1 Projective Geometry

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1 CIS8, Machine Perception Review Problem - SPRING 26 Instructions. All coordinate systems are right handed. Projective Geometry Figure : Facade rectification. I took an image of a rectangular object, and its image measurement is shown in Figure left. The original object shape is shown in Figure right. Let the homography from the rectangle drawing (Figure right) to the image measurement (Figure left) be H. λ i x i = Hx i = h h 2 h 3 h 2 h 22 h 23 x i, h 3 h 32 h 33 where λ i is a scalar, x i and x i are homogeneous coordinates for corresponding points on Figure left and Figure right respectively. The homography, H, maps [ ] the origin to [ ] [ ] to

2 [ ] infinity in the x-direciton to the vanishing point [ ] infinity in the y-direciton to the vanishing point. Compute H with h 33 normalized to. 2. Compute the vanishing line, l, in Figure left. 3. Project the vanishing line l in Figure left to Figure right. Hint: l = T T l, where T is the homography transformation from Figure left to right, l and l are line equation on Figure left and right respectively. 2

3 2 Single View Metrology Figure 2: The cross-ratio. In this problem, we will use cross-ratio to determine possible correspondence between a floor plan (Figure 2 left) and a room photograph (Figure 2 right). This is useful for online apartment finding since we often have a floor plan and photos listed, but no correspondence for matching room photos to those shown on the floor plan. In this example shown in Figure 2 left, the floor-plan shows the width of the two windows and the wall between them are of the same width W. Let a, b, c, d and e be the 2D points measured in pixel coordinates in Figure 2 right.. Express the image cross-ratio in terms of a, b, c, d and e. 2. Express and compute the corresponded scene cross-ratio. 3. How to use this information to verify if this pair of the floor plan and room photo matches? 3

4 3 Rotation (a) Two meanings of rotation matrix (b) Rotation Combination Figure 3: Camera Rotation In this problem, we will familiarize ourselves with the concept of rotation matrix R. Mathematically, the rotation matrix R can be used as following (3D example): x b x a y b = R y a z b z a This equation has two geometrical meanings for the same rotation action, as illustrated in Figure 3(a). Two rotation matrices are given: R =, R 2 = R measure the rotation of θ about y-axis, and R 2 measure the rotation of θ 2 about z-axis.. In the same coordinate system, rotate point from X a = ( 6, 2, 4) T, to point X b for θ about y-axis. Compute point X b position. 2. Rotate the coordinate system from U to V for θ 2 about z-axis. Given the point position in U coordinate X U = ( 2, 2, 4) T, compute the point position in V coordinate X V. 3. First, rotate the coordinate system from U to V for θ 2 about z-axis. Then in the coordinate system V, rotate point from X V a, to point X V b = (8, 3, )T for θ about y-axis. Compute the point position in U coordinate X U a. 4

5 4 Epipolar Geometry When two cameras view a 3D scene from two distinct positions / orientations, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. In this problem, we will use known camera positions and orientations to build essential matrix, fundamental matrix, epipolar line and epipole. The intrinsic parameters of the two cameras K, K 2 and the external parameters of the two cameras R, t and R 2, t 2 are all given, where P = K [R t ] and P 2 = K 2 [R 2 t 2 ]. X is a 3D point.. Express the essential matrix 2 E with respect to R, R 2, t and t 2. Note that for a given point X in 3D world, X T 2 2 E X =, where X and X 2 are the coordinates for X in the first and second camera frame, respectively. 2. Express the fundamental matrix 2 F with respect to R, R 2, t, t 2, K and K 2. [ ] T [ ] x2 Note that for a given point X in 3D world, 2 x F =, where x and x 2 are the 2D projection of X in the first and second camera image plane, respectively. 3. Epipolar line is the intersection line of image plane and epipolar plane. Given the 3D point X and its 2D projection x on image plane, it corresponds to an epipolar line on image plane 2. Express the equation for epipolar line in image 2 of point x in image with respect to x, R, R 2, t, t 2, K and K 2, 4. Epipole is the 2D projection of one camera center in another camera s image plane. Express the coordinate of epipole in image 2 with respect to R, R 2, t, t 2, K and K 2,

6 Parameter Estimation [ ] x = K [ R t ] [ ] X represents perspective projection, where K is camera calibration matrix, R and t are the camera extrinsic parameters, X is 3D point in world coordinate and x is the projected point in image coordinate. Given part of the parameters in the equation, we can solve for the other parameters using linear and non-linear optimization. In this problem, we will discuss about triangulation, PnP, Bundle Adjustment and revisit calibration.. Triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In this question, given the perspective projection parameters R, t, K and projected point x, we need to solve point in world coordinate X. (a) In linear triangulation, we formulate the triangulation problem as a linear least squares problem. [ ] Explicitly, we combine and reduce the projection equation into the form of X A =, where A is build from R, t, K and x, and solve X by SVD. What s the minimum number of cameras needed to triangulate one point? Express matrix A with respect to R, t, K and x. Use R i, t i, K i and x i for the parameters and 2D projection of the i th camera. Note that intermediate steps are needed for this question. (b) In nonlinear triangulation, we minimize the reprojection error of our estimation X with respect to its 2D reprojection. Express reprojection error with respect to X and 2D projection x. Compute the Jacobian matrix for the cost function, when X = X. 2. Perspective-n-Point (PnP) refers to the process of determining the camera extrinsic parameters given camera calibration matrix, several points in world coordinate and corresponding 2D projection. In linear PnP, we formulate the PnP problem as a linear least squares problem. Explicitly, we combine and reduce the projection equation into the form of AT =, where T is a 2 vector representing elements in R and t, A is build from K, X and x, and solve T by SVD. What s the minimum number of points needed to register one camera? Express matrix A with respect to K, X and x. Use X i and x i for 3D coordinates and its 2D projection for the i th point. Note that intermediate steps are needed for this question. 3. In Bundle Adjustment, we refine all the parameters (3D points coordinates and camera external parameters) by minimizing the reprojection error. What kind of optimization method, linear or non-linear can be used for Bundle Adjustment? Why? 4. Camera calibration can also be formulate in the same way. In this question, given the perspective projection parameters R and t, points in world coordinate X and projected point x, we will solve the camera calibration matrix K. Using the similar linear least squares method, what s the minimum number of X and x pairs needed to determine camera calibration matrix? Express matrix A with respect to R, t, X and x. Use X i and x i for 3D coordinates and its 2D projection for the i th point. Note that intermediate steps are needed for this question. 6