convex quadratic programming for object location
DESCRIPTION
Convex Quadratic Programming for Object Location. Hao Jiang, Mark S. Drew and Ze-Nian Li School of Computing Science Simon Fraser University. Introduction. Object localization is an important task in computer vision. Template. Object localization and labeling. - PowerPoint PPT PresentationTRANSCRIPT
Convex Quadratic Programming for Object Location
Hao Jiang, Mark S. Drew and Ze-Nian Li
School of Computing ScienceSimon Fraser University
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Introduction Object localization is an important task in computer vision
Template
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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.
p
q
fp
fq
c(p,fp)
c(q,fq)Template Target Object
Site Label
Neighboring relation
(q-p) (fq-fp)
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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 methodsGraduated Non-Convexity (GNC) [Blake & Zisserman 87] Belief Propagation (BP) [Pearl 88, Weiss 2001].Graph Cut (GC) [Boykov & Zabih 2001].
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Focus of the research
(# of labels in the order of hundred)
(# of labels can be in theorder of thousand or bigger)
Many vision problems such as object matching, large scale motion,
tracking etc will benefit from the solver.
Wellsolved
Hard to solve The focus of
the research
Consistent labeling
Previous methods become slow as the number of labels goes to the order of several thousand.
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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.
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The non-linear optimization problem
The labeling problem can be solved by optimizing:
p
q
fp
fq
c(p,fp)
c(q,fq)
(q-p) (fq-fp)
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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.
t
c(s,t)
J1 J2fs
fs = a * J1 + b * J2
c(s,fs) = a * c(s,J1) + b * c(s,J2)a + b = 1
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Convex relaxation (Cont’)
The L2 norm smoothness terms do not need additional conversions.
j2 j3
j4j5j1j1 s,j1
+ j2 s,j2+
j3 s,j3
+ j4 s,j4
+ j5 s,j5
c(s,j1)
c(s,j2)
c(s,j3)
c(s,j4)c(s,j5)
c(s,j1)s,j1+ c(s,j2) s,j2
+ c(s,j3) s,j3
+ c(s,j4) s,j4
+ c(s,j5) s,j5
s,j1+ s,j2
+ s,j3
+ s,j4
+ s,j5
=1
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Convex quadratic program (CQP)
We have the convex quadratic program:
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In some cases, the CQP is exact
If 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.
t t
c(s,t) c(s,t)
Feasible solution Feasible solution
Continuousextension
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In general it is an approximation
For general problems, the CQP approximates each labeling cost surface with the 3D lower convex hull.
t =(x,y)
c(s,t)
The convexified surfaces are much simpler than the original ones.
x
y
Label cost
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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
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Successive convexification
To improve the approximation we use the successiveconvexification (SC) scheme as follows:
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Experimental results
Matching Random Dots
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Experimental results (Cont’)
Matching Random Dots
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Experimental results (Cont’)
Matching Leaf
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Experimental results (Cont’)
Matching Face
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Experimental results (Cont’)
Matching Hand
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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.
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Thank you!
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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
Convexification Trust Region Shrinking
Template
Target Image
Matching Costs
LowerConvexHull