3_HW3Solutions

3_HW3Solutions - EE 211A Fall Quarter 2011 Instructor John Villasenor Digital Image Processing I Handout 12 Homework 3 Solutions Textbook problems

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Unformatted text preview: EE 211A Fall Quarter, 2011 Instructor: John Villasenor Digital Image Processing I Handout 12 Homework 3 Solutions Textbook problems: 4.3 The spectrum of 4 cos 2π 2 x has delta functions of strength 2 at u = ± 2 and is described by F (u, v) = 2δ (u ± 2, v) : v u 1 The spectrum of cos 2π 3 y is described by F (u, v) = δ (u, v ± 3) : 2 v u The spectrum of the product 4 cos 2π 2 x cos 2π 3 y is the convolution of the spectra above and is described by F (u, v) = δ (u ± 2, v ± 3) : v u 1 Reconstruction cases: (1) Sampling at Δx = Δy = 0.5 is a sampling frequency of 2, so the spectrum F (u, v) will be replicated at intervals of 2, 2 in u, v. The reconstruction filter will only let through those frequencies with |u|, |v| ≤ 1. This gives a spectrum for the reconstructed signal: v u Thus, (2) ~ f ( x, y ) = 8 cos 2πy. Sampling at Δx = Δy = 0.2 ⇒ Fs = 5, cutoff for reconstruction is Fc = ±2.5. This gives a reconstructed spectrum: v u Stated in another way, F (u, v) = δ (u − 2, v − 2) + δ (u + 2, v − 2) + δ (u − 2, v + 2) + δ (u + 2, v + 2) ~ f ( x, y ) = 2 cos 2π (2 x + 2 y ) + 2 cos 2π (2 x − 2 y ) = 4 cos 4πx cos 4πy 4.9 We know, from p.85, that comb ( x, y, Δx, Δy ) = ∑m ∑n δ ( x − mΔx, y − nΔy ) ⎛ྎ 11 1 1 ⎞ྏ 1 1 comb ⎜ྎ u , v, , ⎟ྏ = ⎜ྎ Δx Δ y Δx Δy ⎟ྏ Δx Δy ⎝ྎ ⎠ྏ The function whose transform we want is: has Fourier Transform ⎛ྎ k l ⎞ྏ ∑ ∑ δ ⎜ྎ u − Δx , v − Δy ⎟ྏ . ⎜ྎ ⎟ྏ k l ⎝ྎ ⎠ྏ g ( x, y) = ∑m ∑n δ ( x − 2m, y − 2n) + ∑m ∑n δ ( x − 2m − 1, y − 2n − 1) = g1 ( x, y ) + g 2 ( x, y) 2 1 k l ⎞ྏ ⎛ྎ ∑k ∑l δ ⎜ྎ u − 2 , v − 2 ⎟ྏ = G1 (u, v) 4 ⎝ྎ ⎠ྏ g 2 ( x, y) = g1 ( x − 1, y − 1) 1 k l ⎞ྏ ⎛ྎ g1 ( x − 1, y − 1) ↔ e − j 2π (u +v ) ∑k ∑l δ ⎜ྎ u − , v − ⎟ྏ 4 2 2 ⎠ྏ ⎝ྎ g 1 ( x, y ) ↔ ⎛ྎ k l ⎞ྏ − j 2π ⎜ྎ + ⎟ྏ 1 k l ⎞ྏ ⎛ྎ = ∑k ∑l e ⎝ྎ 2 2 ⎠ྏδ ⎜ྎ u − , v − ⎟ྏ = G2 (u, v) 4 2 2 ⎠ྏ ⎝ྎ −j 2π ( k +l ) When k + l is odd, then e 2 = −1. In this case, G2 (u, v) = −G1 (u, v) and the sum G1 + G2 = 0. When k + l is even, e − jπ ( k +l ) = 1, G2 = G1 , and G1 + G2 = 2G1 . ⇒ G (u, v) = 1 ∑ 2 k +l ⎛ྎ even k l ⎞ྏ ∑ δ ⎜ྎ u − 2 , v − 2 ⎟ྏ ⎝ྎ ⎠ྏ Note that this result can be verified intuitively since g ( x, y ) is a rectangular grid with o 2 and a rotation of 45 . Its transformation is, as expected, a rectangular 1 grid at 45o with spacing . 2 spacing 1. The matrices associated with the functions h(m, n) and x ʹȃ(m, n) are: ⎡ྎ2 0 0⎤ྏ ⎡ྎ 1 1 ⎤ྏ h = ⎢ྎ0 1 0⎥ྏ , x ʹȃ = ⎢ྎ ⎥ྏ . ⎢ྎ ⎥ྏ ⎣ྏ− 1 − 1⎦ྏ ⎢ྎ1 − 1 1⎥ྏ ⎣ྏ ⎦ྏ a. The column vector X is then ⎡ྎ 1 ⎤ྏ ⎢ྎ −1⎥ྏ ⎡ྎ X ⎤ྏ ⎢ྎ ⎥ྏ = ⎢ྎ 0 ⎥ྏ . ⎢ྎ 1 ⎥ྏ ⎢ྎ X 1 ⎥ྏ ⎢ྎ ⎥ྏ ⎣ྏ ⎦ྏ ⎣ྏ −1⎦ྏ b. Since h has 3 columns, there are 3 matrices H n , each with 2 columns (since x ʹȃ has 2 rows) and 4 rows (since y has 3 + 2 – 1 = 4 rows) 3 ⎡ྎ2 ⎢ྎ0 H 0 = ⎢ྎ ⎢ྎ1 ⎢ྎ ⎣ྏ0 0⎤ྏ 0 ⎤ྏ ⎡ྎ 0 ⎡ྎ0 ⎥ྏ ⎢ྎ 1 ⎥ྏ ⎢ྎ0 2⎥ྏ 0 ⎥ྏ , H 1 = ⎢ྎ , H 2 = ⎢ྎ ⎢ྎ− 1 1 ⎥ྏ ⎢ྎ1 0⎥ྏ ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ 1 ⎦ྏ ⎣ྏ 0 − 1⎦ྏ ⎣ྏ0 0⎤ྏ 0⎥ྏ ⎥ྏ . 0⎥ྏ ⎥ྏ 1 ⎦ྏ c. The output y will have 4 columns, therefore ⎡ྎ y 0 ⎤ྏ ⎡ྎ H [0] ⎤ྏ ⎢ྎ ⎥ྏ ⎢ྎ 0 ⎥ྏ ⎢ྎ y1 ⎥ྏ ⎢ྎ H 1 H 0 ⎥ྏ ⎡ྎ X 0 ⎤ྏ . ⎢ྎ y ⎥ྏ = ⎢ྎ H H 1 ⎥ྏ ⎢ྎ X 1 ⎥ྏ ⎣ྏ ⎦ྏ ⎢ྎ 2 ⎥ྏ ⎢ྎ 2 ⎥ྏ ⎢ྎ y ⎥ྏ ⎣ྏ[0] H 2 ⎦ྏ ⎣ྏ 3 ⎦ྏ d. ⎡ྎ 2 0 ⎢ྎ 0 2 ⎢ྎ ⎢ྎ 1 0 ⎢ྎ ⎢ྎ 0 1 ⎢ྎ 0 0 ⎢ྎ 0 ⎢ྎ 1 ⎢ྎ− 1 1 ⎢ྎ ⎢ྎ 0 − 1 Hx = ⎢ྎ 0 0 ⎢ྎ ⎢ྎ 0 0 ⎢ྎ 0 ⎢ྎ 1 ⎢ྎ 0 1 ⎢ྎ 0 ⎢ྎ 0 ⎢ྎ 0 0 ⎢ 0 ⎢ྎ 0 ⎢ྎ 0 0 ⎣ྏ 0 0 ⎤ྏ 0 0 ⎥ྏ ⎥ྏ 0 0 ⎥ྏ ⎥ྏ 0 0 ⎥ྏ 2 0 ⎥ྏ ⎥ྏ 0 2 ⎥ྏ 1 0 ⎥ྏ ⎥ྏ 0 1 ⎥ྏ 0 0 ⎥ྏ ⎥ྏ 1 0 ⎥ྏ ⎥ྏ − 1 1 ⎥ྏ 0 − 1⎥ྏ ⎥ྏ 0 0 ⎥ྏ 0 0 ⎥ྏ ⎥ྏ 1 0 ⎥ྏ 0 1 ⎥ྏ ⎦ྏ 4 ⎡ྎ 2 ⎤ྏ ⎢ྎ− 2⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ 1 ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ − 1 ⎥ྏ ⎢ྎ 2 ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ − 1 ⎥ྏ ⎢ྎ − 1 ⎥ྏ ⎡ྎ 1 ⎤ྏ ⎢ྎ ⎥ྏ ⎢ྎ− 1⎥ྏ ⎢ྎ ⎥ྏ = ⎢ྎ 0 ⎥ྏ = y ⎢ྎ 0 ⎥ྏ ⎢ྎ 1 ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ 1 ⎥ྏ − 1⎦ྏ ⎣ྏ ⎢ྎ ⎥ྏ ⎢ྎ − 1 ⎥ྏ ⎢ྎ 0 ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ 0 ⎥ྏ ⎢ྎ 0 ⎥ྏ ⎢ྎ ⎥ྏ ⎢ྎ 1 ⎥ྏ ⎢ྎ − 1 ⎥ྏ ⎣ྏ ⎦ྏ 2 0 0 ⎤ྏ ⎡ྎ 2 ⎢ྎ− 2 − 1 1 0 ⎥ྏ ⎥ྏ ⇒ y (m, n) = ⇒ y = ⎢ྎ ⎢ྎ 1 − 1 − 1 1 ⎥ྏ ↑ ⎢ྎ ⎥ྏ ⎣ྏ − 1 0 0 − 1⎦ྏ 0 0 1 −1 0 1 −1 0 2 −1 −1 0 2 −2 −1 1 → e. 0 0 n↑ 0 1 1 −1 2 0 1 n↑ * 1 → −1 1 −1 → m m 001 0 0 −1 0 1 −1 0 −1 1 = 0 1 −1 + + 0 −1 1 + 201 − 2 0 −1 20 1 0 − 2 0 −1 0 00 00 1 0 0 −1 0 n↑ 1 −1 0 = 0 1 −1 0 2 2 −1 −1 0 − 2 1 −1 → m ⇒ the two methods agree. 5 2. This function is separable, with u(m, n) = u1 (m)u2 (n) , where − p + 1 ≤ m ≤ p − 1, ⎧ྏ1, u1 (m) = ⎨ྏ N N ⎩ྏ0, − + 1 ≤ m ≤ − p, p ≤ m ≤ . 2 2 and u 2 (n) is identical to u1 (with n replacing m). This function is a square pulse containing 2 p − 1nonzero values. Since it is even, the DFT will be real. One way to get the DFT is to take the DFT of ⎧ྏ1, 0 ≤ m ≤ 2 p − 2 u 3 (m) = ⎨ྏ ⎩ྏ0, otherwise. and then to discard the complex phase factor. v3 ( k ) = = = 1 2 p −2 − j 2π km / N 1 1 − e− j 2π k (2 p −1) / N e = ∑ − j 2π k / N N m−0 N 1− e 1 e− jπ k ( 2 p −1) / N (e jπ k (2 p −1) / N − e− jπ k (2 p −1) / N ) e− jπ k / N (e jπ k / N − e− jπ k / N ) N 1 − jπ k ( 2 p −2) / N sin π k (2 p − 1) / N = v3 ( k ) e sin π k / N N The desired 2D transform is therefore v( k , l ) = 1 sin(π k (2 p − 1) / N ) sin(π l (2 p − 1) / N ) . N sin(π k / N ) sin(π l / N ) 6 sin(π k (2 p − 1) / N ) sin(*) becomes 2 p − 1. Consider . For sin(π k / N ) s in(*) large N, this expression can be written as For k = 0 or l = 0 , the sin π k (2 p − 1) / N sin π k (2 p − 1) / N = (2 p − 1) = (2 p − 1) sinc (k (2 p − 1) / N ) πk / N π k (2 p − 1) / N Therefore, v(k , l ) |large (2 p − 1) 2 sinc(k(2p-1)/N)sinc(l(2p-1)/N) N= N 7 ...
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This note was uploaded on 12/27/2011 for the course EE211A 211A taught by Professor Villasenor during the Fall '11 term at UCLA.

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