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# What energy functions can be minimized via graph cuts?

IEEE Transactions on Pattern Analysis and Machine Intelligence, no. 2 (2004): 147-159

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Abstract

In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular...More

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Introduction

**Introduction and overview**

Many of the problems that arise in early vision can be naturally expressed in terms of energy minimization.- The minimum cut in turn can be computed very efficiently by max flow algorithms
- These methods have been successfully used for a wide variety of vision problems including image restoration [8, 9, 17, 19], stereo and motion [4, 8, 9, 18, 23, 27, 30, 31], voxel occupancy [34], multi-camera scene reconstruction [24] and medical imaging [5, 6, 22].
- Two recent evaluations of stereo algorithms using real imagery with dense ground truth [32, 36] found that the best overall performance was due to algorithms based on graph cuts

Highlights

**Introduction and overview**

Many of the problems that arise in early vision can be naturally expressed in terms of energy minimization- In this paper we focus on two classes of energy functions
- We define the class F2 to be functions that can be written as a sum of functions of up to 2 binary variables at a time, E(x1, . . . , xn) = Ei + Ei,j
- In section 2 we describe how graph cuts can be used to minimize energy functions, and discuss the importance of energy functions with binary variables in the context of two example vision problems, namely stereo and multi-camera scene reconstruction
- We summarize the graph constructions used for regular functions

Conclusion

**Summary of the results**

In this paper the authors focus on two classes of energy functions. Let {x1, . . . , xn}, xi ∈ {0, 1} be a set of binary-valued variables.- The authors define the class F3 to be functions that can be written as a sum of functions of up to 3 binary variables at a time, E(x1, .
- The authors show how to construct a graph G for a regular function E(x1, x2) of two variables.
- It will contain four vertices: V = {v1, v2, s, t}.

Related work

- There is an interesting relationship between regular functions and submodular functions.7 Let S be a finite set and g : 2S → R be a real-valued function defined on the set of all subsets of S. g is called submodular if for any X, Y ⊂ S g(X) + g(Y ) ≥ g(X ∪ Y ) + g(X ∩ Y ).

See [14], for example, for a discussion of submodular functions. An equivalent definition of submodular functions is that g is called submodular if for any X ⊂ S and i, j ∈ S − X g(X ∪ {j}) − g(X) ≥ g(X ∪ {i, j}) − g(X ∪ {i}).

Obviously, functions of subsets X of S = {1, . . . , n} can be viewed as functions of n binary variables (x1, . . . , xn); the indicator variable xi is 1 if i is included in X and 0 otherwise. Then it is easy to see that the second definition of submodularity reduces to the definition of regularity. Thus, submodular functions and regular functions are essentially the same. We use different names to emphasize a technical distinction between them: submodular functions are functions of subsets of S while regular functions are functions of binary variables. From our experience, the second point of view is much more convenient for vision applications.

Funding

- This research was supported by NSF grants IIS-9900115 and CCR-0113371, and by a grant from Microsoft Research

Reference

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