# CAP 5415 Computer Vision Fall 2004

AI and Robotics

Oct 18, 2013 (4 years and 8 months ago)

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Alper Yilmaz, Fall 2004 UCF

CAP 5415 Computer Vision

Fall 2004

Dr. Alper Yilmaz

Univ. of Central Florida

www.cs.ucf.edu/courses/cap5415/fall2004

Office: CSB 250

Alper Yilmaz, Fall 2004 UCF

Recap

Epipolar Geometry

Defines relation between camera coordinates

Essential matrix

Defines relation between image coordinates

Fundamental matrix

epipole

epipole

x
l

x
r

C
l

C
r

P

Co
-
planarity condition

Alper Yilmaz, Fall 2004 UCF

Recap

Essential Matrix

Related to camera extrinsic parameters

Defines relation between to camera coordinates

Essential matrix

Alper Yilmaz, Fall 2004 UCF

Recap

Fundamental Matrix

Related to both intrinsic and extrinsic camera
parameters

Fundamental matrix

(x,y) point on left image

(x’,y’) point on right image

Alper Yilmaz, Fall 2004 UCF

Computing Fundamental Matrix

8
-
Point Algorithm

Requires 8 corresponding points in both images

Write unknown fundamental matrix parameters
(f1..f9) into vector and the rest into equation
(observation) matrix.

Alper Yilmaz, Fall 2004 UCF

8
-
Point Algorithm

Let there be N corresponding points in both images

Alper Yilmaz, Fall 2004 UCF

8
-
Point Algorithm

Due to 1s in the last column of O rank of O is 8

Homogenous equation, solution is given using SVD
(or eigenspace decomposition)

Perform eigenspace decomposition and select
minimum eigenvalued eigenvector as solution

Let solution be
e
min

Alper Yilmaz, Fall 2004 UCF

8
-
Point Algorithm

e
min

should satisfy rank 2 constraint.

Perform SVD on
E
min
.

Alper Yilmaz, Fall 2004 UCF

Enforcing Rank Constraint

Set
s
3

to 0

Recompute
E
min

fundamental matrix

Alper Yilmaz, Fall 2004 UCF

Scaling and Normalization in

8
-
Point Algorithm

It is important to normalize and scale the input points
(tokens on images)

Perform separately for both images

Take mean of x and y values,

x

and

y
.

Subtract mean from every point (translate origin to mean)

Divide x and y of each point by

x
/2 and

y
/2 (on the
average each point is (1,1)

Construct the observation matrix

Compute fundamental matrix

Alper Yilmaz, Fall 2004 UCF

Graph Theoretical Techniques for
Image Segmentation

Alper Yilmaz, Fall 2004 UCF

Region Segmentation

Alper Yilmaz, Fall 2004 UCF

Region Segmentation

Find sets of pixels, such that

All pixels in region
i

satisfy some constraint of
similarity.

Alper Yilmaz, Fall 2004 UCF

Graph

A graph G(V,E) is a triple consisting of a vertex set
V(G) an edge set E(G) and a relation that associates
with each edge two vertices called its end points

Alper Yilmaz, Fall 2004 UCF

Path

A path is a sequence of edges e
1
, e
2
, e
3
, … e
n
. Such
that each (for each i>2 & i<n) edge e
i

e
(i+1)

and e
(i
-
1)
. e
1

2

and e
n

is only
(n
-
1)
.

Alper Yilmaz, Fall 2004 UCF

Connected & Disconnected Graph

A graph
G

is connected if there
is a path from every vertex to
every other vertex in
G
.

A graph G that is not connected
is called disconnected graph.

Alper Yilmaz, Fall 2004 UCF

Graph Representations

a

e

d

c

b

Alper Yilmaz, Fall 2004 UCF

Weighted Graphs and Their
Representations

Weight Matrix: W

Alper Yilmaz, Fall 2004 UCF

Minimum Cut

A cut of a graph
G

is the set
of edges
S

such that
removal of
S

from
G
disconnects
G
.

Minimum cut is the cut of
minimum weight, where
weight of cut <A,B> is given
as

Alper Yilmaz, Fall 2004 UCF

Minimum Cut and Clustering

Alper Yilmaz, Fall 2004 UCF

Image Segmentation & Minimum Cut

Image

Pixels

Pixel

Neighborhood

w

Similarity

Measure

Minimum Cut

Alper Yilmaz, Fall 2004 UCF

Minimum Cut

There can be more than one minimum cut in a given
graph

All minimum cuts of a graph can be found in
polynomial time
1
.

1
H. Nagamochi, K. Nishimura and T. Ibaraki, “Computing all small cuts in an undirected
network. SIAM J. Discrete Math. 10 (1997) 469
-
481.

Alper Yilmaz, Fall 2004 UCF

Drawbacks of Minimum Cut

Weight of cut is directly proportional to the number of
edges in the cut.

Ideal Cut

Cuts with

lesser weight

than the

ideal cut

Alper Yilmaz, Fall 2004 UCF

Normalized Cuts
1

Normalized cut is defined as

N
cut
(A,B) is the measure of dissimilarity of sets A and B.

Minimizing N
cut
(A,B) maximizes a measure of similarity within
the sets A and B

1
J. Shi and J. Malik, “Normalized Cuts & Image Segmentation,” IEEE Trans. of PAMI, Aug 2000.

A

B

Alper Yilmaz, Fall 2004 UCF

Finding Minimum Normalized
-
Cut

Finding the Minimum Normalized
-
Cut is NP
-
Hard.

Polynomial Approximations are generally
used for segmentation

Alper Yilmaz, Fall 2004 UCF

Finding Minimum Normalized
-
Cut

Image

Pixels

Pixel

Neighborhood

w

Similarity

Measure

1

n

3

2

n
-
1

Alper Yilmaz, Fall 2004 UCF

Finding Minimum Normalized
-
Cut

Alper Yilmaz, Fall 2004 UCF

Finding Minimum Normalized
-
Cut

It can be shown that

such that

If
y

is allowed to take real values then the
minimization can be done by solving the generalized
eigenvalue system

Alper Yilmaz, Fall 2004 UCF

Algorithm

Compute matrices W & D

Solve
(D
-
W)y=

Dy

for eigenvectors with the
smallest eigenvalues

Use the eigenvector with second smallest
eigenvalue to bipartition the graph

Recursively partition the segmented parts if
necessary.

Alper Yilmaz, Fall 2004 UCF

Figure from “Image and video segmentation: the normalized cut framework”, by Shi and Malik, 1998

Alper Yilmaz, Fall 2004 UCF

Figure from “Normalized cuts and image segmentation,” Shi and Malik, 2000

Alper Yilmaz, Fall 2004 UCF

Alper Yilmaz, Fall 2004 UCF

Alper Yilmaz, Fall 2004 UCF

Drawbacks of Minimum Normalized Cut

Huge Storage Requirement and time
complexity

Bias towards partitioning into equal
segments

Have problems with textured backgrounds

Alper Yilmaz, Fall 2004 UCF