Planar Piece Research Articles

RECONSTRUCTING BUILDINGS WITH DISCONTINUITIES AND ROOF OVERHANGS FROM OBLIQUE AERIAL IMAGERY

Abstract. This paper proposes a two-stage method for the reconstruction of city buildings with discontinuities and roof overhangs from oriented nadir and oblique aerial images. To model the structures the input data is transformed into a dense point cloud, segmented and filtered with a modified marching cubes algorithm to reduce the positional noise. Assuming a monolithic building the remaining vertices are initially projected onto a 2D grid and passed to RANSAC-based regression and topology analysis to geometrically determine finite wall, ground and roof planes. If this should fail due to the presence of discontinuities the regression will be repeated on a 3D level by traversing voxels within the regularly subdivided bounding box of the building point set. For each cube a planar piece of the current surface is approximated and expanded. The resulting segments get mutually intersected yielding both topological and geometrical nodes and edges. These entities will be eliminated if their distance-based affiliation to the defining point sets is violated leaving a consistent building hull including its structural breaks. To add the roof overhangs the computed polygonal meshes are projected onto the digital surface model derived from the point cloud. Their shapes are offset equally along the edge normals with subpixel accuracy by detecting the zero-crossings of the second-order directional derivative in the gradient direction of the height bitmap and translated back into world space to become a component of the building. As soon as the reconstructed objects are finished the aerial images are further used to generate a compact texture atlas for visualization purposes. An optimized atlas bitmap is generated that allows perspectivecorrect multi-source texture mapping without prior rectification involving a partially parallel placement algorithm. Moreover, the texture atlases undergo object-based image analysis (OBIA) to detect window areas which get reintegrated into the building models. To evaluate the performance of the proposed method a proof-of-concept test on sample structures obtained from real-world data of Heligoland/Germany has been conducted. It revealed good reconstruction accuracy in comparison to the cadastral map, a speed-up in texture atlas optimization and visually attractive render results.

Improved Algorithms for Grid-Unfolding Orthogonal Polyhedra

An unfolding of a polyhedron is a single connected planar piece without overlap resulting from cutting and flattening the surface of the polyhedron. Even for orthogonal polyhedra, it is known that edge-unfolding, i.e., cuts are performed only along the edges of a polyhedron, is not sufficient to guarantee a successful unfolding in general. However, if additional cuts parallel to polyhedron edges are allowed, it has been shown that every orthogonal polyhedron of genus zero admits a grid-unfolding with quadratic refinement. Using a new unfolding technique developed in this paper, we improve upon the previous result by showing that linear refinement suffices. For 1-layer orthogonal polyhedra of genus [Formula: see text], we show a grid-unfolding algorithm using only [Formula: see text] additional cuts, affirmatively answering an open problem raised in a recent literature. Our approach not only requires fewer cuts but yields much simpler algorithms.

Cutting out polygon collections with a saw

(I) Given a segment-cuttable polygon P drawn on a planar piece of material Q, we show how to cut P out of Q by a (short) segment saw with a total length of the cuts no more than 2.5 times the optimal. We revise the algorithm of Demaine et al. (2001) so as to achieve this ratio.(II) We prove that any collection R of n disjoint11For brevity, a collection of disjoint polygons refers to a collection of polygons with pairwise disjoint interiors. axis-parallel rectangles drawn on a planar piece of material Q is cuttable by at most 4n rays and present an algorithm that runs in O(nlogn) time for computing a suitable cutting sequence. In particular, the same result holds for cutting with an arbitrary segment saw (of any length).(III) Given a collection P of segment-cuttable polygons drawn on a planar piece of material such that no two polygons in P touch each other, P is always cuttable by a sufficiently short segment saw. We also show that there exist collections of disjoint polygons that are uncuttable by a segment saw.(IV) Given a collection P of disjoint polygons drawn on a planar piece of material Q, we present a polynomial-time algorithm that computes a suitable cutting sequence to cut the polygons in P out of Q using ray cuts when P is ray-cuttable and otherwise reports P as uncuttable.

Reprint of: Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.

Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.

Graph Minors. XXII. Irrelevant vertices in linkage problems

In the algorithm for the disjoint paths problem given in Graph Minors XIII, we used without proof a lemma that, in solving such a problem, a vertex which was sufficiently “insulated” from the rest of the graph by a large planar piece of the graph was irrelevant, and could be deleted without changing the problem. In this paper we prove the lemma.

WireWarping: A fast surface flattening approach with length-preserved feature curves

This paper presents a novel approach — WireWarping for computing a flattened planar piece with length-preserved feature curves from a 3D piecewise linear surface patch. The property of length-preservation on feature curves is very important to industrial applications for controlling the shape and dimension of products fabricated from planar pieces. WireWarping simulates warping a given 3D surface patch onto plane with the feature curves as tendon wires to preserve the length of their edges. During warping, the surface-angle variations between edges on wires are minimized so that the shape of a planar piece is similar to its corresponding 3D patch. Two schemes — the progressive warping and the global warping schemes are developed, where the progressive scheme is flexible for local shape control and the global scheme gives good performance on highly distorted patches. Experimental results show that WireWarping can successfully flatten surface patches into planar pieces while preserving the length of edges on feature curves.

Grid Vertex-Unfolding Orthogonal Polyhedra

An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a net, a connected planar piece with no overlaps. A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertex-unfolding allows faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of “gridding” of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n 2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3×1×1 refinement of the vertex grid.

Epsilon-Unfolding Orthogonal Polyhedra

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as ɛ = 1/2Ω(n).