Digital Image Processing Fundamentals of Digital Image Processing, A. K. Jain
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
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
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
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
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
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
DFT Properties: Symmetric Unitary Periodic Extension Sampled Fourier Fast Conjugate Symmetry Circular Convolution 8
Basis of DFT (Real and Imaginary): 9
Basis of DFT (Real and Imaginary): 0
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
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
DCT Basis: 3
4
DCT Basis: 5
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
7
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
Welsh-Hadamard Transform (WHT): N=2 n D Cases: Walsh Function H 2 H H H H H H H 0 7 3 4 H3 8 6 2 5 n n n n n 2 Hn H n 9
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
Welsh-Hadamard Transform Properties: Real, Symmetric, and orthogonal: H=H * =H T = H - Ultra Fast Transform (±) Good-Very Good energy compactness 2
Welsh-Hadamard Basis: 22
Welsh-Hadamard Basis 23
Welsh-Hadamard Basis 24
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 2 2 0 O.W. m x, m 0,,, N N 25
Haar Transform N=2 n D Cases: H r 0 2 2 2 2 0 0 0 02 0 0 0 0 2 2 2 22 8 2 2 0 0 0 0 0 0 2 0 0 2 2 0 0 0 02 0 0 0 0 2 2 0 0 2 0 0 0 0 0 0 2 2 2 2D Cases: Hr*A*Hr T 26
Haar Basis Function 27
Haar Transform Properties Real and Orthogonal: Hr=Hr *, Hr - =Hr T Fast Transform Poor energy compactness 28
Slant Transform (N=2 n ) S S 2 0 0 a b a b S 0 b2 a2 b2 a2 2 2 2 2 2 2 2 0 0 02 S 29
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 0 2 2 0 0 0 Sn 0 0 bn an bn an 0 I( N/ 2) 2 0 I( N/ 2) 2 2 2 2 2 n 3N N N 2, an, b 2 n 2 4N 4N 30
Slant Basis Function 3
Slant Transform Properties: Real and Orthogonal S=S * S - =S T Fast Very Good Compactness 32