UNIVERSITY OF TORONTO Faculty of Applied Science and Engineering. ROB501H1 F: Computer Vision for Robotics. Midterm Examination.

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1 UNIVERSITY OF TORONTO Faculty of Applied Science and Engineering ROB501H1 F: Computer Vision for Robotics October 26, 2016 Student Name: Student Number: Instructions: 1. Attempt all questions. 2. The value of each question is indicated in the table opposite. 3. Write the answers only in the space provided for each question. 4. You are permitted one double-sided notes sheet. 5. No calculator is permitted. 6. There are 8 pages and 5 problems in this test paper. For Examiner Only Problem Value Mark Total 80 1 of 8

2 Problem 1: True/False Statements Determine if the following statements are true or false and indicate by T (for true) and F (for false) in the box beside the question. The value of each question is 2 points. 1. a) Projective transformations preserve ratios of lengths. 1. b) Solving Bayes Rule for the posterior distribution depends on knowledge of the evidence term. 1. c) The computational complexity of RANSAC is independent of the percentage of outliers in the data. 1. d) Chromatic aberration is caused by lens foreshortening effects. 1. e) An ROC curve directly measures the accuracy of, e.g., a feature matching algorithm, as a function of the threshold value. 2 of 8

3 Problem 2: Short Answers Provide short but detailed answers. 2. a) What are the three characteristics of a well-posed problem, according to Jacques Hadamard? (3 points) 2. b) The solutions to regularization problems typically involve two (mathematical) terms; what are they and what are their roles? (4 points) 2. c) What is the aperture problem? (3 points) 2. d) Is the Harris corner detector invariant to changes in image gain (brightness scaling)? And/or to image bias (constant brightness offset)? Why or why not? Give a brief mathematical argument. (4 points) 2. e) What does the Hessian matrix, used by the SIFT feature detector, represent? And what information does it provide about the local image region? (3 points) 2. f) What are two difficulties encountered when applying nonlinear optimization algorithms, in general? (3 points) 3 of 8

4 Problem 3: Long Answer Provide a full answer with all relevant information. (20 points) 3. a) Consider the 2D line in homogeneous form l 1 = (a, b, c). We can normalize the line coordinates by dividing by (a 2 +b 2 ) 1/2 to give (n x, n y, d) = (ˆn, d), with ˆn = 1. Show that the vector ˆn is perpendicular to the line, and that the distance to the origin is d. (10 points) l y n^ d x 4 of 8

5 3. b) Given two parallel 2D lines l 1 = (a, b, c) and l 2 = (d, e, f) (i.e., with the same slope), determine their intersection point according to projective geometry. (10 points) x = l 1 l 2 5 of 8

6 Problem 4: Long Answer Provide a full answer with all relevant information. (10 points) 4. The 4 4 camera matrix P can be used to map directly from augmented world coordinates p w = (x w, y w, z w, 1) to image plane coordinates, x s = (x s, y s, 1, d), P = K 0 C t = KE 0 T 1 0 T 1 where E is a 3D rigid-body transformation and K is the full rank calibration matrix. To determine the image plane coordinates, x s P p w, the vector is normalized by dividing by the third element after multiplying by P. There is a remaining value, d, in the vector give the mathematical expression for this value (in terms of the world-to-image transform), and explain what it represents physically. 6 of 8

7 Problem 5: Long Answer Provide a full answer with all relevant information. (20 points) 5. a) In the lecture, it was shown the that the nonlinear least squares (NLS) error update for a small change in the parameter vector is E NLS ( p) = J(x i ; p 0 ) p r i 2 i [ ] [ = p T J T J p 2 p T i i J T r i ] + i r i 2 Derive this result (i.e. the second formula) directly. (8 points) 7 of 8

8 5. b) Instead of minimizing the sum-of-squared-errors (residuals), other error functions are possible, E RNLS ( p) = i ρ ( r i ), where ρ( ) is what is known as a robust cost. Consider the robust cost function ρ( r i ) = r i 2 σ 2 + r i 2, for some σ 1. If there are outliers in the data, what effect will the use of this function have on their influence on the overall solution? Give a qualitative answer and explain your reasoning. (8 points) 5. c) The Levenberg-Marquardt algorithm for solving NLS problems uses the iterative update step (A + λ diag A) p = b. What is the purpose of the additional term (diag A)? And what is the damping factor λ designed to do? (4 points) 8 of 8