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1 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 417 Principles of Signal Analysis Spring 14 EXAM 3 SOLUTIONS Friday, May 9, 14 This is a CLOSED BOOK exam. There are a total of 1 points in the exam. Each problem specifies its point total. Plan your work accordingly. You must SHOW YOUR WORK to get full credit. Problem Score Total Name:

2 NAME: Exam 3 Solutions Page Useful Angles θ cos θ sin θ e jθ 1 1 π/6 3/ 1/ 3/ + j/ π/4 / / / + j / π/3 1/ 3/ 1/ + j 3/ π/ 1 j π π/ 1-1 j π 1 1 Gaussian Probability Densities (to Two Significant Figures) x 1 π e x / Other Possibly Useful Formulas X(e jω ) = x[n = 1 π h[n = sin ω cn πn n= π π H(e jω ) = x[ne jωn X(e jω )e jωn dω { 1 ω < ωc otherwise ω(n 1) j sin(ωn/) u[n u[n N e sin(ω/) δ[n 1 e jαn πδ(ω α) X[k = x[n = 1 N S = N 1 n= N 1 k= x[ne jπkn/n X[ke jπkn/n n ( x k m)( x k m) T k=1

3 NAME: Exam 3 Solutions Page 3 Problem 1 (1 points) You are given a 64x48 B/W input image, x[n 1, n for integer pixel values n 1 639, n 479. You wish to interpolate the given pixel values in order to find the value of the image at location (5.3, 3.8). Specify the formula used to calculate x[5.3, 3.8 using each of the following algorithms. Be certain that your formula clearly states which pixels from the input image are used. (a) Piece-wise constant interpolation. (b) Bilinear interpolation. x[5.3, 3.8 = x[5, 31 (c) Sinc interpolation. x[5.3, 3.8 = (.7)(.)x[5, 3 + (.7)(.8)x[5, 31 + x[5.3, 3.8 = n 1 = n = (.3)(.)x[51, 3 + (.3)(.8)x[51, 31 x[n 1, n sinc (π(5.3 n 1 )) sinc (π(3.8 n ))

4 NAME: Exam 3 Solutions Page 4 Problem (4 points) The images y[ η and x[ m are related by an affine transformation, where η = [η, ξ, 1 T and m = [m, n, 1 are coordinate vectors of the input and output image, respectively, m is the row index, and n is the column index. (a) The affine transformation η = A m is a rotation by π 3 radians. Find A. A = (b) The affine transformation η = B m consists of scaling the height of the image (m) by a factor of 5, while keeping the width (n) unchanged. Find B. B = (c) The affine transformation η = C m consists of shifting all pixels to the left (negative n direction) by columns. Find C. C = (d) The affine transformation η = D m consists of performing parts (a) through (c) of this problem, one after the other, in order. Specify the matrix D in terms of the matrices A, B, and C. There should be no numbers in your answer to this part. D = CBA

5 NAME: Exam 3 Solutions Page 5 Problem 3 (11 points) A particular triangle has corner coordinates at [ [ x 1 =, x = 1 [ 1, x 3 = Let λ = [λ 1, λ, λ 3 T be the barycentric coordinate vector corresponding to pixel x = [ 3, 1 3 T. Find λ. λ = 1/3 /3 Problem 4 (1 points) The images y[ η and x[ m are related by an affine transformation η = A m, where η = [η, ξ, 1 T and m = [m, n, 1 T are coordinate vectors of the input and output images, respectively. It is known that under this transformation, the origin swaps places with the point [,, thus [ [, and [ Specify the A matrix as completely as you can. There should be two scalar variables in your answer; you may use the variables names α and β. A = α 1 α β 1 β 1 [

6 NAME: Exam 3 Solutions Page 6 Problem 5 (16 points) You are creating a recommender system that tries to recommend songs that will be considered to be similar to a given query. Each song is characterized by a two-dimensional vector x k = [b k, v k T where b k is the number of beats per minute, and v k is the fraction of air-time during which there is a human voice. Your customer considers the following four songs to be similar: [ [ x 1, x, x 3, x 4 = You are given two more test data, x 5 = [b 5, v 5 T and x 6 = [b 6, v 6 T, and you are asked whether or not x 5 and x 6 should be considered similar. Write formulas for the Mahalanobis distance between x 5 and x 6 under the following conditions: (a) Estimate a diagonal data covariance matrix directly from the data, and use it to write the squared Mahalanobis distance d Σ ( x 5, x 6 ). d Σ( x 5, x 6 ) = (b 5 b 6 ) 1 + (v 5 v 6 ).1 (b) Estimate a diagonal data covariance matrix from the data, then regularize it using regularization parameter λ =.1 before using the result to write the squared Mahalanobis distance d Σ ( x 5, x 6 ). d Σ( x 5, x 6 ) = (b 5 b 6 ) (v 5 v 6 ).

7 NAME: Exam 3 Solutions Page 7 Problem 6 (16 points) A particular 6 megapixel image contains 3 million red pixels ([r, g, b = [55,, ) and 3 million blue pixels ([r, g, b = [,, 55). Define its 8-quantile color histogram h[k R, k G to be an 8 8 table of numbers, specifying the number of pixels having redshift in the kr th quantile (where smaller k R indicates smaller redshift, k R 7), and greenshift in the kg th quantile ( k G 7). (a) Find h[k R, k G k R = 7, k G = h[k R, k G = k R =, k G = otherwise (b) Suppose that there is another 6 megapixel image with 3 million black pixels ([r, g, b = [1, 1, 1) and 3 million white pixels ([r, g, b = [55, 55, 55). Say that the color histogram of this image is called g[k R, k G. What is g[k R, k G h[k R, k G, the distance between the color histogram of the black-white image and the color histogram of the red-blue image? { k g[k R, k G = R =, k G = otherwise Correct answer can be either the l 1 or l norm: g[k R, k G h[k R, k G = g[k R, k G h[k R, k G 1 = 1 1 6