Higher-order Graph Cuts

Transcription

1 ACCV2014 Area Chairs Workshop Sep. 3, 2014 Nanyang Technological University, Singapore 1 Higher-order Graph Cuts Hiroshi Ishikawa Department of Computer Science & Engineering Waseda University

2 Labeling problem Original Noise-added Denoised Assign a label to each pixel Pixel Label 2

3 Energy minimization Original Noise-added Denoised, Globally optimizeable using graph cuts Greig, Porteous and Seheult 89 3

4 First-order (pairwise) energy Good (Low Energy) Bad (High Energy) 12 Bad 12 Bad 40 Good 40 Good 4

5 Higher-order energy Good (Low Energy) Bad (High Energy) Better (Lower Energy) Worse (Higher Energy) A B C D 5

6 Higher-order energy Better (Lower Energy) Worse (Higher Energy) A B C D A: 15 B: 12 C: 1 D: 0 A: 16 B: 6 C: 5 D: 1 6

7 Higher-order energy,,,, Better (Lower Energy) Worse (Higher Energy) A B C D 7

8 Functions of binary variables Pseudo-Boolean function (PBF) Function of binary ( or ) variables Can always write it uniquely as a polynomial One variable : Two variables : Three variables : th order binary MRF = th degree PBF 8

9 Reducing higher-order energy Convert any higher-order binary energy C into an equivalent first-order energy Adds variables More than 2 labels Fusion moves Ishikawa CVPR 2009, PAMI

10 New! Convert many higher-order binary energies C into an equivalent first-order energy Without adding variables How? Ishikawa CVPR

11 Example: a cubic term :cubic (3 rd degree),, coefficient: Define new function: i.e., value is added only when 11

12 Example: a cubic term Define new function: i.e., value is added only when New coefficient (replace with ): So is now quadratic (2 nd degree) Similarly, we can reduce the degree by changing one of the 8 possible values But and are different functions! 12

13 When can we do this?,, Different only when Suppose is a potential in are three of the variables in If, and For minimizer of,, then, we can replace with without changing the minimizer 13

14 ELC Excludable Local Configuration (ELC) A (usually) locally-testable sufficient condition for local configuration to be not part of global minimizer Excludable as a part of global minimizer ELC may not exist May take time to find Approximation Just use the local configuration the largest value with 14

15 Experiment: 4 th deg. FoE denoising Original Noise added 4 th degree 15

16 Experiment: 4 th deg. FoE denoising Number of variables after conversion 0 ELC+HOCR ELC+Fix et. al Approx. ELC HOCR Fix et. al 16

17 Experiment: 4 th deg. FoE denoising Energy ELC+HO CR ELC Approx. HOCR Fix et al Iterations 17

18 Experiment: 4 th deg. FoE denoising Energy HOCR ELC+HO CR Fix et al. ELC Approx Time (sec.) 18

19 Experiment: 4 th deg. FoE denoising % pixels labeled ELC Approx. 100 ELC+HOCR Fix et al HOCR Iterations 19

20 Experiment: 4 th deg. FoE denoising ELC exists: 3 rd degree: 96.12% 4 th degree: 99.60% Approximation: Guessed configuration is in fact an ELC 3 rd degree: 88% of the time 4 th degree: 97% of the time Even if it is not an ELC, it is not part of maximizer 3 rd degree: 99.98% 4 th degree: % % of (labeled) pixels correctly labeled 20

21 Conclusion Higher-order energy minimization Unary, Pairwise, Triple,. Binary labels: reduce to first order Then use graph cuts Multiple labels: use Fusion Move Reducing to first order Before: add variables New: no variables added Faster Much less memory Code available: 21