Detection of constrictions on closed polyhedral surfaces Franck H - - PowerPoint PPT Presentation

Detection of constrictions

• n closed polyhedral surfaces

Franck H´ etroy

• , Dominique Attali
• ✁

Laboratoire des Images et Signaux, I.N.P. Grenoble, France

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Context

Decompose 3D objects in several components connected by narrower parts detect these narrower parts

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Objects

Boundary = closed triangulated surface

• manifold without boundary

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Other works

Other works to decompose polyhedral surfaces:

Decomposition into meaningful patchs: Tal et al. 1995 to 2003, Gregory et al. 1998

Topological decomposition: Erickson and Har-Peled 2002, Colin de Verdière and Lazarus 2002

Dynamical systems approach: Dey et al. 2002

...

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Plan

• 1. Definition and characterization of constrictions
• 2. Algorithm
• 3. Results and comments

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Definition

A constriction is a locally

• length-minimizing

simple, closed curve on a constriction is a closed geodesic

• curve
• for the Hausdorff distance
• follows a shortest path between any two sufficiently

close points

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What do constrictions look like

pivot vertices = vertices through which a constriction goes

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What do constrictions look like

between two successive pivots, a constriction intersects a sequence of faces

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What do constrictions look like

the planar unfolding of a constriction in the planar unfolding of the sequence is a straight line segment

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What do constrictions look like

the angle made by a constriction at a pivot vertex is

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What do constrictions look like

constriction simple, closed curve with angle

at each pivot and geodesic curve between any two successive pivots References about geodesics:

Sharir/Schorr 1986

Mitchell/Mount/Papadimitriou 1987

Chen/Han 1990

Mitchell 1998 (survey)

...

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Plan

• 1. Definition and characterization of constrictions
• 2. Algorithm
• 3. Results and comments

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Algorithm

Idea to detect constrictions: surface will disconnect in their area when simplified

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Overview of algorithm

seed curve detection surface simplification curve construction

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Surface simplification

Edge collapse operator:

• ■❍
• ■❏
• Several existing algorithms

Used algorithm: Garland and Heckbert’s (SIGGRAPH’97)

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Surface simplification

Stop when a seed curve

is found

the triangle

is not a face of

c b a

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Seed curve

After one more simplification step, the surface is not a manifold anymore

• ❯

2 different edges between

and

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Overview of algorithm

seed curve detection surface simplification curve construction

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Curve construction

Case 1: the curve is not modified

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Curve construction

Case 2: the curve is modified between 2 pivot vertices

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Curve construction

Geodesic computation between the two pivot vertices

Pham-Trong et al.’s algorithm (Num. Algo. 2001)

needs an initial sequence of faces

two geodesics: we keep the shortest one

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Curve construction

Case 3: the curve must have at least one pivot vertex

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Curve construction

and/or

will be pivot vertices of the new curve

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Curve constructions

Conclusion about this stage:

• each

is a closed piecewise geodesic

• the curve is only partially modified at each step

(except one special case)

• we are not sure the final curve

• n

is a constriction: constriction = straight between two successive pivot vertices + angle

at each pivot vertex first condition OK, but what about the second one ??

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Plan

• 1. Definition and characterization of constrictions
• 2. Algorithm
• 3. Results and comments

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Results on synthetic surfaces

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Results on synthetic surfaces

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Remarks

Detection of several constrictions:

• forbid the collapse of the 3 edges of seed curves and

continue surface simplification

• construct a sequence of closed piecewise geodesic

for each seed curve Constrictions can cut the object or its complementary

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Results on scanned data

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Results on scanned data

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Conclusion

Decomposition of a 3D object in separate components:

• surface approach (boundary of the object)
• progressive simplification (almost) until

deconnection of the surface

• curve construction by iterative geodesic computation
• final curve is a closed piecewise geodesic but may

not be a constriction

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Perspectives

“Loosening” algorithm to slide the curve to a constriction

done, other paper

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Perspectives

Not all constrictions are detected:

• seed curves can degenerate to a single point during

reconstruction

• some constrictions are not simplified to seed curves

interesting to decompose the object in a few number of components (noisy surfaces) but may not be enough

replace seed curves by another configuration ?

theoretical justification ?

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Perspectives

Results highly depend on the simplification algorithm must preserve as much as possible the shape of the

• bject

theoretical study, comparison between existing algorithms

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Thank you

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