12 décembre 2013 13h30 à 16h20 PLT-2501 Deuxième examen partiel A 2013 Toute documentation permise sauf Internet QUESTION 1 (15 points au total) Camera calibration. When a camera is calibrated, a target with n object points of known coordinates in the world reference frame is observed by a camera that is modeled by a non-inverting pinhole. The image coordinates of the points on the target are estimated and a match is done between each object point and its corresponding image point. The overdetermined linear system of equations (1) allows to estimate vector m containing the elements of the camera matrix: A 2n 12 m 12 1 = 0 with n» 6 (1) A) (5 points) Explain the basic principle of the minimization of the algebraic error for the estimation of the elements of the camera matrix. B) (5 points) Explain the basic principle of the minimization of the geometric error for the estimation of the elements of the camera matrix. C) (5 points) In your opinion, which error minimization approach (algebraic or geometric) is the best in real camera calibration experiments? Justify your answer. QUESTION 2 (20 points au total) Homography. A) (10 points) Describe briefly the basic principle of homography and describe the conditions under which it can be applied. B) (10 points total) The two images in Figure 1 are used for estimating a homography.
2/6 Figure 1 Images for Question 2 B Left image Right Image A minimum of four point matches is required for estimating the homography. Four different persons select these four points manually according to the experiments shown in Figure 2. 1) (5 points) Which person makes the best matches and why? Explain the weakness of the matches made by the three other persons? 2) (5 points) Would it be better to select more than four points for estimating the homography? Justify your answer. Figure 2 Point matches made par different people. (a) Person 1 (b) Person 2 (c) Person 3 (d) Person 4
3/6 QUESTION 3 (20 points) Radiometry. Assume the geometry shown in Figure 3. Figure 3 Geometry for Question 3. Source I = 100 W/sr d 1 =1m d 2 =2m D1, R = 10 cm D2, R = 10 cm D3, R = 10 cm s = 1 m A uniform point source of intensity I = ---- illuminates a Lambertian disc D1 with radius R = 10cm. The 100 W sr point source also illuminates disc D3 with radius R = 10cm. Disc D1 diffuses part of the energy received from the source towards Lambertian disc D2 with radius R = 10cm. Lambertian disc D2 diffuses itself part of the energy received from D1 towards disc D3. What is the total illuminance received by disc D3? QUESTION 4 (15 points) Radiometry. Fundamental image formation equation. In Figure 4, a uniform point source with intensity I = ---- illuminates a lambertian disc D with radius R = 10cm. What is the illuminance of the image of the disc on the image plane of a camera with a lens with a diameter of 10mm and a focal length of F = 10mm? 100 W sr Figure 4 Geometry of Question 4. Source I = 100 W/sr y F=10 mm 1m D 50 cm z 1 m Plan image (Image plane) Lentille (lens)
4/6 QUESTION 5 (10 points) Principal component analysis and eigenfaces. A database containing n images of faces with resolution l lines and c columns (with m = l c) is available. The database is stored in a matrix where each column is an image of the database. Let i : R m n i n 1 = -- Rij n i = 1 m j = 1 (2) be the mean value of the i th line of R. Let also vector e be defined as: e = 1 1 n T. (3) A) (2 points) What is the meaning of matrix X = R e T? B) (2 points) Let C = COV X be the covariance matrix of matrix X. Matrix C has dimensions m m. It is possible to find a matrix V m m that diagonalises C and produces a diagonal matrix D m m which contains the eigenvalues of C on its diagonal elements: 1 V m m C m m V m m = D m m (4) Columns of V are the eigenvectors of C. The elements of D (i.e. the eigenvalues of C ) are reordered such that they appear in decreasing order of magnitude and the columns of V associated with each eigenvalue are reordered accordingly. Keeping only the first columns of, it is possible to build matrix such that: l V W m l Wpq = Vpq avec p = 1 m, q = 1 l. (5) Is each column of C) (2 points) Consider vector s : W an image? Justify your answer. s = s m = Cpq, p = q = m = 1 m (6) What does this vector represent? D) (2 points) Matrix can be defined as: Z m n
5/6 Z = X ------- se T (7) where division. e T is defined as in Equation (3) and where the division operator implements an element by element What does matrix Z mean? Justify your answer. E) (2 points) Matrix can be defined as: Y l n Y = W * Z (8) where W * is the conjugate transpose of W in Equation 5. What does matrix Y represent? QUESTION 6 (20 points) Image processing. Assume signals s 1 x, s 2 x, s 3 x, and s 4 x in Figure 5. A) (5 points) What is the gradient of each signal (a handrawing is enough)? B) (5 points) What is the Laplacian of each signal (do a handrawing is enough)? Figure 5 Signals for Questions 6 A et B C) (10 points total)
6/6 Given the signal in Figure 6. Figure 6 Signal for Question 6 C. s(x) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 x Give: 1) (5 points) The signal resulting form a median filtering with a 1 3 kernel. 2) (5 points) The signal resulting from an average filtering with a uniform 1 3 kernel.