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Convex Quadratic Programming for Object Location

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Title: Convex Quadratic Programming for Object Location


1
Convex Quadratic Programming for Object Location
  • Hao Jiang, Mark S. Drew and Ze-Nian Li
  • School of Computing Science
  • Simon Fraser University

2
Introduction
  • Object localization is an important task in
    computer vision

Template
3
Object 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
4
Previous 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.

5
Focus 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
6
The 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.

7
The 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)
8
Convex 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
9
Convex 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
10
Convex quadratic program (CQP)
We have the convex quadratic program
11
In 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
12
In 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.
13
The 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
14
Successive convexification
To improve the approximation we use the
successive convexification (SC) scheme as follows
15
Experimental results
Matching Random Dots
16
Experimental results (Cont)
Matching Random Dots
17
Experimental results (Cont)
  • Matching Leaf

18
Experimental results (Cont)
  • Matching Face

19
Experimental results (Cont)
Matching Hand
20
Conclusion
  • 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.

21
Thank you!
22
In 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
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