1. Revision 4
CSMSC5711
Revision 4: CSMC5711 v.9a

2. Q1a
Camera model: A 3D point M is at [X,Y,Z]T=[0.5,0.5,2.5]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: CSMC5711 v.9a

3. Q1a
The image point of M is at the (600,600) pixel position. 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 3D 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: CSMC5711 v.9a

4. Q2b 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: CSMC5711 v.9a

5. Q3
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, H1 = (G-B)/(max-min)
if G = max, H1 = 2 + (B-R)/(max-min)
if B = max, H1 = 4 + (R-G)/(max-min)
H = H1 * 60
Revision 4: CSMC5711 v.9a

6. Q4
Corner feature: An image IMG with grey level values is shown in the figure below. The matrix E is defined as
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 Y
IMG 1 x
Revision 4: CSMC5711 v.9a

7. Q5
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 [u1,v1]T on the left image is corresponding to an image point [u2,v2]T on the right image.
Write a formula to find the relation among these terms: f11, f12, f13, f21, f22, f23, f31, f32, f33, (elements of F), u1,v1, u2,v2.
Describe how to find F if 8 corresponding image points are given.
Write a formula for the essential matrix E using F, Kint1, Kint2 .
Revision 4: CSMC5711 v.9a

8. Q6. Structure estimation (SFM2):
A 3D 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=[T1, T2, T3 ]T is the translation of the camera with respective to the world coordinate system. The horizontal image position of the 3D point after perspective image projection is [ut,vt]T.
d) If the 3D 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 3D model reconstruction algorithm.
e) Describe the 3D model reconstruction algorithm (SFM2) procedure to find the 3D model point X.
Revision 4: CSMC5711 v.9a