3DCV Homework 2
Name:
Student ID:
1. Perspective Projection
Use the perspective projection equations to explain why, in a picture of a face
taken frontally and from a very small distance, the nose appears much larger
than the rest of the face. Can this effect be reduced by acting on the focal
length?
2. Pinhole Camera
A. How does an image change if the focal length is varied?
B. A scene point at coordinate (400,600,1200) is perspectively projected
into an image at coordinate (24,36), where both coordinates are given
in millimeters in the camera coordinate frame and the camera principle
point is at coordinates (0,0,f). Assuming the aspect ratio of the pixels in
the camera is 1, what is the focal length of the camera?
3. Eigen Values
For the 2 by 2 case, obtain the relationship between the eigenvalues of 𝐴 and
of 𝐴2 . Does the result you obtained hold for 𝐴 and𝐴𝑛 in general?
4. Singular Value Decomposition
Find the camera center of the following projection matrix using Singular Value
Decomposition. You may use MATLAB for your calculations.
5. Homography Matrix
The following corresponding points of two images are given. 𝐴𝑖 points are the
points in the first image and 𝐵𝑖 points are the points in the second image.
Calculate a Homography matrix that maps 𝐴𝑖 points to 𝐵𝑖 points using
Singular Value Decomposition.
6. Camera Calibration (bonus)
You are required to calibrate a camera using a cardboard with a size of 50𝑚𝑚
×50𝑚𝑚. This intrinsic parameters of this camera can be written as 𝐾 =
𝑓𝑥 0 𝑐𝑥
( 0 𝑓𝑦 0 ). The rotation transformation from the world coordinate system
0 1 𝑐𝑦
1.0 0
0
to the camera coordinate system is denoted by 𝑅 = ( 0 1.0 0 ) and the
0
0 1.0
camera center in world is represented by 𝐶 = (0,0, −500)𝑇 . The world
coordinates of four corners of the cardboard are (0,0,0)𝑇 , (50,0,0)𝑇 , (0,50,0)𝑇 ,
(50,50,0)𝑇 respectively and their corresponding coordinates in image are
(160,120)𝑇 , (200,120)𝑇 , (160,160)𝑇 , (200,160)𝑇 . Please try to estimate 𝐾 and
the list of detailed equations in your estimation.
Notes:
 The deadline for HW2 is 23:59 of 16 Aban. The penalty for late submission is
20% each day.
 Put all your files including your document and your code file in folder named
HW2_STDID and send it to [email protected]
 Any case of plagiarism would result in a negative grade equal to homework’s
grade.
 Please feel free to contact me if you have any questions:
[email protected]