Title: Convex Quadratic Programming for Object Location
1Convex Quadratic Programming for Object Location
- Hao Jiang, Mark S. Drew and Ze-Nian Li
- School of Computing Science
- Simon Fraser University
2Introduction
- Object localization is an important task in
computer vision
Template
3Object localization and labeling
- Object localization can be formulated as labeling
problems. - Consistent labeling
- Find small cost label assignment.
- Enforce labeling consistency of neighboring
sites.
Label
Site
c(p,fp)
p
fp
Neighboring relation
(fq-fp)
(q-p)
q
fq
c(q,fq)
Template
Target Object
4Previous methods for consistent labeling
- Consistent labeling, in general form, is NP-hard.
- Polynomial time schemes exist for special cases
- Dynamic Programming (DP).
- Max-flow Ishikawa 2000, Roy 98.
- Approximation schemes
- Greedy (local searching) methods
- Relaxation labeling (RL) Rosenfeld 76.
- Iterated Conditional Modes (ICM) Besag 86.
- Need good initialization and easily
trapped in local minimum. - Global searching methods
- Graduated Non-Convexity (GNC) Blake Zisserman
87 - Belief Propagation (BP) Pearl 88, Weiss 2001.
- Graph Cut (GC) Boykov Zabih 2001.
5Focus of the research
- Previous methods become slow as the number of
labels goes to the order of several thousand.
( of labels can be in the order of thousand or
bigger) Many vision problems such as object
matching, large scale motion, tracking etc will
benefit from the solver.
Hard to solve
The focus of the research
( of labels in the order of hundred)
Well solved
Consistent labeling
6The trick of the proposed scheme
- In stead of working on the original label space,
we represent the label set with a small number
basis labels. - We convert the hard problem into a sequence of
simpler problems built using only the basis
labels. - In this way, the size of the approximation
problem is largely decoupled from the original
label set. - Each sub-problem is a convex problem and can be
globally optimized. - A successive relaxation implementation is used to
zero in the target.
7The non-linear optimization problem
- The labeling problem can be solved by optimizing
p
fp
c(p,fp)
(fq-fp)
(q-p)
q
fq
c(q,fq)
8Convex relaxation
- To convert the labeling cost term into linear
functions, for each site, we define a basis label
set. - Each label can then be represented as a linear
combination of the basis labels. - The cost of the label is approximated by the
linear combination of the costs of these basis
labels.
c(s,t)
fs a J1 b J2 c(s,fs) a c(s,J1) b
c(s,J2) a b 1
t
J1
J2
fs
9Convex relaxation (Cont)
- The L2 norm smoothness terms do not need
additional conversions.
c(s,j1)xs,j1 c(s,j2) xs,j2 c(s,j3) xs,j3
c(s,j4) xs,j4 c(s,j5) xs,j5
c(s,j3)
c(s,j2)
xs,j1 xs,j2 xs,j3 xs,j4 xs,j5 1
j2
j3
c(s,j4)
c(s,j1)
c(s,j5)
j4
j5
j1
j1 xs,j1 j2 xs,j2 j3 xs,j3 j4 xs,j4 j5 xs,j5
10Convex quadratic program (CQP)
We have the convex quadratic program
11In some cases, the CQP is exact
- If x are binary numbers, the mixed integer
program is exactly equivalent to the original
problem. - If c(s,t) is convex over t for each s 2 S, the
CQP is equivalent to the continuous extension of
the original problem.
c(s,t)
c(s,t)
Continuous extension
t
t
Feasible solution
Feasible solution
12In general it is an approximation
- For general problems, the CQP approximates each
labeling cost surface with the 3D lower convex
hull.
Label cost
c(s,t)
y
t (x,y)
x
The convexified surfaces are much simpler than
the original ones.
13The CQP can be greatly simplified
- The most compact basis labels correspond to the
lower convex hull vertices. - This shows a way to simplify the label set in
labeling. - We can safely discard many labels without
worrying about the problems met in previous
methods.
of original label 400 of basis label 20
14Successive convexification
To improve the approximation we use the
successive convexification (SC) scheme as follows
15Experimental results
Matching Random Dots
16Experimental results (Cont)
Matching Random Dots
17Experimental results (Cont)
18Experimental results (Cont)
19Experimental results (Cont)
Matching Hand
20Conclusion
- Set out an object localization method which can
deal with textureless objects in strong
background clutter. - Propose a successive convex quadratic method
- CQP large decouples the number of target image
points with the size of the convex program It
searches the whole target image quickly. - Successive convexification is studied to improve
the result recursively. - Successive convexification is a general method
can be applied to any convex regularization
problem.
21Thank you!
22In summery
- Label space is approximated with small number of
basis labels. - Original problem is converted to a sequence of
much easier convex programs and solved by
successive relaxation. - The size of the convex program is largely
decoupled with the number of candidate labels
Matching Costs
Lower Convex Hull
Template
Target Image
Convexification Trust Region
Shrinking