E27: Computer Vision Spring 2016 - HOMEWORK 1
1. Why the cross product has no inverse.
For all vectors a and b in R3, we can write the cross product a x b as a matrix operation
a x b = [a]x b
where [a]x is a 3 x 3 skew-symmetric matrix depending only on a. A matrix A is skew-symmetric if
AT = -A
a. What are the elements of [a]x? Note that each element of (a x b) is linear in the elements of b. Gather the appropriate coefficients and place them into the matrix.
b. Explain why the cross product has no inverse. Starting from your definition of [a]x, given any vector c, indicate why is there no unique vector b such that a x b = c. Hint: a good place to start is to compute the determinant of [a]x or to show that one of its columns is a linear combination of the other two.
2. Rigid transformations.
A 2D rigid transformation is an invertible transformation which preserves distances. For any point p ∈ R2, we can write the transformation as
P' = Rp + t
where R is a 2 x 2 rotation matrix, t ∈ R2 is a translation vector, and p' is the transformed point.
a. Matrix representation. Show that the transformation can be represented as a 3 x 3 homogenous matrix M˜ (which itself may depend on R and t) such that p¯' = M˜p¯. What is the matrix?
b. Matrix inverse. Solve for p in terms of R, t, and p'. What form must M˜-1 therefore take?
c. Rigid transformation of a line. Given a rigid transformation specified by (R, t) and a line l˜, what is the corresponding line l˜', such that for all p¯,
l˜' ·p¯ = l˜ ·p¯
Express l˜' in terms of M˜ and l˜. Note: you won't be able to complete this question until after we cover the homogenous representation of lines on 1/26.
3. OpenCV "Hello world".
Read and run the tutorial.py program from the sample code on the course website. Then modify it or write your own program to transform an image in a different way. You can use a different image than the purple flower one used in the tutorial if you want. Here are some example ideas:
- flip the image horizontally and/or vertically (easier)
- swap the red and blue channels of the image (medium)
- blur only the non-flower parts of the image (harder)
Be creative! You may find it helpful to look over the OpenCV documentation to see what some of the possibilities are.
Attachment:- Assignment.zip