Creating virtual models of real spaces and objects is cumbersome and time consuming. This paper focuses on the problem of geometric reconstruction from sparse data obtained from certain image-based modeling approaches. A number of elegant and simple-to-state problems arise concerning when the geometry can be reconstructed. We describe results and counterexamples, and list open problems.
There is a rich history of work on reconstructing polyhedra from partial descriptions. See Lucier @cite_23 for a survey. There is little work, however, on reconstructing polyhedra from sparse point-plane or point-normal data, let alone more complex metadata. Biedl discuss several polygon reconstruction problems based on point-normal data and related data @cite_25 . Their reconstruction results are limited to two dimensions.
@cite_25 @cite_23
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This thesis covers work on two topics: unfolding polyhedra into the plane and reconstructing polyhedra from partial information. For each topic, we describe previous work in the area and present an array of new research and results. Our work on unfolding is motivated by the problem of characterizing precisely when overlaps will occur when a polyhedron is cut along edges and unfolded. By contrast to previous work, we begin by classifying overlaps according to a notion of locality. This classification enables us to focus upon particular types of overlaps, and use the results to construct examples of polyhedra with interesting unfolding properties. The research on unfolding is split into convex and non-convex cases. In the non-convex case, we construct a polyhedron for which every edge unfolding has an overlap, with fewer faces than all previously known examples. We also construct a non-convex polyhedron for which every edge unfolding has a particularly trivial type of overlap. In the convex case, we construct a series of example polyhedra for which every unfolding of various types has an overlap. These examples disprove some existing conjectures regarding algorithms to unfold convex polyhedra without overlaps. The work on reconstruction is centered around analyzing the computational complexity of a number of reconstruction questions. We consider two classes of reconstruction problems. The first problem is as follows: given a collection of edges in space, determine whether they can be rearranged by translation only to form a polygon or polyhedron. We consider variants of this problem by introducing restrictions like convexity, orthogonality, and non-degeneracy. All of these problems are NP-complete, though some are proved to be only weakly NP-complete. We then consider a second, more classical problem: given a collection of edges in space, determine whether they can be rearranged by translation and or rotation to form a polygon or polyhedron. This problem is NP-complete for orthogonal polygons, but polynomial algorithms exist for nonorthogonal polygons. For polyhedra, it is shown that if degeneracies are allowed then the problem is NP-hard, but the complexity is still unknown for non-degenerate polyhedra. We formulate multi-view 3D shape reconstruction as the computation of a minimum cut on the dual graph of a semi- regular, multi-resolution, tetrahedral mesh. Our method does not assume that the surface lies within a finite band around the visual hull or any other base surface. Instead, it uses photo-consistency to guide the adaptive subdivision of a coarse mesh of the bounding volume. This generates a multi-resolution volumetric mesh that is densely tesselated in the parts likely to contain the unknown surface. The graph-cut on the dual graph of this tetrahedral mesh produces a minimum cut corresponding to a triangulated surface that minimizes a global surface cost functional. Our method makes no assumptions about topology and can recover deep concavities when enough cameras observe them. Our formulation also allows silhouette constraints to be enforced during the graph-cut step to counter its inherent bias for producing minimal surfaces. Local shape refinement via surface deformation is used to recover details in the reconstructed surface. Reconstructions of the Multi- View Stereo Evaluation benchmark datasets and other real datasets show the effectiveness of our method. Accurate recovery of 3D geometrical surfaces from calibrated 2D multi-view images is a fundamental yet active research area in computer vision. Despite the steady progress in multi-view stereo (MVS) reconstruction, many existing methods are still limited in recovering fine-scale details and sharp features while suppressing noises, and may fail in reconstructing regions with less textures. To address these limitations, this paper presents a detail-preserving and content-aware variational (DCV) MVS method, which reconstructs the 3D surface by alternating between reprojection error minimization and mesh denoising. In reprojection error minimization, we propose a novel inter-image similarity measure, which is effective to preserve fine-scale details of the reconstructed surface and builds a connection between guided image filtering and image registration. In mesh denoising, we propose a content-aware @math -minimization algorithm by adaptively estimating the @math value and regularization parameters. Compared with conventional isotropic mesh smoothing approaches, the proposed method is much more promising in suppressing noise while preserving sharp features. Experimental results on benchmark data sets demonstrate that our DCV method is capable of recovering more surface details, and obtains cleaner and more accurate reconstructions than the state-of-the-art methods. In particular, our method achieves the best results among all published methods on the Middlebury dino ring and dino sparse data sets in terms of both completeness and accuracy. We present a new volumetric method for reconstructing watertight triangle meshes from arbitrary, unoriented point clouds. While previous techniques usually reconstruct surfaces as the zero level-set of a signed distance function, our method uses an unsigned distance function and hence does not require any information about the local surface orientation. Our algorithm estimates local surface confidence values within a dilated crust around the input samples. The surface which maximizes the global confidence is then extracted by computing the minimum cut of a weighted spatial graph structure. We present an algorithm, which efficiently converts this cut into a closed, manifold triangle mesh with a minimal number of vertices. The use of an unsigned distance function avoids the topological noise artifacts caused by misalignment of 3D scans, which are common to most volumetric reconstruction techniques. Due to a hierarchical approach our method efficiently produces solid models of low genus even for noisy and highly irregular data containing large holes, without loosing fine details in densely sampled regions. We show several examples for different application settings such as model generation from raw laser-scanned data, image-based 3D reconstruction, and mesh repair.