2. (Affine and metric rectification) Consider the image attached.
Read the image into matlab and use matlab to perform the calculations
necessary for the following questions:
(a) Calculate two vanishing points.
(b) Find the image of the line at infinity
(

Stereopsis
Mark Twain at Pool Table", no date, UCR Museum of Photography
Woman getting eye exam during immigration procedure at Ellis
Island, c. 1905 - 1920, UCR Museum of Phography
Why Stereo Vision?
2D images project 3D points into 2D:
P
Q
P=Q
O
3DPoi

Some useful linear algebra
Linearly independent vectors
1v1 + 2 v2 + 3v3 + + i vi = 0 only for 1 = 2 = = i = 0
span(V): span of vector space V is all linear combinations of vectors vi,i.e. v
i i i
Ax = x (I - A) x = 0 hence (I - A) is singular
The eigenv

Resampling
Image Scaling
This image is too big to
fit on the screen. How
can we reduce it?
How to generate a halfsized version?
Image sub-sampling
1/8
1/4
Throw away every other row and
column to create a 1/2 size image
Why does this look so crufty?
Call

Projective geometry
Ames Room
Slides from Steve Seitz and Daniel DeMenthon
Projective geometrywhats it good for?
Uses of projective geometry
Drawing
Measurements
Mathematics for projection
Undistorting images
Focus of expansion
Camera pose estimation, mat

Motion from normal flow
Optical flow difficulties
The aperture problem
Depth discontinuities
Translational Normal Flow
u tr
un =
n
Z
In the case of translation each
normal flow vector constrains
the location of the FOE to a
half-plane.
Intersection of h

Image Motion
The Information from Image
Motion
3D motion between observer and scene + structure of the
scene
Wallach OConnell (1953): Kinetic depth effect
http:/www.biols.susx.ac.uk/home/George_Mather/Motion/KDE
.HTML
Motion parallax: two static point

Image formation 2
Blur circle
Points a t distance z are brought into focus at distance z '
A point at distance z
is imaged at point z ' from the lens
1
1
1
+
=
z' z f
and so
-z
z
P
Q
Q
d
b
f
f
z ' z ' =
( z z)
(z + f ) (z + f )
Thus points at distance z w

Image formation
Image Formation
Vision infers world properties form images.
How do images depend on these properties?
Two key elements
Geometry
Radiometry
We consider only simple models of these
Camera Obscura
"When images of illuminated objects . p

Vision infers world properties form images.
How do images depend on these properties?
Two key elements
Geometry
Radiometry
We consider only simple models of these
"When images of illuminated objects . penetrate through a
small hole into a very dark

Fourier Transform
Analytic geometry gives a coordinate
system for describing geometric objects.
Fourier transform gives a coordinate system
for functions.
Decomposition of the image
function
The image can be decomposed into a weighted sum of
sinusoids a

Filte ring
Most slides from Steve Seitz
1
What is an imag e ?
We can think of an imag e as a function, f, from R2 to R:
f( x, y ) gives the inte ns ity at position ( x, y )
Realistically, we expect the image only to be
defined over a rectangle, with a f

Epipolar geometry
Three questions:
(i) Correspondence geometry: Given an image point x in the first view, how does this constrain the position of the corresponding point x in the second image? Camera geometry (motion): Given a set of corresponding image p

3D Motion Estimation
3D model construction
3D model construction
Video Manipulation
Visual Motion
Allows us to compute useful properties of the 3D world, with very little knowledge. Example: Time to collision
Time to Collision
v L
l(t)
L t=0
An obje of h