1 Fast Primal-Dual Strategies for MRF Optimization (Fast PD) Robot Perception Lab Taha Hamedani Aug 2014

Overview A new efficient MRF optimization algorithm generalizes α-expansion at least 3-9 times faster than α-expansion used for boosting the performance of dynamic MRFs, i.e. MRFs varying over time guarantee an almost optimal solution for a much wider class of NP-hard MRF problems 2

Energy Function weighted graph G (with nodes V, edges E and weights w pq ), one seeks to assign a label x p (from a discrete set of labels L) to each p ∈ V, so that the following cost is minimized: p(·), d(·, ·) determine the singleton and pairwise MRF potential functions 3

Primal-dual MRF optimization algorithms Theorem 1 (Primal-Dual schema). Keep generating pairs of integral-primal, dual solutions (x k, y k ), until the elements of the last pair, (say x, y), are both feasible and have costs that are close enough, e.g. their ratio is ≤ f app : Then x is guaranteed to be an f app -approximate solution to the optimal integral solution x ∗, i.e. cTx ≤ f app · cTx ∗. 4

The primal dual schema 5

Fast primal-dual MRF optimization In the above formulation, θ ={ {θ p } p ∈ V, {θ pq } pq ∈ E } represents a vector of MRF-parameters that consists of all unary θp = {θ p (·)} and pairwise θpq = {θ pq (·, ·)} x ={{x p } p ∈ V, {x pq } pq ∈ E } denotes a vector of binary MRF- variables consisting of all unary subvectors xp = {x p (·)} and pairwise subvectors x pq = {x pq (·, ·)}. (0,1 variables) i.e, they satisfy x p (l) = 1 ⇔ label l is assigned to p, while x pq (l, l′) = 1 ⇔ labels l, l′ are assigned to p, q 6

MRF constraints (first constraint) simply express the fact that a unique label must be assigned to each node p (second constraint) since they ensure that if x p (l) = x q (l′) = 1, then x pq (l, l′) = 1 as well (marginal polytope) 7

local marginal polytope connected with the linear programming (LP) relaxation, which is formed by replacing the integer constraints x p (·), x pq (·, ·) ∈ {0, 1} with the relaxed constraints x p (·), x pq (·, ·) ≥ 0 8

The original (possibly difficult) optimization problem decomposes into easier sub problems (called the slaves) that are coordinated by a master problem via message exchanging 9

decompose the original MRF optimization problem, which is NP-hard (since it is defined on a general graph G ) decompose into a set of easier MRF sub problems, each one defined on a tree T ⊂ G. Needed to transform our problem into a more appropriate form by introducing a set of auxiliary variables. let T (G ) be a set of sub trees of graph G (cover at least one node and edge of graph G) 10

For each tree T ∈ T (G ) we will then imagine that there is a smaller MRF defined just on the nodes and edges of tree T We will associate to it a vector of MRF- parameters θT. as well as a vector of MRF-variables xT (these have similar form to vectors θ and x of the original MRF, except that they are smaller in size) (Decomposition) 11

Redundancy MRF-variables contained in vector xT will be redundant initially assume that they are all equal to the corresponding MRF-variables in vector x, i.e it will hold xT= x|T x|T represents the sub vector of x containing MRF-variables only for nodes and edges of tree T 12

all the vectors {θT} will be defined so that they satisfy the following conditions: Here, T (p) and T (pq) denote the set of all trees of T (G ) that contain node p and edge pq respectively. 13

Energy Decomposition The first constraints can reduced by : MRF problem can decomposed as : 14

It is clear that without constraints xT= x|T, this problem would decouple into a series of smaller MRF problems (one per tree T ) Lagrangian dual form : Eliminate vector x by minimizing over it : 15

The resulting lagrangian dual form is simplified as : Dual from by maximizing over feasible set : Master Slave 16

According to Lemma 1 : λ T must first be updated as Sub gradient of gt is equal to optimal solution of slave problem : 17

Fast PD procedure 18

References [1] Komodakis, N.; Paragios, N.; Tziritas, G., "MRF Energy Minimization and Beyond via Dual Decomposition," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol.33, no.3, pp.531,552, March [2] Chaohui Wang, Nikos Komodakis, Nikos Paragios, Markov Random Field modeling, inference & learning in computer vision & image understanding: A survey, Computer Vision and Image Understanding, Volume 117, Issue 11, November 2013, Pages , ISSN Thank You ?