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Image Transform

Image Transform. Fundamentals of digital image processing Anil K. Jain, chap. 5. Introduction. Topics Unitary transform Discrete Fourier Transform (DFT) Discrete Cosine Transform (DCT) Discrete Sine Transform (DST) Discrete Walsh Transform (DWT) Discrete Hadamard Transform (DHT)

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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’)

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