Skeletal representations of orthogonal shapes PhD. defense Advisors - - PowerPoint PPT Presentation

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

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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
• ne, as it behaves well with orthogonal shapes.

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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.).

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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.

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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.

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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.

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Contributions

Extract B-Rep Refine topology Compute L∞ Voronoi diagram Compute CS Simplify CS

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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
• rthogonal 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.

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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.

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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.

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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.

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Introduction: skeleton properties

• Dimension reduction.
• Homotopy equivalence.
• Hierarchical shape representation and abstraction.
• Stability and smoothness.

Unstable skeleton More stable skeleton

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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

is O

• n3+ǫ

, for ǫ > 0.

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Introduction: approximation paradigm

O O P

• M
• O
• M
• O
• Approximate M

Approximate shape Prune Compute M

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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

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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

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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.

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Preprocess: B-Rep extraction

Extract B-Rep Refine topology Compute L∞ Voronoi diagram Compute CS Simplify CS

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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
• f edge loops and the set of faces (topological relation

face:loop and loop:vertices). Lemma (O’Rourke) Let Lxy be the lexicographically xy-sorted list of vertices

• f an orthogonal manifold polygon. Then, the pairs of consecutive

vertices sorted in Lxy are the vertical polygon edges.

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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.

xor

step j step j+1

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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.

+X +Y +Z

(a) (b) (c) (d) (e) (j) (i) (h) (g) (f) (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) (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)

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Preprocess: topology refinement

Extract B-Rep Refine topology Compute L∞ Voronoi diagram Compute CS Simplify CS

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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.

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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.

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Preprocess: topology refinement

Input polyhedron A possible solution

((fx1 = fx2) ∧ (fy1 > fy2) ∧ (fz1 > fz2)) ∨ ((fx1 < fx2) ∧ (fy1 = fy2) ∧ (fz1 < fz2)) ∨ ((fx1 > fx2) ∧ (fy1 < fy2) ∧ (fz1 = fz2))

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Preprocess: topology refinement

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Preprocess: topology refinement

• A solution to the trihedralization may not exist.
• Represents an almost negligible amount of the tested randomly

generated datasets.

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L∞ Voronoi diagram computation

Extract B-Rep Refine topology Compute L∞ Voronoi diagram Compute CS Simplify CS

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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.

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2D L∞ Voronoi diagram: overview

Computed Voronoi diagram Wavefront Sweep line Ray x y

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2D L∞ Voronoi diagram: edge-based algorithm

• Edge event: the sweep-line meets a vertical polygon edge.

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2D L∞ Voronoi diagram: edge-based algorithm

• Ray-ray junction event: the intersection of two rays that

could induce a Voronoi vertex.

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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.

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2D L∞ Voronoi diagram: robustness

• Degeneracy by collinearity: a polygonal area whose points

are L∞ equidistant to two or more collinear edges.

• Degeneracy by coincidence: a Voronoi vertex L∞

equidistant to more than three non-collinear edges.

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2D L∞ Voronoi diagram: experimental results

Our algorithm Thinning algorithm CGAL implementation CGAL exact implementation

• Our approach was significantly faster than thinning and CGAL approaches.

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3D L∞ Voronoi diagram: overview

• Based on the sweep-plane technique instead of propagating the

faces inwards (Barequet et al. [1]).

• Video demonstration.

Computed Voronoi diagram Wavefront Sweep plane Ray

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3D L∞ Voronoi diagram: events

• Vertex event: the occurrence of a trihedral vertex of the input

polyhedron.

• Ray-ray junction event: the intersection of two rays.

z x y z x y

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3D L∞ Voronoi diagram: ray-face junction events

• Ray-face junction event: the intersection of a ray and a face.

z y x z y x

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3D L∞ Voronoi diagram: ray-face junction event

z x y z y x

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3D L∞ Voronoi diagram: algorithm complexity

• W has size O (n). Wavefront update O (log n) per event.
• Find ray-ray junction events O (1) per event. Find ray-face

junction events O (n) per event.

• Overall O (kn), where k is the total number of processed

events.

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3D L∞ Voronoi diagram: robustness

• Degeneracy by coplanarity: a three-dimensional volume

containing points that are L∞ equidistant to two or more coplanar faces.

• Degeneracy by coincidence: a Voronoi vertex L∞

equidistant to more than four non-collinear faces.

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3D L∞ Voronoi diagram: experimental results

Our algorithm Thinning algorithm Voronoi vertices

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Cube skeleton

Extract B-Rep Refine topology Compute L∞ Voronoi diagram Compute CS Simplify CS

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Cube skeleton: motivation

• Could the L∞ Voronoi diagram serve as a skeletal

representation?

• Problems (dimension reduction, homotopy equivalence)

specially arise in three dimensions.

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Cube skeleton: definition

• Boundary element: A vertex or set of open collinear/coplanar

edges/faces of an orthogonal shape.

• Cube skeleton: set of points in which have at least two

closest boundary elements.

• The closest boundary elements are defined recursively.

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Cube skeleton: definition

• If only one maximal dimension boundary element is closer,

recursion is performed.

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Cube skeleton: homotopy equivalence

• We proved the homotopy equivalence between the cube

skeleton and the input orthogonal shape.

• We defined a deformation retract between the orthogonal

shape and the cube skeleton.

• We studied the continuity during the deformation.

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Cube skeleton: homotopy equivalence

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Cube skeleton: computation

• Some L∞ Voronoi edges/faces belong to the cube skeleton.
• The remaining ones can be computed efficiently from the

diagram.

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Interior cube skeleton 2D

• A homotopy equivalent subset of the 2D cube skeleton, which

does not intersect the shape boundary.

Cube skeleton Interior cube skeleton

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Scale cube skeleton

Extract B-Rep Refine topology Compute L∞ Voronoi diagram Compute CS Simplify CS

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Scale cube skeleton: motivation

• The cube skeleton is sensitive to boundary noise.
• The scale axis transform of Giesen et al. [2] could be adapted.

Grow balls Compute MA Shrink balls

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Scale cube skeleton: definition

• Consider the union of cubes induced by the cube skeleton.
• Scale cube skeleton: the cube skeleton of the scaled union of

cubes.

• The scaled shape is orthogonal when is grown. It corresponds

to the union of axis-aligned boxes.

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Scale cube skeleton: results

Medial axis Straight skeleton Cube skeleton Scale cube skeleton

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Scale cube skeleton: results

s=1 s=1.2 s=1.6 s=2.5 s=4

• Video demonstration.

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Conclusions

• Although the constrained structure of orthogonal shapes, few

skeletal representations specifically designed for them are found.

• We provided a complete framework to compute the L∞

Voronoi diagram of pseudomanifold orthogonal shapes. All the presented methods are meant to be practical.

• The cube and scale cube skeletons could serve as a skeletal

representations with linear/planar structure and constrained combinatorial complexity.

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Work in progress

• Definition of the 3D interior cube skeleton.
• Computation of the 3D scale cube skeleton.

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Future work

• The B-Rep extraction may identify and classify 3D shells.
• Settle the complexity of the trihedralization problem in our

restricted setting.

• The performance of 3D status data structure could be

improved.

• The recursive formulation of the cube skeleton and scale cube

skeleton could be extended to higher dimensional orthogonal polytopes.

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Bibliography

Barequet, G., Eppstein, D., Goodrich, M. T., and Vaxman, A. Straight skeletons of three-dimensional polyhedra. Lecture Notes in Computer Science 5193 (2008), 148 – 160. Giesen, J., Miklos, B., Pauly, M., and Wormser, C. The scale axis transform. In Proceedings of the 25th annual symposium on Computational geometry (2009), pp. 106–115. Papadopoulou, E., and Lee, D. T. The L∞ Voronoi diagram of segments and VLSI applications. International Journal of Computational Geometry and Applications 11 (2001), 503 – 508.

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Acknowledgments

Thank you for your attention.

This work has been partially supported by the project TIN2008-02903 and TIN2011-24220 of the Spanish government.

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