) G(3) g(6) _ G(4) g(1) _ G(5) g(3) N/2-point DWT _ G(6) g(5) _ G(7) g(7) Walsh Transform (DWT) • 1-dim. DWT (cont.) • Fast algorithm (cont.) EE535 Digital Image Processing (Spring 2000’)

  30. Walsh Transform (DWT) • 2-Dim. DWT • Basis Functions (Separable) EE535 Digital Image Processing (Spring 2000’)

  31. x 5 6 7 0 1 2 3 4 k 0 + + + + + + + + - + 1 + + - - + - + 2 + - - - - + + - - + - 3 + + - + + - + - + + 4 - - - - - + 5 - + + + + 6 + - - + - - + - - - + + + + 7 - Hadamard Transform (DHT) • 1-dim. DHT • Definition • Inverse DHT • Forward DHT • Basis • : k-th bit of z EE535 Digital Image Processing (Spring 2000’)

  32. DHT (cont.) • 1-dim. DHT (cont.) • Property of Hadamard kernels EE535 Digital Image Processing (Spring 2000’)

  33. DHT (cont.) • 1-dim. DHT (cont.) • Fast algorithm of Hadamard transform EE535 Digital Image Processing (Spring 2000’)

  34. G(0) g(0) G(2) g(2) N/2-point DHT G(4) g(4) G(6) g(6) _ G(1) g(1) _ G(3) g(3) N/2-point DHT _ G(5) g(5) _ G(7) g(7) DHT (cont.) • 1-dim. DHT (cont.) • Fast algorithm of Hadamard transform (cont.) EE535 Digital Image Processing (Spring 2000’)

  35. k k’ 0 1 2 3 4 5 6 7 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 0 1 3 2 6 7 5 4 DHT (cont.) • 1-dim. DHT (cont.) • Fast algorithm of Hadamard transform (cont.) • Reordering (Ordered Hadamard Transform) where EE535 Digital Image Processing (Spring 2000’)

  36. x x 5 5 6 6 7 7 0 0 1 1 2 2 3 3 4 4 k k 0 0 + + + + + + + + + + + + + + + + + - + + 1 1 + + - + - - + - + - - - + + 2 2 + + - - - - + - + - + - + - + - - - + + - + 3 3 + + - + - - + - + + - - - + + - + + + - 4 4 - - - + - + - - - - + + 5 5 - + + - + + + - - + 6 6 + + + - + - - + - - - - + + - + - - + - + - + - + + + + 7 7 - - DHT (cont.) • 1-dim. DHT (cont.) • Fast algorithm of Hadamard transform (cont.) • Ordered Hadamard Transform (cont.) EE535 Digital Image Processing (Spring 2000’)

  37. DHT (cont.) • 2-Dim. DHT • Basis Functions (Separable) EE535 Digital Image Processing (Spring 2000’)

  38. Harr transform • 1-dim. Harr Transform • Definition EE535 Digital Image Processing (Spring 2000’)

  39. G(0) g(0) g(1) g(2) g(3) g(4) g(5) g(6) g(7) G(4) -1 G(2) -1 G(5) -1 G(1) -1 G(6) -1 G(3) -1 G(7) -1 Harr Transform(Cont.) • 1-dim. Harr Transform • Example • Fast algorithm EE535 Digital Image Processing (Spring 2000’)

  40. Slant transform • Definition • Example EE535 Digital Image Processing (Spring 2000’)

  41. KL Transform (or Hotelling Transform) • Definition (1-dim.) • Basis vector : • KL transform of u, and inverse transform • KL transform depends on the (second-order) statistics of the data EE535 Digital Image Processing (Spring 2000’)

  42. KL Transform (cont.) • Definition (2-dim.) • autocovariance of N X N image u(m,n) • Basis images : • If is separable EE535 Digital Image Processing (Spring 2000’)

  43. KL Transform (cont.) • Properties of the KL transform • Decorrelation • Proof is diagonal matrix containing the eigenvalues of R EE535 Digital Image Processing (Spring 2000’)

  44. KL Transform (cont.) • Properties of KL transform(cont.) • Distribution of variances • Among all the unitary transforms v=Au, the KL transform packs the maximum average energy in samples of v • Rate-distortion function • For each fixed D(distortion), the KL transform achieves the minumum rate among all unitary transforms. EE535 Digital Image Processing (Spring 2000’)

  45. y : reproduced value x : Gaussian r.v of variance Rate distortion function for a Gaussian source : Gaussian r.v.’s : reproduced values KL Transform (cont.) • Properties of KL transform(cont.) • Rate distortion function (cont.) • Distortion • Rate distortion function of x • For a fixed average distortion D where is determined by solving EE535 Digital Image Processing (Spring 2000’)

  46. KL Transform (cont.) • Properties of KL transform(cont.) • Rate-distortion function (cont.) EE535 Digital Image Processing (Spring 2000’)

  47. where Singular Value Decomposition • Definition EE535 Digital Image Processing (Spring 2000’)

  48. Singular Value Decomposition (cont.) • Properties • Unitary SVD transform • Best Approximation of U : • Application areas • to find the generalized inverse of singular matrices. EE535 Digital Image Processing (Spring 2000’)