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Exact and Partial Energy Minimization in Computer Vision Alexander - - PowerPoint PPT Presentation
Exact and Partial Energy Minimization in Computer Vision Alexander - - PowerPoint PPT Presentation
Ph.D. Thesis Exact and Partial Energy Minimization in Computer Vision Alexander Shekhovtsov Supervisor: Vclav Hlav Supervisor-specialist: Tom Werner Czech Technical University in Prague Faculty of Electrical Engineering, Department
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Minimum Cut Problem
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Introduction
CPU Mem CPU Mem Quick Quick Slow Slow
Distributed Sequential
CPU Mem Quick Quick Slow Slow Disk
Distributed Parallel
Distributed Model Distributed Model – Divide Computation AND Memory Divide Computation AND Memory
Solve large problem on a single computer
- n more computers
Split data in parts:
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Introduction
Sequential Algorithms
Parallel Algorithms Our goal is to improve on thi Our goal is to improve on this
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Main Result
Main Result: Main Result:
t
s
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Main Idea
New distance function length of the path = number of boundary edges distance = length of a shortest path to the sink
d∗B(u) = 2 d∗B(v) = 0 t
corresponds to costly operations Algorithm: push-relabel between regions, augmenting path inside regions
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Experimental Confirmation
push-relabel (with heuristics) proposed method Synthetic instances: Grid graph with random capacities, partitioned into 4 regions sweep = synchronously send messages on all boundary arcs
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Instances in Computer Vision
Dataset Published by Vision Group Dataset Published by Vision Group at University of Western Onta at University of Western Ontario rio
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speedup: Ours/BK
Sequential Variant for Limited Memory Model
sometimes faster than BK (CPU time excluding I/O)
robust over partition size
uses BK (Boykov and Kolmogorov) inside regions
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Messages (sweeps) speedup over push-relabel (distributed version of Delong and Boykov 2008)
Sequential Variant for Limited Memory Model
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Parallel Variant
Competitive with shared memory model methods
Speedup bounded by memory bandwidth
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Conclusion
Ne New distribute w distributed algo d algorithm rithm
Terminates in at most B Terminates in at most B2+1 sweeps (few in practice) +1 sweeps (few in practice)
Sequential Sequential Algorithm Algorithm 1) competitive with sequential solvers 2) uses few sweeps (= loads/unloads of regions) 3) suitable to run in the limited memory model
Pa Parallel Algorithm rallel Algorithm 1) competitive with shared memory algorithms 2) uses few sweeps (= rounds of message exchange) 3) suitable for execution on a computer cluster
Implementation can be specialized for regular grids (less memory/faster)
(?) no good worst case complexity bound in terms of elementary operations
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Discrete Energy Minimization Problem
s
t
fst(1, 1)
ft(1) fs(0)
fst(0, 0)
fst(1, 0) fst(0, 1)
x
t0
s
t
xs
xt
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Partial Optimality
Energy model for stereo, minimization NP-hard Find optimal solution in some some pixels solution unknown solution globally optimal
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s
t t0
y
Partial Optimality
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Partial Optimality (Multilabel)
s
t t0
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Overview
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Dead End Elimination
Desmet et al. (1992), Goldstein (1994), Lasters et al. (1995),Pierce et al. (2000), Georgiev et al. (2006)
s
t t0
y
t00
α
β
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Improving Mapping
s
t t0
1 2 3
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Linear Embedding
s
t
x
t0
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Linear Embedding
s
t
x
t0
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Linear Embedding of Maps
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Linear Embedding of Maps
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Linear Embedding of Maps
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Linear Embedding of Maps
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More General Projections/Maps
s
t α 3
6
7
6 7
3
0.5 0.5 Example of fractional map
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Λ-Improving Characterization
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Special Cases
DEE conditions by Desmet (1992) and Goldstein (1994)
(Weak/strong) Persistency in Quadratic Pseudo-Boolean Optimization (QPBO) by Nemhauser & Trotter (1975), Hammerr et al. (1984), Boros et
- al. (2002)
Multilabel QPBO Kohli et al. (2008), Shekhovtsov et al. (2008)
Submodular Auxiliary problems by Kovtun (2003, 2010)
Iterative Pruninig by Swoboda et al. (2013) Methods that can be explained by the proposed condition: Common properties, Only (M)QPBO was previously related to LP relaxation
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Maximum Λ-Improving Projections
Problem: Problem: Find the mapping that maximizes domain reduction
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MINCUT [1] Nemhauser & Trotter (1975), Hammerr et al. (1984), Boros et al. (2002) [2] Picard & Queyranne (1977) (Vertex Packing)
Maximum Λ-Improving Projections
Follow-up work, submitted to CVPR
Thesis
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Conclusions
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Higher Order
- Adams, W. P., Lassiter, J. B., and Sherali, H. D. (1998). Persistency in
0-1 polynomial programming.
- Kolmogorov, V. (2012). Generalized roof duality and bisubmodular
functions.
- Kahl, F. and Strandmark, P. (2012). Generalized roof duality.
- Lu, S. H. and Williams, A. C. (1987). Roof duality for polynomial 0-1
- ptimization.
- Ishikawa, H. (2011). Transformation of general binary MRF
minimization to the first-order case.
- Fix, A. et al. (2011). A graph cut algorithm for higher-order Markov
random fields.
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Generalized Potts (5 labels) Fully Random (4 labels)
Experiments: solution completeness on random problems
Algorithm proposed:
New
Follow-up Work
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