Segmentation with Pairwise Attraction and Repulsion
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1 Segmentation with Pairwise Attraction and Repulsion Stella. Yu Jianbo Shi Robotics Institute Carnegie Mellon Universit {stella.u, Funded b DARPA HumanID ONR N4---95
2 Grouping: decomposition into regions of coherent properties Figure-ground: decomposition into foreground and background [Nitzberg et al 93] [Cann 86] [Horowitz & Pavlidis 74] [Geman & Geman 84] [Blake & Zisserman 87] [Zhu & Yullie, 95] [Shi & Malik 97] [Grossberg & Mingolla 85] [Belhumeur 96] Grouping Figure-ground [Herault and Horaud, 93] [Chou, 95] [Geiger & Kumaran, 96] [Williams & Jocobs, 97] [Amir & Lindenbaum, 98] [Liu & Wang, 99] Issues Fundamental: segmentation and figure-ground reasoning is circular Practical: lack of efficient computational techniques 7/3/3
3 Figure-ground cues for segmentation Figure-ground organization Segmentation Figure-ground labelling Between-region relations can influence the internal organization of a region Roles of figure-ground cues: Link incoherent background or foreground together Global binding through local cues 7/3/3 3
4 alk overview Segmentation in a graph framework Ordered graph partitioning Representing pairwise relationships Attraction and repulsion Criteria Optimize global configuration of figure vs ground Energ formulation Raleigh quotient of Hermitian matrices Efficient solution technique Sparse matrix eigendecomposition Results 7/3/3 4
5 Segmentation in a graph framework G(V, E, W V: each node denotes a pixel E: each edge denotes a pixel-pixel relationship W: each weight measures pairwise affinit Segmentation vertex partitioning break V into disjoint sets V, V Pairwise affinit, W: similarit of pixel attributes relative order of figure-ground Goal: ordered partitioning V figure, V ground Criteria: within-region similarit: high between-region similarit: low order from figure to ground: high 7/3/3 5
6 7/3/3 6 Generalized affinit: attraction and repulsion Attraction A feature similarit feature location, brightness, textureness, motion, the larger the A jk, the more likel j and k in one region Directional repulsion R relative order of figure-ground the larger the R jk, the more likel j and k in different regions Generalized Affinit W A i R A 5-node graph example W is Hermitian image A b proxmit A b brightness similarit R for a figure pixel R for a ground pixel Relative depth from -junctions A R A is smmetric R is skew-smmetric
7 Criteria: cuts, associations and difference flows within-region similarit within-group attraction associations assoc ( P u P, v P A( u, v between-region similarit between-group attraction cuts cut ( P, Q u P, v Q A( u, v, P Q Φ order from figure to ground figure-to-ground repulsion difference flow difflow ( P, Q R( u, v, u P, v Q P Q Φ 7/3/3 7
8 Criteria: generalized normalized association Maximize within-region similarit and figure-to-ground order within figuresimilarit Nassoc total figure affinit within ground similarit total ground affinit figure ~ ground order averagetotal affinit assoc( V Nassoc ( V, V deg( V assoc( V deg( V difflow deg( V ( V, V deg( V 7/3/3 8
9 Criteria: generalized normalized cuts Minimize between-region similarit and ground-to-figure order figure~groundsimilarit figure~groundsimilarit Ncut totalfigureaffinit totalgroundaffinit ground~figureorder averagetotalaffinit cut ( V, V Ncut ( V, V deg( V cut ( V, V deg( V difflow deg( V ( V, V deg( V 7/3/3 9
10 Properties of the criteria Dualit to achieve both goals at the same time Ncut Nassoc arg min Ncut constant arg max Nassoc Normalization for global structures Min-cut Ncut 7/3/3
11 7/3/3 Energ formulation, ( D A Nassoc D A D D R Variables: group indicators Energ functions of indicator variables l l l V u V u u,, (, n n n n [ ] [ ] [ ]
12 7/3/3 Energ formulation deg( deg(, ( V V k k k.. ( D t s D A Nassoc, ( D A Nassoc D A D D R deg( deg(, V V k k i k.. ( ( D t s D R i A Nassoc H H Change of variables Raleigh quotient
13 hree aspects of solutions Efficient solutions in the continuous domain: Eigenvector corresponding to the largest eigenvalue of (Ai R, D H ( A i R opt arg max ( A i R H opt Little increase in complexit Hermitian matrices preserve the most important properties of real smmetric matrices in eigendecomposition. Sparse matrix eigendecomposition O(n 3/, n # of pixels D λ D opt Interpretation of complex-valued solutions Separation in phase plane indicates the separation of groups Relative phase advance indicates figure-ground ordering 7/3/3 3
14 Segmentation as embedding Assign region identit. Ever pixel corresponds to a point in a high dimensional feature space. color location opt u M u n grouping Segmentation is about finding an embedding of manifolds into some D or D space. u textureness figure-ground v brightness opt u M u n i v i v n u 7/3/3 4
15 Interaction of attraction and repulsion Image Attraction Repulsion Phase Result: Plot A Result: R Result: A and R 7/3/3 5
16 Objects ordered in depth /3/3 6
17 Figure-ground example Oversegmentation Image Solutions: A based on A R Figure Ground Ground Phase plot 7/3/3 7
18 Conclusions Representation Attraction directional repulsion Formulation Ordered graph partitioning for figure and ground Raleigh quotient of Hermitian matrices Solution Efficient algorithm b eigendecomposition Embedding in the phase plane of complex numbers Benefits Grouping and figure-ground in one step Computational efficienc 7/3/3 8
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