12th International Fall Workshop VISION, MODELING, AND VISUALIZATION 2007 November 7-9, 2007 Saarbrücken, Germany

Using Quadtrees for Energy Minimization Via Graph Cuts Cristina N. Vasconcelos1, Asla M. Sá2, Paulo Cezar Carvalho3, Marcelo Gattass1,2 (PUC-Rio1, Tecgraf2, IMPA3)

Graph Cuts Largely used for solving optimization problems formulated as labeling problems. The energy functions to be optimized are generally represented as [Boykov & Kolmogorov]:

E ( L) = ∑ D p ( L p ) + p∈P

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∑V

p , q∈N

p ,q

( L p , Lq )

2

Graph Structure Nodes: • a node for each pixel of the image. • special nodes (terminals), for each possible label. Edges: • connecting each pixel with each label. • connecting each pixel to its neighbors.

Max-flow : binary labeling case 14:12

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Grouping similar neighbors Why should we group similar neighbors? • we assume that similarity between neighbors means that we want to associate them with the same label; • having similar data, they have similar Dp cost; • when associated to the same label, the value of the edges connecting two neighbors is 0 (Vp,q).

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Similarity regions

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Grouping similar neighbors Adjusting the Energy Function:

E ( L) = ∑ α r . Dr ( Lr ) + where:

r∈R

∑β

r , s∈N

r ,s

.Vr , s ( Lr , Ls )

α: similarity region r area β: common border perimeter between similarity regions r and s

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Grouping similar neighbors • Problems with grouping similar neighbors into maximal similarity areas: How to efficiently... (ordered in increase order of difficult) – compute them? – compute their areas? – compute the perimeters of common borders? – compute their neighborhood system?

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Quadtrees Our proposal:

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Quad Graph

Nodes: • terminals nodes. • a node representing each leaf. Edges: • connecting each leaf with each label. • connecting each leaf to its neighbors. 14:12

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Graph Cuts using Quadtree Leaves Energy Function:

E ( L) = ∑ α .Dt ( Lt ) + t∈T

∑ β.V

t ,u∈N

t ,u

( Lt , Lu )

where: T: is the set of the leaves of the quadtree N: is the quadtree leaves neighborhood system

α = (2 ) β = 2 min( level (t ), level (u )) level 2

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Quadtree on GPU • Image Pyramid

(simpler than [Ziegler et al]: we do not construct a leaves list)

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Quadtree on GPU • Identifying the real leaves:

Bottom-up approach: checking for each leaf candidate, its parent textel.

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Quadtree leaves leaves size: • • • • • • • •

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1x1 2x2 4x4 8x8 16x16 32x32 64x64 128x128

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Sample application: background/foreground segmentation

• Active illumination used to produce initial seed [Sá et al.]:

E ( L) = ∑ R p ( L p ) + p∈P

∑B

p , q∈N

p ,q

( L p , Lq )

• Rp(xp) measures the fitness to the background or the foreground. • Bp(xp,xq) penalizes discontinuities in the label assignment 14:12

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Sample application The Regional Term: ⎧⎪− log( PO ( p )) R(l p ) = ⎨ ⎪⎩− log( PB ( p )) PB ( p ) =

1 2π σ L

exp(

if l p = 1 if l p = 0

− LI 2 ( p ) − LI1 ( p ) 2σ

2

2 L

)

nk PO ( p ) = nO

The Boundary term: B( p, q ) = l p − lq log(1 − exp(

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− Lab( p ) − Lab(q ) 2σ

2 C

2

))

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Sample application • Background Probability & Seed

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Sample application • Input image in Lab space(ab) & Histogram

(adapting [Scheuermann and Hensley])

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Sample Application • Stratified Texture:

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Quadtree leaves leaves size: • • • • • • • •

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1x1 2x2 4x4 8x8 16x16 32x32 64x64 128x128

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Sample application processing pipeline lit image unlit image

RGB to Lab Conversion stratified texture

luminance difference (Gaussian)

GPU

object region seed

construct quadtree object distribution (histogram)

CPU construct Quad-Cut graph

energy minimization

GPU image reconstruction

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image composition

20

Results

• Input images 14:12

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Results

• Seed mask 14:12

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Results

• Output mask 14:12

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Results • Composition 800x600 pixels 9,556 leaves(2%)

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Results

• Input images 14:12

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Results

• Seed Mask 14:12

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Results

• Output mask 14:12

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Results

• Composition 800x600 pixels 30,036 leaves (6%) 14:12

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Results Processing step Energy function on GPU:

in milliseconds

RGB to Lab Background Probability Histogram Quadtree on GPU: Pyramid Construction Quadtree Leaves Isolation Quadtree on CPU: Reading Texture to CPU Leaf list Neighborhood Graph-Cut Minimization Answer Reconstruction

<1 <1 15

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4.7 <1 15 16 14 1 16

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Results Quality measure: Thresholding the similarity metric: – in the limit, the traditional graph cut problem is reached; Sample application: – errors due to active segmentation formulation;

Number of generated leaves: • choosing de similarity metric appropriately; 14:12

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References Y. Boykov and V. Kolmogorov. An Experimental Comparison of MinCut/Max-Flow Algorithms for Energy Minimization in Computer Vision. In Proc. of Int’l Workshop Energy Minimization Methods in Computer Vision and Pattern Recognition, 2001. G. Ziegler, R. Dimitrov, C. Theobalt and H.-P. Seidel. Real-time Quadtree Analysis using HistoPyramids. In IS&T and SPIE Conference on Electronic Imaging, 2007. A. Sá, M. B. Vieira, A. Montenegro, P. C. Carvalho and L. Velho. Actively Illuminated Objects using Graph-Cuts. In Proceedings of SIBGRAPI, 2006. T. Scheuermann and J. Hensley. Efficient histogram generation using scattering on GPUs. In SI3D ’07: Proceedings of the 2007 symposium on Interactive 3D graphics and games, 33–37, 2007. O. Fluck, S. Aharon, D. Cremers and M. Rousson. GPU histogram computation. In SIGGRAPH ’06: ACM SIGGRAPH 2006 Research posters, 53, 2006.

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Using Quadtrees for Energy Minimization Via Graph Cuts

How to efficiently... (ordered in increase order of difficult). – compute them? – compute their areas? – compute the perimeters of common borders? – compute their neighborhood system? Page 8. 14:12. 8. Quadtrees. Our proposal: Page 9. 14:12. 9. Quad Graph. Nodes: • terminals nodes. • a node representing each leaf.

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