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

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[Illustrate projection here]

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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.)

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Transforms in 3D

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Rotation in 3D

This is a rotation around the z axis:

What axis is this a rotation around? In what direction?

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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.

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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

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