Minimizing Sparse Higher Order Energy Functions of Discrete Variables (CVPR’09) Namju Kwak Applied Algorithm Lab. Computer Science Department KAIST 1Namju Kwak, AALab, KAIST

Topics Abstract Labeling Problem Notations Transforming Multi-label Functions (Simple Case) Transforming Multi-label Functions (General Case) Transforming Multi-label Functions (Compact Parameterization) Transforming Pseudo-Boolean Functions Conclusion Namju Kwak, AALab, KAIST2

Abstract The higher order functions in CV are very often “sparse”. Here, we address the problem of minimizing such sparse higher order energy functions. –Transforming the problem into an equivalent quadratic function minimization problem –The resulting quadratic function can be minimized using popular message passing or graph cut based algorithms for MAP inference. Namju Kwak, AALab, KAIST3

Labeling Problem Computer vision problems such as object segmentation, disparity estimation, and 3D reconstruction can be formulated as pixel or voxel labeling problems. Classical approach: pairwise CRF/MRF formulations (finding MAP solutions using BP, GC, and TRW message passing) a.Transformation of the higher order energy into a quadratic function b.Minimization of the resulting function using efficient inference algorithms The higher order functions used in CV have sparseness property (making them easy to handle). 4Namju Kwak, AALab, KAIST

Notations Consider a random field defined over … –A set of latent variables –The label set – : a set of subset of (i.e. cliques), over which the higher random field is defined 5Namju Kwak, AALab, KAIST

Notations Consider a random field defined over … –Energy function (corresponding to higher order random field) where is the set of random variables included in –Higher order potential which assigns a cost to each possible labeling for 6Namju Kwak, AALab, KAIST

Transforming Multi-label Functions (Simple Case) Multi-label function where and denotes a particular labeling of the variables 7Namju Kwak, AALab, KAIST

Transforming Multi-label Functions (Simple Case) Multi-label function The minimization problem of the function is transformed as follows: where and, and 8Namju Kwak, AALab, KAIST

Transforming Multi-label Functions (General Case) List of possible labelings of and their corresponding costs with such that Higher-order multi-label function 9Namju Kwak, AALab, KAIST

Transforming Multi-label Functions (General Case) Higher-order multi-label function The minimization problem of the function is transformed as follows: where, and 10Namju Kwak, AALab, KAIST

Transforming Multi-label Functions (Compact Parameterization) Many low cost label assignments tend to be close to each other in terms of the difference between labelings of pixels. (labeling fffb is closer to ffbb than bbbb.) where is defined as, is the cost added to the deviation function, and is 1 if, 0 otherwise. Namju Kwak, AALab, KAIST11

Transforming Multi-label Functions (Compact Parameterization) Compact parameterized multi-label function The minimization problem of the function is transformed as follows: where, and 12Namju Kwak, AALab, KAIST

Transforming Pseudo-Boolean Functions The minimization problem of the function is transformed as follows: –Type-I Transformation Namju Kwak, AALab, KAIST13

Transforming Pseudo-Boolean Functions The minimization problem of the function is transformed as follows: –Type-II Transformation where Namju Kwak, AALab, KAIST14

Conclusion This paper provides a method for minimizing sparse energy functions. (C. Rother, P. Kohli, W. Feng, and J. Jia. Minimizing Sparse Higher Order Energy Functions of Discrete Variables. CVPR 2009.) Namju Kwak, AALab, KAIST15