1. CS-498 Computer Vision Week 7, Day 1 3-D Geometry
4/24/2017
CS-498 Computer Vision
Week 7, Day 1
3-D Geometry
Projection onto a camera image
Projection as a matrix
Rotation and Translation in 3D
Homography as an “image of image” transform
Print
Dr. Josiah Yoder

2. [Illustrate projection here]

3. The equations for projection onto the image plane are:
i = x/z j = y/z These can be written as a homography… 1. Write a transform that maps x to i, j to y, and does not destroy z 2. Treat the result as a homographic point. (Note that the original wasn’t.)

4. Transforms in 3D
𝑥 𝑛𝑒𝑤 𝑦 𝑛𝑒𝑤 𝑧 𝑛𝑒𝑤 = ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ 𝑥 𝑜𝑙𝑑 𝑦 𝑜𝑙𝑑 𝑧 𝑜𝑙𝑑

5. Rotation in 3D
This is a rotation around the z axis: cos 𝜃 −sin 𝜃 0 sin 𝜃 cos 𝜃 What axis is this a rotation around? In what direction? cos 𝜃 0 sin 𝜃 −sin 𝜃 0 cos 𝜃

6. We can have homographic 3D points, too
Exercise: Consider the equations xnew = xold + tx ynew = yold + ty znew = zold + tz Write the right-hand side of this equation as a matrix multiplication.

7. Homography as a “picture of a picture”
Suppose we take a picture of a picture.
The original picture is on a plane, and we can represent points on that plane as
𝑖 𝑐𝑎𝑚 𝑗 𝑐𝑎𝑚 "1" = 𝑖 𝑜𝑙𝑑 𝑗 𝑜𝑙𝑑 0 1