Revision 4 CSMSC 5711 Revision 4 CSMC 5711

Revision 4 CSMSC 5711 Revision 4: CSMC 5711 v. 9 a 1

Q 1 a • Camera model: A 3 D point M is at [X, Y, Z]T=[0. 5, 2. 5]T meters in the world coordinate system. The intrinsic parameters (Kint) and characteristics of a camera are shown below: • The image centre is at (Ox , Oy)=(500, 495) pixel position. The camera (1, 1) pixel position is at the right bottom corner of the sensor. The horizontal coordinate is increasing from the right to the left, and the vertical coordinate is increasing from the bottom to the top. • Horizontal pixel width sx = vertical pixel width sy =5 m. • The image centre is at the centre of the image plane. • You may assume the camera coordinates are the same as the world coordinates. Revision 4: CSMC 5711 v. 9 a 2

The image point of M is at the (600, 600) pixel position. Q 1 a • Estimate the horizontal and the vertical focal lengths in pixels • • • of the camera. Calculate the horizontal and vertical focal lengths (fx, fy) in meters. Discuss the purpose of having parameters Ox and Oy of a camera. A 3 D point P is at [0. 4, 0. 6, 2. 4] T in meters. M is being rotated around P first, the rotation angles are small and are (1, 2, 1. 5) in degrees (remember to convert degrees into radians before use). Then, it is translated by T= [0. 2, 0. 1, 0. 4]T meters. Find the new 3 -D position (M’) of M in pixels. Find the 2 -D image position (in pixels) of M’. You may round off to the results to the nearest integers. Repeat (d) if M is being translated by T first, and then rotated around P. Find the new 3 -D position (M”) of M in pixels Revision 4: CSMC 5711 v. 9 a 3

Q 2 b • Convolution, edge mask and edge detection: • An image S and a second order mask m are shown below • Find the convolution result of S and m, the result should include all partially overlapping cases. Revision 4: CSMC 5711 v. 9 a 4

Q 3 • Color processing: The RGB_raw values of a pixel are (45, 20, 10), the maximum value of each R, G, B channel is 255. • Remember to normalize R, G. B values before use, ie. If R_raw=100, R=100/255, etc. • Calculate the color values in the HSV format. • Calculate the color values in the HSL format. • Assume you have a histogram-equalization program that is able to equalize the grey level distribution of a picture. Describe how to use this program to improve the image quality of a picture in RGB format. • • • max=max_value(R, G, B) min=min_value(R, G, B) if R = max, H 1 = (G-B)/(max-min) if G = max, H 1 = 2 + (B-R)/(max-min) if B = max, H 1 = 4 + (R-G)/(max-min) H = H 1 * 60 Revision 4: CSMC 5711 v. 9 a 5

Q 4 • Corner feature: An image IMG with grey level values is shown in the figure below. The matrix E is defined as Y 0 IMG 0 1 1 1 0 0 0 0 0 x • Calculate the matrix E for the picture Img. Show your calculation steps. • Discuss how to check if this window has a corner feature or not. There is no need to find the actual Eigen values involved, just describe the procedure Revision 4: CSMC 5711 v. 9 a 6

Q 5 • Stereo vision: Given that F is the fundamental matrix between two stereo cameras, one on the left and another on the right. An image point [u 1, v 1]T on the left image is corresponding to an image point [u 2, v 2]T on the right image. • Write a formula to find the relation among these terms: f 11, f 12, f 13, f 21, f 22, f 23, f 31, f 32, f 33, (elements of F), u 1, v 1, u 2, v 2. • Describe how to find F if 8 corresponding image points are given. • Write a formula for the essential matrix E using F, Kint 1, Kint 2. Revision 4: CSMC 5711 v. 9 a 7

Q 6. • Structure estimation (SFM 2): • A 3 D model point X= [X, Y, Z]T is in the world camera coordinates, f is a known focal length. At time t, Rt is the rotation and Tt=[T 1, T 2, T 3 ]T is the translation of the camera with respective to the world coordinate system. The horizontal image position of the 3 D point after perspective image projection is [ut, vt]T. • d) If the 3 D model point X is observed by the camera at 4 different positions indexed by t=1, 2, 3, 4, such that Rt=1, 2, 3, 4 and Tt=1, 2, 3, 4 are known. Write the Jacobean of the 3 D model reconstruction algorithm. • e) Describe the 3 D model reconstruction algorithm (SFM 2) procedure to find the 3 D model point X. Revision 4: CSMC 5711 v. 9 a 8