Midterm Exam. CS283, Computer Vision Harvard University. Nov. 20, 2009

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1 Midterm Exam CS283, Computer Vision Harvard University Nov. 2, 29 You have two hours to complete this exam. Show all of your work to get full credit, and write your work in the blue books provided. Work written on this document will not be evaluated. (Possibly) Useful Information Given vectors u = [u, u 2, u 3 ], v = [v, v 2, v 3 ] we can write the following. u v = u v + u 2 v 2 + u 3 v 3 ; u v = v u = u 2v 3 u 3 u 2 u 3 v u v 3 u v 2 u 2 v A one-dimensional signal x[n], n =,...,N and its discrete Fourier transform X[u] are related by X[u] = N n= x[n] exp( j2πnu/n), x[n] = N N u= X[u] exp(j2πnu/n) Also, as you proved in Assignment Four, if y[n] = ( ) n x[n], their Fourier transforms are related by Y [u] = X[N/2 u], where N is the length of signal x[n]. Below are expressions for the general multi-variate Gaussian distribution for random variable x R d having mean µ and covariance matrix Σ, as well as the special case in which the covariance matrix is a scaled identity matrix Σ = σ 2 I. p(x) = exp (2π) d/2 Σ /2 >> help ones ONES Ones array. ONES(N) is an N-by-N matrix of ones. [ ] 2 (x µ) Σ x µ), p(x) = ONES(M,N) or ONES([M,N]) is an M-by-N matrix of ones. [ (2π) d/2 σ exp d ] x µ 2 2σ 2 >> help zeros ZEROS Zeros array. ZEROS(N) is an N-by-N matrix of zeros. ZEROS(M,N) or ZEROS([M,N]) is an M-by-N matrix of zeros. >> help repmat REPMAT Replicate and tile an array. B = repmat(a,m,n) creates a large matrix B consisting of an M-by-N tiling of copies of A. The size of B is [size(a,)*m, size(a,2)*n]. The statement repmat(a,n) creates an N-by-N tiling. B = REPMAT(A,[M N]) accomplishes the same result as repmat(a,m,n).

2 Question (2 points) The equation for a conic in the plane using inhomogeneous coordinates (x, y) is ax 2 + bxy + cy 2 + dx + ey + f =. () a. Suppose you are given a set of inhomogeneous points x i = (x i, y i ), i =,...,N. Derive an expression for the least squares estimate of the conic c = (a, b, c, d, e, f) passing through those points. (Your expression may take the form of a null vector or eigenvector of a matrix.) b. What is the minimum value of N that allows a unique solution for c? c. Homogenize Eq. by making the substitutions x x /x 3, y x 2 /x 3, and show that in terms of homogeneous coordinates (x = (x, x 2, x 3 )) the conic can be expressed in matrix form, with a symmetric matrix C. x Cx =, d. Suppose we apply a projective transformation to our points: x i = Hx i. The transformed points x i will lie on a transformed conic represented by a new symmetric matrix C. What is the relation between C and C? Question 2 (2 points) Consider a camera with intrinsic parameter matrix 3 3 K = 3 2 and complete camera matrix P = Suppose we add a new camera P with the same orientation as that of camera P. The camera centre of this second camera is located at [3 2] (an inhomogeneous point in R 3 ), and it has a focal length that is one-third that of P. a. What is the camera center for the first camera (P) in inhomogeneous coordinates? b. Compute the camera matrix P. c. Compute the epipole in each camera, expressed in inhomogeneous coordinates. d. Are the epipolar lines in the first camera parallel to one another? Justify your answer.. 2

3 Question 3 (2 points) Consider a surface patch with BRDF f r (ŝ, ˆv, ˆn) =, ˆn ŝ ˆn ˆv where ˆn, ˆv, and ŝ are the surface normal, view direction and light source direction, respectively. Here, the BRDF is expressed in global coordinates, instead of writing the input and output directions in a local coordinate system relative to the surface normal. (The two representations are equivalent. That is, at a small surface patch with normal vector ˆn, given surface irradiance due to radiance from direction ŝ, this tells us the the value of the radiance that is emitted in direction ˆv.) Suppose we view such a surface patch from a known direction ˆv, and suppose we capture two radiance measurements E and E 2 under unit-strength distant lighting from known directions ŝ and ŝ 2. a. Write expressions for the measurements E and E 2 in terms of the view, normal and source directions. b. Show that you can recover the the surface normal from these two measurements. (Hint: derive an expression for n up to scale, and then argue that the sign ambiguity can be resolved by requiring visibility from direction ˆv.) Question 4 (2 points) a. Let x z [n], n {,...,2N } be a one-dimensional image of length 2N with zeros at every alternate pixel. That is, x z [n] = for every odd n. Now suppose we down-sample x z [n] by a factor of two to obtain x dz [n] = x z [2n], which is of length N. Give an expression for X dz [u], u {,...N } in terms of X z [u], where X dz and X z are the one-dimensional discrete Fourier transforms of x dz [n] and x z [n] respectively. b. Next, consider a general one-dimensional image x[n] of length 2N (where all elements can now be non-zero). Suppose we downsample x[n] to get x d [n] = x[2n], essentially throwing away the odd pixels in x[n]. For this case, what is the expression for X d [u] in terms of X[u]? Question (8 points) According to the principal of trichromacy, given three primaries (i.e., light sources with fixed spectral distributions) P (λ), P 2 (λ), P 3 (λ) a typical person can adjust the weights (the brightness) of these light sources so that the resulting mixture looks the same as any given test light T(λ). We write this using algebraic notation as: T(λ) w P (λ) + w 2 P 2 (λ) + w 3 P 3 (λ), where means looks the same as. An important caveat is that subtractive matching must be allowed, meaning that the person needs to have the ability to add some of the primaries to the test light. This can be viewed as adjusting a primary to a negative brightness, if we are willing to apply the algebraic manipulation T(λ) + w P (λ) w 2 P 2 (λ) + w 3 P 3 (λ) = T(λ) w P (λ) + w 2 P 2 (λ) + w 3 P 3 (λ) Explain why the need for subtractive matching implies that if the primaries P i (λ) are all positive functions of λ (which they are if we are using real lights) the corresponding color matching functions must be negative at some wavelengths. 3

4 Question (2 points) Suppose we are constructing a binary (two-category) classifier to discriminate between classes ω and ω 2 based on measurements x R d. We are interested in zero-one loss, so our decision rule is Rule : Decide ω if p(ω x) > p(ω 2 x). Another way of characterizing a classifier is through discriminant functions, and when there are two categories, there are two equivalent ways to do this. We can define two discriminant functions g (x) and g 2 (x) and use the rule Decide ω if g (x) > g 2 (x) or we can define a single discriminant function g(x) g (x) g 2 (x) and use the rule Decide ω if g(x) >. a. Assuming that the class conditional densities p(x ω i ) and prior distributions p(ω i ) are known, show that the discriminant function for Rule can be written g(x) = log p(x ω ) p(x ω ) + log p(ω ) p(ω 2 ). b. Suppose the two classes are equally probable, so p(ω ) = p(ω 2 ), and suppose the class conditional densities are Gaussian distributions with means µ and µ 2 and covariance matrices that are diagonal and equal: Σ = Σ 2 = σ 2 I. Show that the discriminant function for Rule can now be written g(x) = w x + b. where w,b R d are vectors that depend on the means µ i and the variance parameter σ. (Hint: expand the quadratic forms x µ i 2 = x x 2µ i x + µ i µ i and think about which terms can be ignored.) c. The decision rule Rule induces a decision surface in the measurement space, with measurements being assigned to one class or the other depending on which side of the surface they lie. Based on part (b), provide a geometric interpretation of the decision surface. Question 7 (2 points) The next page contains Matlab code that clusters three-dimensional points using the Expectation-Maximization algorithm and a mixture-of-gaussian model for the data. a. Which two lines of this code must be modified so that the function performs the k-means algorithm instead? b. Substitute new code for these lines to implement this change. (While this can be done by inserting only two new lines of code, you are free to insert multiple lines of code in place of each of the two removed lines.) 4

5 function Zo=EM(X,Mo,Co) 2 %EM Expectation-Maximization for Gaussian mixtures in 3D. 3 % Input: X = (numpts) x 3 array of points 4 % Mo= (numclusters) x 3 array of initial cluster means % Co= 3 x 3 x (numclusters) array of initial cluster % covariance matrices 7 % Output: Zo= (numpts) x vector with cluster number 8 % (i.e., one of,2,...(numclusters)) for each point 9 numpts=size(x,); numclusters=size(mo,); 2 3 % support maps for each point 4 Z=zeros(numclusters,numpts); % mixture weights are initially assumed uniform 7 weights=ones(numclusters,)/numclusters; 8 9 % Allocate space for mean and covariance at each iteration. 2 % Initialize to Mo and Co. 2 M=Mo; C=Co; % repeat for ten iterations 24 for n=: 2 2 % E-step 27 for c=:numclusters 28 Z(c,:)=weights(c)*gaussian(X,M(c,:),C(:,:,c)) ; 29 end 3 Z=Z./repmat(sum(Z),[numclusters,]); 3 32 % M-step 33 weights = mean(z,2); 34 for c=:numclusters 3 Xm=X-repmat(M(c,:),[numpts,]); 3 C(:,:,c)=(repmat(Z(c,:),[3,]).*Xm )*Xm/sum(Z(c,:)); 37 M(c,:)=sum(repmat(Z(c,:),[3,]).*X,2)/sum(Z(c,:)); 38 end 39 end 4 4 % Final label for each point is cluster with maximum support 42 [y,zo]=max(z); 43 Zo=Zo ; 44 4 %%% SUB-ROUTINES 4 47 function G=gaussian(X,M,C) 48 % Evaluate multi-variate Gaussian with mean M and covariance C 49 % at points X. ndims=length(m); 2 numpts=size(x,); 3 X=X-repmat(M(:),[numpts,]); 4 G=exp(-(sum(X.*(inv(C)*X ))) /.)/sqrt(((2*pi)^ndims)*det(c));

6 Question 8 (2 points) Consider the six textures below, which are numbered (i) (vi). Below the textures are six sets of graphs, labeled (a) (f). Each set of graphs corresponds to one of the six textures, and each set contains a normalized histogram (pz (z)) for the gray levels in the texture as well as two functions computed from the Fourier spectrum: (r, θ) S(θ) = S(r) = R rmax Rπ r F (r, θ)dr θ F (r, θ)dθ where F (r, θ) is a centered Fourier spectrum written in polar coordinates as shown above-right. Match each texture to it s corresponding set of graphs by writing a label ((a)-(f)) for each texture ((i)-(vi)). (i) (a) (ii) (iii) (b).. (iv) (v) (c). (vi) x x (d) 2 2 (e) (f) x x 2 x