Final CS556: Exam Questions
1. What are the main steps in David Marrs framework of computational vision? What are the key differences between Marrs paradigm and Intrinsic images? 2. Specify the criterion for identifying brightness changes in a pixel neighborhood, used by the (a) Harris detector. (b) Hessian detector. (c) Kadir-Brady detector. 3. Derive the criterion for detecting interest points, used by the (a) Harris detector. (b) Hessian detector. (c) Kadir-Brady detector. 4. Specify the Difference-of-Gaussians (DoG) detector? 5. Specify the main steps of extracting Maximally Stable Extremal Regions? 6. Derive a linear transform that maps a detected interest point onto a canonical, normalized form. 7. Explain differences between the SIFT and DAISY descriptors. 8. Explain the main steps of the Canny edge detector. In one of the steps, the Canny detector connects broken edges in the image, by lling the gaps with straight lines. Why is this postprocessing necessary? What is the underlying Gestalt principle for connecting the edges? 9. Suppose we applied the Canny edge detector to an image. Can we use intersections of detected Canny edges as interest points? 10. The goal is to extract scale invariant DAISY descriptors. Which of the following two algorithms would you use? Explain your choice.
Algorithm 1: For each scale S Compute DAISY gradient field of the image; Detect interest points at scale S; Compute the DAISY descriptor for each detection; End
1
Algorithm 2: Detect interest points that are stable over different scales; For each detected interest point Identify the scale of a region associated with the interest point; Use this scale to define the basic radius of the DAISY descriptor; Compute the DAISY descriptor; End
11. What is the difference between progressive scanning and interlaced scanning? 12. What is a vanishing point? Can there be more than one vanishing point in the image? 13. Homogeneous coordinates of point P in the image are P = [40 164 2]T . Which row and column contains P if homogeneous coordinates are measured from the top left corner of the image? 14. The intrinsic parameters of two cameras are given by 0.6 0.3 1.4 0.4 0.3 0.9 0.9 , K1 = 0 0.4 K2 = 0 0.6 1.4 0 0 1 0 0 1 Which of the two cameras would you prefer to use? Explain your choice. 15. Point P in the image is mapped to point P by rst translating P to P1 15 rows in the same direction of y axis, and 102 columns in the opposite direction of x axis, then, by rotating P1 to P2 around the top left image corner by 30 counter clockwise, and nally by scaling the coordinates of P2 to P by 1.3 along x axis, and 1.6 along y axis. Find the afne transformation matrix that captures these transformation steps from P to P . 16. Point P in the 3D scene is mapped to point P by rst translating P to P1 using the translation vector t = [45 56 78]T , and, then, by rotating P1 to P around the y axis of the world coordinate center by 45 clockwise. Find the afne transformation matrix that captures these transformation steps from P to P . 17. State the camera calibration problem. 18. Given a stereo pair of images, any two points x and x , from the two images respectively, can be related by a homography x = H x. These points can also be related via the fundamental matrix xT F x = 0. Derive an expression that relates F and H . What is the rank of H and F? 19. The intersection of two epipolar lines l1 and l2 is the epipole e in image 1 of a stereo pair. Suppose we are given the image coordinates of two non-planar pairs of points (x1 , x1 ) and (x2 , x2 ) and their respective homographies x1 = H1 x1 and x2 = H2 x2 . Derive the expressions of epipole e and fundamental matrix F in terms of these points and their homographies. 2
20. In a stereo pair of images, corresponding 2D points x and x that are generated from 3D coplanar points X that belong to a plane, , can be related by a homography, x = H x. These points can also be related via the fundamental matrix xT F x = 0. Given coordinates of two corresponding pairs of points (x1 , x1 ) and (x2 , x2 ), where x1 = H1 x1 , and x2 = H2 x2 , and 1 = 2 , and given homographies H1 and H2 , compute F ? 21. What is the motion between two images of a stereo pair whose epipoles are located: (a) in the center of the images, (b) in the innity; 22. Given the epipoles e and e of image 1 and image 2 of a stereo pair, and their fundamental matrix F , compute eT F =? and F e =? 23. Suppose we are given: (i) the projection camera matrices M and M for image 1 and image 2 of a stereo pair, (ii) the coordinates of two corresponding points x = MX and x = M X , and (iii) the point in the 3D scene, C , that projects onto the camera center of image 1 (i.e. MC = 0). Derive the equations of the epipolar lines l and l , and fundamental matrix F , in terms of M, M , x, x , C . What is the expression for F when the camera centers of image 1 and image 2 are the same (i.e., the stereo images are related only through a rotation at the same camera center)? 24. Given the projection camera matrices M and M for image 1 and image 2 of a stereo pair, the fundamental matrix F , and N pairs of corresponding points (x, x ) derive the least squares error-minimization algorithm for 3D scene reconstruction. 25. Perform Ncuts based foreground background segmentation of an image, represented as a graph of pixels. The graph contains three groups of nodes, A, B , and C . A contains 20 nodes, B contains 10 nodes, and C contains 5. All nodes in A are connected to each other with a weight of 1. Likewise, all nodes within B and C , respectively, are connected to each other with a weight of 1. Subgraphs A and B are connected only via one edge between node a in A and node b in B , whose weight is wab 0. Also, subgraphs A and C are connected only via one edge between node a in A and node c in C , whose weight is wac 0. Finally, subgraphs B and C are connected only via one edge between node b in B and node c in C , whose weight is wbc 0. Explain your Ncut results as a function of increasing wab , especially for wab values = 0, 1, 30, when: (i) wac = 0 and wbc = 0; (ii) wac = 1 and wbc = 1; and (iii) wac = 25 and wbc = 15; 26. Suppose we are given a graph G = (V, E, w ), where V is the set of nodes, E is the set of edges, and w : E R+ is a function that denes non-negative edge weights. If D is the degree matrix and W is the adjacency matrix of G, prove that the Laplacian of G, dened as L = D W is positive semidenite matrix. Hint: Prove that for any arbitrary column vector the following holds: T L = T D T W =
i
d i 2 i
ij
wij i j 0.
3
27. Finding the minimum Ncut in a graph G = (V, E, w ) with the Laplacian matrix L = D W can be formalized as 1T L1G1 G , min T 1 G1 1 D 1 G1 G1 where 1G1 is the indicator vector whose elements are equal to 1 over all nodes of G included in the subgraph G1 . Apply relaxation to this discrete problem, and formulate the corresponding continuous problem. 28. Suppose we are given a graph G = (V, E, w ), where V is the set of nodes, E is the set of edges, and w : E R+ is a function that denes non-negative edge weights. If D is the degree matrix and W is the adjacency matrix of G, then the symmetrically normalized Laplacian is dened as Ls = D 1/2 (D W )D 1/2 . Consider the solution of the following problem min 1T 1 Ls 1G1 , G
G1
where G1 is a subgraph of G. Is this solution equivalent to the one obtained by using the Ncut algorithm? Explain. 29. Suppose we are given a set of data points D = {x1 , x2 , . . . , xN }, dened in some feature space. This feature space can be partitioned into a number of bins, Bj , j =1, . . ., M . Let nj denote the number of data points within Bj , and let bj denote the representative point of Bj (e.g., mean value, or median value of all data points within Bj ). Then, the non-parameteric probability density estimate, f (x), of our data points D can be computed as f (x ) =
1 N M j =1 nj KHj (x
b j ),
where KHj (x bj ) is a kernel function with bandwidth matrix Hj that varies over different bins Bj . An example of the kernel function is the Gaussian function KHj (x bj ) = exp (x bj )T Hj 1 (x bj ) . Prove that the meanshift vector, m(x), for the Gaussian kernel with variable bandwidths Hj has the following form: m (x ) = H (x ) where
M
f ( x ) , f (x )
[H (x)]1 =
j =1
nj KHj (x bj )
M j =1
nj KHj (x bj )
Hj 1 ,
and f (x) =
d f (x ) . dx
30. A pixel has the following color values R = 28, G = 28, B = 28. Compute the corresponding HSI color values. 31. Which procedure is better for detecting salient edges in the image: 4
(a) Compute the image gradients and then smooth these gradients using the Gaussian lter; (b) Smooth the image pixel values and then compute the image gradients; (c) Filter the image using the gradient of the Gaussian lter. 32. The Canny edge detector uses two thresholds of image gradients to identify edges. Why? Which Gestalt principles of grouping are modeled by the Canny edge detector? 33. Suppose we are given a set of mutually dependent elements, A = {a1 , a2 , . . . , aN }, and a set of their labels, = {1 , 2 , . . . , M }, and the compatibility function rij (, ) that any pair of elements ai and aj have labels and , respectively. Specify the relaxation labeling update rule for computing the likelihoods ai , pi() = P (ai |). 34. Suppose the goal is to cluster edges in the image, A = {a1 , a2 , . . . , aN }, into two clusters, {0, 1}, and we are given the similarity values between the edges, sij = s(ai , aj ), and the spatial distances between the edges dij = d(ai , aj ). Specify the clustering algorithm using (a) The relaxation labeling algorithmi.e., specify the compatibility function rij (, ) that edges ai and aj have labels and , respectively, in terms of sij and dij , and then the update rule for computing the likelihoods ai , pi () = P (ai|). (b) The Ncut algorithmi.e., specify the cut and volume of a graph whose nodes are edges of A, in terms of sij and dij , and then the Ncut optimization function (discrete problem). 35. Specify the algorithm for the Voronoi image tessellation. 36. The Voronoi diagram is dened for a set of points. How can it be generalized for a set of edges? 37. State the Gestalt laws of closure, good continuation, and gure-ground.
Figure 1: Graph for Problem 38 38. Suppose we are given a graph G = (V, E, w ), as shown in Fig. 1. V is the set of nodes, E is the set of edges, and w : E R+ is a function that denes non-negative edge weights. Let D denote the degree matrix, and W denote the adjacency matrix of G. Also, let G1 and G2 be two subgraphs of G marked by dashed lines in Fig. 1. Compute: (a) 1T 1 D 1G1 D =? G 5
(b) 1T 1 D 1G1 W =? G (c) 1T 1 (D W )1G1 =? G (d) (1 1G1 )T (D W )(1 1G1 ) =? (e) D 1/2 (D W )D 1/2 =? 39. Suppose we are given a dataset D = {i : i = 1, 6}. List all clusters of D , obtained in the agglomerative clustering, when the distance between the clusters is dened as (a) Single linkage, (b) Complete linkage, (c) Average linkage. 40. Suppose we are give two open sequences of points: s1 = {s1,1 , . . . , s1,4 } and s2 = {s2,1 , . . . , s2,5 }, and their cost matrix, C . The rows of C correspond to the points in s1 , and the columns of C correspond to the points in s2 , and each element of C represents the cost of matching the corresponding pair of points C (i, j ) = d(s1,i , s2,j ). The cost matrix is specied as 1302 2 4 1 1 C= 0 1 4 4 1042 Compute the output of Dynamic Time Warping algorithm (DTW). To this end, transform C into an auxiliary matrix DTW that will record DTW(i, j ) = C (i, j ) + min [DTW(i 1, j ), DTW(i, j 1), DTW(i 1, j 1)] and then compute the optimal path in matrix DTW. 41. Let P (x, y ) denote the joint distribution of feature vector x and its associated label y . Similarly, let P (y |x) denote their posterior distribution. What is the difference between discriminative versus generative learning, in terms of P (x, y ) and P (y |x)? 42. What is a visual dictionary? What is a bag-of-words model? Describe the relationship between classication accuracy and the number of dictionary words.
6