Skeletal representations of orthogonal shapes PhD. defense Advisors Candidate Dra. N´ uria Pla Jon` as Mart´ ınez Dr. Marc Vigo Grup de recerca d’Inform` atica a l’Enginyeria Departament de Llenguatges i Sistemes Inform` atics Universitat Polit` ecnica de Catalunya 12th December 2013 1 / 60
Objectives • Our main objective was to obtain a simple skeletal representation of shapes. • We considered orthogonal shapes in order to approximate the input shape. • The L ∞ metric was also considered, to substitute the Euclidean one, as it behaves well with orthogonal shapes. 2 / 60
Objectives • The first objective was to design and implement a robust framework to compute the L ∞ Voronoi diagram of orthogonal polygons and polyhedra. • We found several problems in addressing degenerate cases and pseudomanifold features (brep extraction, non-trihedral vertices, etc.). 3 / 60
Objectives • The second objective was to introduce skeletal representations based on the L ∞ Voronoi diagram. • We found that the L ∞ Voronoi diagram may not reduce the dimension or be homotopical equivalent to the input shape. • Moreover, it is highly sensitive to perturbations on the shape boundary. 4 / 60
Contributions • A practical and robust approach to compute the L ∞ Voronoi diagram of orthogonal shapes: 1. B-Rep extraction of 2D/3D orthogonal pseudomanifolds from several solid representation schemes (voxel, EVM, etc.). 2. Refinement of the topology of 3D orthogonal pseudomanifolds, which is required by the Voronoi computation. 3. Computation of the L ∞ Voronoi diagram of 2D/3D orthogonal pseudomanifolds. 5 / 60
Contributions • The cube skeleton : a skeletal representation derived from the L ∞ Voronoi diagram of orthogonal shapes with dimension reduction, homotopy equivalence and linear/planar structure. • The scale cube skeleton : a skeletal representation derived from the cube skeleton, which is more stable. 6 / 60
Contributions Extract B-Rep Refine topology Compute L ∞ Voronoi diagram Simplify CS Compute CS 7 / 60
List of publications • Martinez, J., Vigo, M., Pla, N., and Ayala, D. Skeleton computation of an image using a geometric approach. In Proceedings of Eurographics (2010), 13–16 • Martinez, J., Vigo, M., and Pla, N. Skeleton computation of orthogonal polyhedra. Computer Graphics Forum 30, 5 (2011), 1573–1582 • Vigo, M., Pla, N., Ayala, D., and Martinez, J. Efficient algorithms for boundary extraction of 2D and 3D orthogonal pseudomanifolds. Graphical Models 74 (2012), 61–74 • Martinez, J., Pla-Garcia, N., and Vigo, M. Skeletal representations of orthogonal shapes. Graphical Models 75 (2013), 189–207 • D. Lopez Monterde, J. Martinez, M. Vigo, N. Pla. A practical and robust method to compute the boundary of three-dimensional axis-aligned boxes. Accepted to GRAPP 2014. 8 / 60
Introduction: skeletal representations • Skeletal representations reduce the dimension of shapes and try to capture their geometric and topological properties. • Skeletons are compact shape representations that emphasize shape properties. 9 / 60
Introduction: skeleton applications • Skeletons are powerful multidisciplinary tools used in a broad number of scientific fields, such as biology, medical imaging, motion analysis and animation, visual perception, etc. • The first application of skeletal representations was as a natural descriptor for shapes such as cells and body tissues. 10 / 60
Introduction: BioCAD applications • In recent years, the spectacular development of bioengineering has yield to the apparition of a new application field of computer aided design, the BioCAD. • One of the challenges of the BioCAD is to understand the morphology of the pore space of bone, biomaterials, rocks, etc. Several approaches rely on the skeleton computation. 11 / 60
Introduction: skeleton properties • Dimension reduction. • Homotopy equivalence. • Hierarchical shape representation and abstraction. • Stability and smoothness. Unstable skeleton More stable skeleton 12 / 60
Introduction: medial axis problems • Medial axis : the set of points which have at least two closest points on the shape boundary. • High sensitivity to small changes in the shape boundary and difficult to compute exactly. • For polygons and polyhedra can be non linear or non planar. • For polyhedra, the upper bound on its combinatorial complexity n 3+ ǫ � � is O , for ǫ > 0. 13 / 60
Introduction: approximation paradigm Approximate shape O O Approximate M Compute M Prune � � �� � � P M O M O 14 / 60
Introduction: orthogonal shapes • Polygons and polyhedra with their edges and faces axis-aligned, respectively. • Their constrained structure has enabled advances on problems for arbitrary shapes. Pseudomanifold polygon Pseudomanifold polyhedron 15 / 60
Introduction: the L ∞ Voronoi diagram • The Voronoi sites are restricted to be open edges and faces of polygons and polyhedra, respectively. • The set of closest sites only considers the L ∞ distance to a point in a site and not as an infimum. Classical definition Proposed definition 16 / 60
Main idea • The shape and the distance function are simplified together. • The shape is approximated by an orthogonal shape. • The Euclidean metric is substituted by the L ∞ or Chebyshev distance. 17 / 60
Preprocess: B-Rep extraction Extract B-Rep Refine topology Compute L ∞ Voronoi diagram Simplify CS Compute CS 18 / 60
Preprocess: B-Rep extraction • None of the existing previous approaches was designed for the pseudomanifold case. • Input : the set of vertices of the orthogonal shape and some neighborhood information (3D). • Output : B-Rep model composed by the set of vertices, the set of edge loops and the set of faces (topological relation face:loop and loop:vertices). Lemma (O’Rourke) Let L xy be the lexicographically xy-sorted list of vertices of an orthogonal manifold polygon. Then, the pairs of consecutive vertices sorted in L xy are the vertical polygon edges. 19 / 60
Preprocess: 2D B-Rep extraction • The events correspond to the vertical edges of the polygon. • The face:loop relationship is recovered by the status data structure. step j step j+1 xor 20 / 60
Preprocess: 3D B-Rep extraction • Sort vertices using the six possible lexicographical orders and apply 2D algorithm. • Indicate the number of times a vertex must be repeated in each list. +Z +Y +X (1,0,1,0,1,0) (2,0,1,1,1,1) (1,1,1,1,1,1) (1,0,1,0,1,0) (0,1,0,1,1,0) (a) (b) (c) (d) (e) (2,1,1,2,2,1) (2,0,0,2,2,0) (2,0,1,1,2,0) (1,1,1,1,1,1) (2,2,2,2,2,2) (f) (g) (h) (i) (j) 21 / 60
Preprocess: topology refinement Extract B-Rep Refine topology Compute L ∞ Voronoi diagram Simplify CS Compute CS 22 / 60
Preprocess: topology refinement • The Voronoi computation becomes more simpler if we modify the topology of the orthogonal polyhedron, such as it only has trihedral vertices and remains self-intersection free ( trihedralization problem ) • Local vertex trihedralization can be seen as the dual problem of planar polygon triangulation. 23 / 60
Preprocess: topology refinement • Consider that some orthogonal polyhedron faces are translated inwards an infinitesimal amount, such that non-trihedral vertices disappear. • The restrictions between coplanar faces can be expressed as equality and inequality integer constraints. • Finding a valid trihedralization is equivalent to solving the constraint satisfaction problem over the set of constraints induced by some vertex configurations. 24 / 60
Preprocess: topology refinement Input polyhedron A possible solution (( f x 1 = f x 2 ) ∧ ( f y 1 > f y 2 ) ∧ ( f z 1 > f z 2 )) ∨ (( f x 1 < f x 2 ) ∧ ( f y 1 = f y 2 ) ∧ ( f z 1 < f z 2 )) ∨ (( f x 1 > f x 2 ) ∧ ( f y 1 < f y 2 ) ∧ ( f z 1 = f z 2 )) 25 / 60
Preprocess: topology refinement 26 / 60
Preprocess: topology refinement • A solution to the trihedralization may not exist. • Represents an almost negligible amount of the tested randomly generated datasets. 27 / 60
L ∞ Voronoi diagram computation Extract B-Rep Refine topology Compute L ∞ Voronoi diagram Simplify CS Compute CS 28 / 60
2D L ∞ Voronoi diagram • Two specific implementations of the sweep line algorithm of Papadopoulou and Lee [3] for planar straight graphs. • The sweep line events correspond to vertical edges or polygon vertices ( edge event and vertex event ), and candidate Voronoi vertices ( junction event ). • The algorithm that processes the vertices has been developed in order to extend it to three dimensions. • Video demonstration. 29 / 60
2D L ∞ Voronoi diagram: overview Computed Voronoi diagram Wavefront Sweep line Ray y x 30 / 60
2D L ∞ Voronoi diagram: edge-based algorithm • Edge event : the sweep-line meets a vertical polygon edge. 31 / 60
2D L ∞ Voronoi diagram: edge-based algorithm • Ray-ray junction event : the intersection of two rays that could induce a Voronoi vertex. 32 / 60
2D L ∞ Voronoi diagram: vertex-based algorithm • Vertex event : the sweep-line meets a polygon vertex. • Ray-edge junction event : the intersection of a ray and an edge that could induce a Voronoi vertex. 33 / 60
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