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158
Graph Cuts and Efficient ND Image Segmentation
, 2006
"... Combinatorial graph cut algorithms have been successfully applied to a wide range of problems in vision and graphics. This paper focusses on possibly the simplest application of graphcuts: segmentation of objects in image data. Despite its simplicity, this application epitomizes the best features ..."
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Cited by 307 (7 self)
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Combinatorial graph cut algorithms have been successfully applied to a wide range of problems in vision and graphics. This paper focusses on possibly the simplest application of graphcuts: segmentation of objects in image data. Despite its simplicity, this application epitomizes the best features of combinatorial graph cuts methods in vision: global optima, practical efficiency, numerical robustness, ability to fuse a wide range of visual cues and constraints, unrestricted topological properties of segments, and applicability to ND problems. Graph cuts based approaches to object extraction have also been shown to have interesting connections with earlier segmentation methods such as snakes, geodesic active contours, and levelsets. The segmentation energies optimized by graph cuts combine boundary regularization with regionbased properties in the same fashion as MumfordShah style functionals. We present motivation and detailed technical description of the basic combinatorial optimization framework for image segmentation via s/t graph cuts. After the general concept of using binary graph cut algorithms for object segmentation was first proposed and tested in Boykov and Jolly (2001), this idea was widely studied in computer vision and graphics communities. We provide links to a large number of known extensions based on iterative parameter reestimation and learning, multiscale or hierarchical approaches, narrow bands, and other techniques for demanding photo, video, and medical applications.
Convexity and Steinitz’s Exchange Property
 ADVANCES IN MATHEMATICS, 124 (1996), 272–311
, 1997
"... “Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave func ..."
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Cited by 64 (28 self)
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“Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the Steinitz exchange property if and only if it can be extended to a concave function ω such that the maximizers of (ω+any linear function) form an integral base polytope. A Fencheltype minmax theorem and discrete separation theorems are established which imply, as immediate consequences, Frank’s discrete separation theorem for submodular functions, Edmonds’ intersection theorem, Fujishige’s Fencheltype minmax theorem for submodular functions, and also Frank’s weight splitting theorem for weighted matroid intersection.
Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case
, 2002
"... We analyze an infinite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are identically distributed random variables that are independent of each other and their distributions depend on the produ ..."
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Cited by 60 (12 self)
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We analyze an infinite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are identically distributed random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to maximize expected discounted, or expected average profit over the infinite planning horizon. We show that a stationary (s, S, p) policy is optimal for both the discounted and average profit models with general demand functions. In such a policy, the period inventory is managed based on the classical (s, S) policy and price is determined based on the inventory position at the beginning of each period. 1
Bisubmodular Function Minimization
 Mathematical Programming
, 2000
"... This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scali ..."
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Cited by 47 (4 self)
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This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scaling algorithm naturally leads to the rst combinatorial polynomialtime algorithm for testing membership in deltamatroid polyhedra. Unlike the case of matroid polyhedra, it remains open to develop a combinatorial strongly polynomial algorithm for this problem. Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 5608531, Japan (fujishig@sys.es.osakau.ac.jp). Research partly carried out while at Forschungsinstut fur Diskrete Mathematik, Universitat Bonn. y Department of Mathematical Engineering and Information Physics, University of Tokyo, Tokyo 1138656, Japan (iwata@sr3.t.utokyo.ac.jp). 1 1 Introduction Let V be a nite none...
Approximating Submodular Functions Everywhere
"... Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., ..."
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Cited by 45 (4 self)
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Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a nonnegative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function ˆ f such that, for every set S, ˆ f(S) approximates f(S) within a factor α(n), where α(n) = √ n + 1 for rank functions of matroids and α(n) = O ( √ n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids. Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω ( √ n / log n), even for rank functions of a matroid.
Mconvex function on generalized polymatroid
, 1997
"... The concept of Mconvex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of DressWenzel (1990). In this paper, we extend this concept to functions on generalized polymat ..."
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Cited by 44 (23 self)
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The concept of Mconvex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of DressWenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems. The restriction of a function to fx 2 ZV j x(V) = kg for k 2 Z is called a layer. We prove the Mconvexity of each layer, and reveal that the minimizers in consecutive layers are closely related. Exploiting these properties, we can solve the optimization on layers efficiently. A number of equivalent exchange axioms are given for Mconvex function on generalized polymatroid.
Phase unwrapping via graph cuts
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2007
"... Phase unwrapping is the inference of absolute phase from modulo2π phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are firstorder Markov random fields. We provide an exact energy minimization algorithm, whenever the correspo ..."
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Cited by 42 (9 self)
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Phase unwrapping is the inference of absolute phase from modulo2π phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are firstorder Markov random fields. We provide an exact energy minimization algorithm, whenever the corresponding clique potentials are convex, namely for the phase unwrapping classical L p norm, with p ≥ 1. Its complexity is KT(n, 3n), where K is the length of the absolute phase domain measured in 2π units and T (n, m) is the complexity of a maxflow computation in a graph with n nodes and m edges. For nonconvex clique potentials, often used owing to their discontinuity preserving ability, we face an NPhard problem for which we devise an approximate solution. Both algorithms solve integer optimization problems, by computing a sequence of binary optimizations, each one solved by graph cut techniques. Accordingly, we name the two algorithms PUMA, for phase unwrapping maxflow/mincut. A set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with stateoftheart phase unwrapping algorithms.
The Theory of Discrete Lagrange Multipliers for Nonlinear Discrete Optimization
 Principles and Practice of Constraint Programming
, 1999
"... In this paper we present a Lagrangemultiplier formulation of discrete constrained optimization problems, the associated discretespace firstorder necessary and sufficient conditions for saddle points, and an efficient firstorder search procedure that looks for saddle points in discrete space. Our ..."
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Cited by 42 (21 self)
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In this paper we present a Lagrangemultiplier formulation of discrete constrained optimization problems, the associated discretespace firstorder necessary and sufficient conditions for saddle points, and an efficient firstorder search procedure that looks for saddle points in discrete space. Our new theory provides a strong mathematical foundation for solving general nonlinear discrete optimization problems. Specifically, we propose a new vectorbased definition of descent directions in discrete space and show that the new definition does not obey the rules of calculus in continuous space. Starting from the concept of saddle points and using only vector calculus, we then prove the discretespace firstorder necessary and sufficient conditions for saddle points. Using welldefined transformations on the constraint functions, we further prove that the set of discretespace saddle points is the same as the set of constrained local minima, leading to the firstorder necessary and sufficient conditions for constrained local minima. Based on the firstorder conditions, we propose a localsearch method to look for saddle points that satisfy the firstorder conditions.
Submodular Function Minimization
 BASED ON CHAPTER 7 OF THE HANDBOOK ON DISCRETE OPTIMIZATION [54] VERSION 3
, 2007
"... This survey describes the submodular function minimization problem (SFM); why it is important; techniques for solving it; algorithms by Cunningham [7, 11, 12], by Schrijver [69] as modified by Fleischer and Iwata [20], by Orlin [64], by Iwata, Fleischer, and Fujishige [45], and by Iwata [41, 43] for ..."
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Cited by 29 (0 self)
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This survey describes the submodular function minimization problem (SFM); why it is important; techniques for solving it; algorithms by Cunningham [7, 11, 12], by Schrijver [69] as modified by Fleischer and Iwata [20], by Orlin [64], by Iwata, Fleischer, and Fujishige [45], and by Iwata [41, 43] for solving it; and extensions of SFM to more general families of subsets.
Continuous Multiclass Labeling Approaches and Algorithms
 SIAM J. Imag. Sci
, 2011
"... We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific r ..."
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Cited by 28 (5 self)
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We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity – one can be used to tightly relax any metric interaction potential, while the other one only covers Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent DouglasRachford scheme, and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other firstorder methods, the approach shows competitive performance on synthetical and realworld images. By combining the method with an improved binarization technique for nonstandard potentials, we were able to routinely recover discrete solutions within 1%–5 % of the global optimum for the combinatorial image labeling problem. 1 Problem Formulation The multiclass image labeling problem consists in finding, for each pixel x in the image domain Ω ⊆ Rd, a label `(x) ∈ {1,..., l} which assigns one of l class labels to x so that the labeling function ` adheres to some local data fidelity as well as nonlocal spatial coherency constraints. This problem class occurs in many applications, such as segmentation, multiview reconstruction, stitching, and inpainting [PCF06]. We consider the variational formulation inf `:Ω→{1,...,l} f(`), f(`):= Ω s(x, `(x))dx ︸ ︷ ︷ ︸ data term + J(`). ︸ ︷ ︷ ︸ regularizer