A Convex Solution to Spatially-Regularized Correspondence Problems
We propose a convex formulation of the correspondence problem between two images with respect to an energy function measuring data consistency and spatial regularity. To this end, we formulate the general correspondence problem as the search for a minimal
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Abstract. We propose a convex formulation of the correspondence problem between two images with respect to an energy function measuring data consistency and spatial regularity. To this end, we formulate the general correspondence problem as the search for a minimal twodimensional surface in R4 . We then use tools from geometric measure theory and introduce 2-vector fields as a representation of two-dimensional surfaces in R4 . We propose a discretization of this surface formulation that gives rise to a convex minimization problem and compute a globally optimal solution using an efficient primal-dual algorithm.
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Introduction
The establishment of spatially dense correspondence is one of the central computational challenges in computer vision with a wide range of applications including stereo disparity estimation [1,2], optical flow estimation [3], shape matching [4] and medical image registration [5–7]. Correspondence estimation is often cast as an energy minimization problem including a (generally) non-convex data consistency term and a spatial regularizer. Although such optimization problems have been intensively studied in computer vision, to date an algorithm that finds a global optimum in polynomial time is not known. 1.1
Problem Statement: Diffeomorphic Matching
We consider a general correspondence estimation problem where the aim is to compute an optimal diffeomorphic image matching defined as follows. Let Ω ⊂ R2 denote the image plane, i.e. an open simply-connected subset of R2 . We describe the data consistency between points on both images as a map g : Ω × Ω → R≥0 , where g(p, q) measures the consistency between brightness, color, depth or some high level features at point p ∈ Ω in image 1 and at point q ∈ Ω in image 2. We define the desired correspondence between both images as the optimal solution to the constrained minimization problem g p, f (p) + W df (p) dp. min (1) + f ∈Diff (Ω,Ω)
Ω
Here Diff + (Ω, Ω) is the set of all orientation-preserving diffeomorphisms from Ω to Ω and df (p) ∈ R2×2 denotes the Jacobian (or more general the differential) c Springer International Publishing AG 2016 B. Leibe et al. (Eds.): ECCV 2016, Part II, LNCS 9906, pp. 853–868, 2016. DOI: 10.1007/978-3-319-46475-6 52
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T. Windheuser and D. Cremers
Id Id of f at p. W : R2×2 → R≥0 , W (df ) = det( df df ), measures the deviation 2×2 from isometry, Id ∈ R being the identity matrix. Solving (1) therefore leads to a diffeomorphic transformation of Ω favoring data consistency via the term g(p, f (p)) and spatial regularity (local isometry) via W (df (p)), with ∈ R≥0 determining the trade-off. 1.2
Related Work
While the correspondence model (1) is not the only conceivable choice, most works on correspondence finding propose similar energies that include data consistency and a spatial regularizer. Moreover, we believe that the proposed solution via geometric measure theory can be generalized to other classes of cost functions. Due to the non-convexity of the data term coupled with the spatial regularizer,
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