Image Transforms. Digital Image Processing Fundamentals of Digital Image Processing, A. K. Jain. Digital Image Processing.

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1 Digital Image Processing Fundamentals of Digital Image Processing, A. K. Jain

2 2D Orthogonal and Unitary Transform: Orthogonal Series Expansion: {a k,l (m,n)}: a set of complete orthonormal basis: N N * Orthonormality: ak, l m, nak, lm, n k k, l l Completeness: N N v k, l u m, n a m, n 0 k, l N m0 n0 NN kl, * u m, n v k, l a m, n 0 m, n N k0 l0 m0 n0 NN k0 l0 kl, *,,, a m n a m n m m n n k, l k, l 2

3 2D Orthogonal and Unitary Transform: v (m,n): Transformed coefficients V={v (m,n)}: Transformed Image Orthonormality requires: P Q * u m, n v k, l a m, n P N, Q N P, Q k, l k0 l0 N N u,, arg min, ˆ PQ m n u m n u m, n uˆ m0 n0 2 3

4 Separable Unitary Transform: Computational Complexity of former: O(N 4 ) With Separable Transform: O(N 3 ) ak, l m, n ak mbl n ak, mbl, n a m b n k, : One Dimensional Complete Orthonormal Basis Orthonormality and Completeness: A={a(k,m)} and B={b(l,n)} are unitary: Usually B is selected same as A (A=B): Unitary Transform! l * T T * A A A A I 4

5 Separable Unitary Transform: Basis Images: N N T, ak, mu m, na l, n V = AUA AAU v k l m0 n0 NN * *,,,, U = A u m n a k m v m n a l n * A a a U A * k,l T * * k l N N k0 l0 v T k k0 l0 a * k : k th Column * T * VA * 2, l Ak,l : Linear Combination of N matrices T T 5

6 Properties of Unitary Transform: v = Au v = u N N N N 2 2, vk, l u m n 2 2 m0 n0 k 0 l0 6

7 Two Dimensional Fourier Transform: N N km ln j2 vk, l um, nwn WN, WN exp m0 n0 N NN km ln u m, n v 2 k, l WN WN N k0 l0 N N km vk, l um, nwn W N m0 n0 NN km u m, n vk, lwn W N k0 l0 ln N ln N Unitary DFT Pair Matrix Notation: V = FUF U = F VF F= W N kn N N kn, 0 * * 7

8 DFT Properties: Symmetric Unitary Periodic Extension Sampled Fourier Fast Conjugate Symmetry Circular Convolution 8

9 Basis of DFT (Real and Imaginary): 9

10 Basis of DFT (Real and Imaginary): 0

11 Discrete Cosine Transform (DCT): D Cases: C c k, n c k, n k 0, 0 n N N 2 2n cos k k N, 0 n N N 2N N 2n k vk k u ncos, 0 k N n0 2N 2 0, k, k N N N N 2n k u n k vk cos, 0 n N k 0 2N

12 Properties of DCT: Real and Orthogonal: C=C * C - =C T Not! Real part of DFT Fast Transform Excellent Energy compaction (Highly Correlated Data) Two Dimensional Cases: A=A * =C 2

13 DCT Basis: 3

14 4

15 DCT Basis: 5

16 Discrete Sine Transform (DST): D Cases: ψ kn, nk 2 k, n sin, 0 k, n N N N 2D Case: A=A * =A T =Ψ nk N 2 vk u nsin, 0 k N N n0 N nk N 2 u n vk sin, 0 n N N k 0 N 6

17 7

18 Properties of DST: Real, Symmetric and Orthogonal: Ψ = Ψ * = Ψ T =Ψ - Forward and Inverse are identical Not! Imaginary part of DFT Fast Transform 8

19 Welsh-Hadamard Transform (WHT): N=2 n D Cases: Walsh Function H 2 H H H H H H H H n n n n n 2 Hn H n 9

20 Welsh-Hadamard Transform (WHT): N=2 n n v = Hu u = Hv, H = H n, N 2 vk N b k, m u m N m0 0 k N u m N b k, m v k N 0 m N n i0 k 0 b k, m k m k, m 0, i i i i n n i i i0 i0 k k 2, m m 2 2D Cases: A=A * =A T =H i i 20

21 Welsh-Hadamard Transform Properties: Real, Symmetric, and orthogonal: H=H * =H T = H - Ultra Fast Transform (±) Good-Very Good energy compactness 2

22 Welsh-Hadamard Basis: 22

23 Welsh-Hadamard Basis 23

24 Welsh-Hadamard Basis 24

25 Haar Transform N=2 n D Cases: h x, x 0,, k 0,, N k p k 2 q 0 p n ; q 0, p 0 p q 2 p 0 h0 x h0,0 x, x 0, N p 2 q q 2 2 x p p 2 2 p 2 q 2 q hk x hp, q x 2 x p p N O.W. m x, m 0,,, N N 25

26 Haar Transform N=2 n D Cases: H r D Cases: Hr*A*Hr T 26

27 Haar Basis Function 27

28 Haar Transform Properties Real and Orthogonal: Hr=Hr *, Hr - =Hr T Fast Transform Poor energy compactness 28

29 Slant Transform (N=2 n ) S S a b a b S 0 b2 a2 b2 a S 29

30 Slant Transform (N=2 n ) S n 0 0 a 0 0 n bn an b n 0 I( N/ 2) 2 0 I ( N/ 2) 2 Sn Sn 0 0 bn an bn an 0 I( N/ 2) 2 0 I( N/ 2) n 3N N N 2, an, b 2 n 2 4N 4N 30

31 Slant Basis Function 3

32 Slant Transform Properties: Real and Orthogonal S=S * S - =S T Fast Very Good Compactness 32

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