Semidefinite Programming Heuristics for Surface Reconstruction Ambiguities

Abstract

Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface.

Template-Based Monocular 3D Shape Recovery Using Laplacian Meshes

“…This problem can be avoided by computing the 3D shape directly from the correspondences, which amounts to solving a degenerate linear system and requires either reducing the number of degrees of freedom or imposing additional constraints [2]. The first can be achieved by various dimensionality reduction techniques [3], [4] while the second often involves assuming the surface to be either developable [5], [6], [7] or inextensible [8], [9], [10], [4]. These two approaches are sometimes combined and augmented by introducing additional sources of information such as shading or textural clues [11], [12] or physicsbased constraints [13].…”

“…These two approaches are sometimes combined and augmented by introducing additional sources of information such as shading or textural clues [11], [12] or physicsbased constraints [13]. The resulting algorithms usually require solving a fairly large optimization problem and, even though it is often well behaved or even convex [10], [8], [14], [15], [4], it remains computationally demanding. Closed-form approaches to directly computing the 3D shape from the correspondences have been proposed [16], [12] but they also involve solving…”

Laplacian Meshes for Monocular 3D Shape Recovery

“…The most popular ones involve preserving Euclidean or Geodesic distances as the surface deforms and are enforced either by solving a convex optimization problem [9,7,8,12,13,3] or by solving in closed form sets of quadratic equations [14,11]. The latter is typically done by linearization, which results in very large systems and is no faster than minimizing a convex objective function, as is done in [3] which is one of the best representatives of this class of techniques.…”

“…The first can be achieved by various dimensionality reduction techniques [2,3] while the second often involves assuming the surface to be either developable [4][5][6] or inextensible [7][8][9]3]. These two approaches are sometime combined and augmented by introducing additional sources of information such as shading or textural clues [10,11].…”

“…These two approaches are sometime combined and augmented by introducing additional sources of information such as shading or textural clues [10,11]. The resulting algorithms usually require solving a fairly large optimization problem and, even though it is often well behaved or even convex [9,7,12,13,3], it remains computationally demanding. Closed-form solutions have been proposed [14,11] but they also involve solving very large equation systems and to make more restrictive assumptions than the optimization-based ones, which can lower the performance [3].…”