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

[Illustrate projection here]

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.)

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

Rotation in 3D This is a rotation around the z axis: cos 𝜃 −sin 𝜃 0 sin 𝜃 cos 𝜃 0 0 0 1 What axis is this a rotation around? In what direction? cos 𝜃 0 sin 𝜃 0 1 0 −sin 𝜃 0 cos 𝜃

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.

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