Basic Multi-View Geometry - Lecture Notes | CAP 5415, Study notes of Computer Science

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Lecture 15: Basic Multi-View Geometry

Stereo

● If I needed to find out how far point is away from me, I could use triangulation and two views scene pointscene point optical centeroptical center image plane image plane (Graphic from Khurram Shaffique)

Today

● For the rest of the lecture we will talk about the geometry of multiple views ● To begin we will talk about epipolar geometry (Image from Forsyth and Ponce)

Epipoles

● The projection of the optical centers of each camera (Image from Forsyth and Ponce) epipole epipole

When the cameras are calibrated

● The vectors , , and are coplanar ● Can be expressed as (Image from Forsyth and Ponce)

How you can see this

● The vector returned by the cross product is perpendicular to the two vectors ● Can be thought of as a normal to a plane ● If the lines in the plane, it should also be perpendicular to that normal (Image from Forsyth and Ponce)

Now, rewrite constraint in terms of

the coordinates of the left camera

For simplicity:

The essential matrix

● Starting with: ● A cross product can be rewritten as a matrix multiplication, leading to the constraint is called the essential matrix

What if the cameras aren't

calibrated

● The relationship still holds, but we have to calibrate the cameras first. ● Those calibration matrices, combined with the essential matrix are known as the fundamental matrix ● Encodes information from the intrinsic and extrinsic parameters ● Also Rank 2

Finding the fundamental matrix

● Basic algorithm: 8-point algorithm ● Find 8 corresponding points in the images ● Once you have the corresponding p and p' points, ● Is linear in F c

Adding more views

● Text also describes the geometry of 3 or more views (From Forsyth and Ponce)

How do we use the fundamental

matrix?

● It can tell us where to look for points in the other image ● The quantity is a vector ● So, defines a line ● This tells us where to look for the point that corresponds to p ● (Demo) c

The Goal

● Interpolate between views to get new view Figure from Dyer and Seitz

Can't use normal interpolation

Figure from Dyer and Seitz