� � ✁ Detection of constrictions on closed polyhedral surfaces Franck H´ etroy , Dominique Attali Laboratoire des Images et Signaux, I.N.P. Grenoble, France VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.1
Context Decompose 3D objects in several components connected by narrower parts detect these narrower parts VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.2
✂ ✄ Objects Boundary = closed triangulated surface -manifold without boundary VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.3
☎ ☎ ☎ ☎ 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 ... VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.4
Plan 1. Definition and characterization of constrictions 2. Algorithm 3. Results and comments VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.5
� � � � � � 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 VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.6
✍ ✎ ✓ ✝ ✟ ✒ ✠ ✡ ✑ ☛ ☞ ✏ ✌ ✆ What do constrictions look like ✍✞✎ ☞✞✌ ✡✞☛ ✆✞✝ ✟✞✠ ✏✞✑ ✒✞✓ pivot vertices = vertices through which a constriction goes VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.7
✜ ✢ ✦ ✕ ✖ ✥ ✗ ✘ ✤ ✙ ✚ ✣ ✛ ✔ What do constrictions look like ✜✞✢ ✚✞✛ ✘✞✙ ✔✞✕ ✖✞✗ ✣✞✤ ✥✞✦ between two successive pivots, a constriction intersects a sequence of faces VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.8
✱ ✭ ✧ ✰ ✴ ✯ ✮ ✵ ✬ ✲ ✶ ✫ ✪ ✷ ✩ ★ ✸ ✳ What do constrictions look like ✯✞✰ ✭✞✮ ✫✞✬ ✧✞★ ✩✞✪ ✵✞✶ ✷✞✸ ✱✞✲ ✳✞✴ the planar unfolding of a constriction in the planar unfolding of the sequence is a straight line segment VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.9
❃ ✿ ❆ ✹ ❂ ❇ ❁ ❀ ❈ ✾ ❅ ❉ ✽ ✼ ❊ ✻ ✺ ❋ ❄ What do constrictions look like ❅✞❆ ❃✞❄ ❁✞❂ ✽✞✾ ✿✞❀ ❇✞❈ ❉✞❊ ✹✞✺ ✻✞✼ the angle made by a constriction at a pivot vertex is VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.10
☎ ☎ ☎ ❋ ☎ ☎ 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) ... VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.11
Plan 1. Definition and characterization of constrictions 2. Algorithm 3. Results and comments VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.12
Algorithm Idea to detect constrictions: surface will disconnect in their area when simplified VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.13
Overview of algorithm surface simplification seed curve detection curve construction VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.14
● Surface simplification Edge collapse operator: ●■❏ ●■❍ Several existing algorithms Used algorithm: Garland and Heckbert’s (SIGGRAPH’97) VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.15
✂ ▲ ❚ ◆ ❖ ❙ ❖ P ▲ ❘ Surface simplification ❑▼▲ Stop when a seed curve ◆◗P is found the triangle ◆◗P is not a face of a c b VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.16
❱ ✂ P ● ❯ ❱ ❱ ❲ ❳ ❱ Seed curve After one more simplification step, the surface is not a manifold anymore 2 different edges between and VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.17
Overview of algorithm surface simplification seed curve detection curve construction VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.18
❳ ❭ ❪ Curve construction ❨❬❩ Case 1: the curve is not modified VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.19
❭ ❪ ❳ ❫ ❪ Curve construction ❨❬❩ ❫❬❴ Case 2: the curve is modified between 2 pivot vertices VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.20
❜ ❜ ❣ ❜ ❤ ❵ ❢ ❞ ❡ ❞ ❢ ❵ ❜ ❢ ❡ ❵ ❛ ❡ ❵ ❛ ❣ ❛ ❜ ❡ ❞ ✂ ❜ ✂ ❣ ❵ ❛ ❞ ❡ ❜ ❞ ❵ ❛ ❡ ❢ ❜ ❢ ❡ ❞ ❡ ❜ 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 VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.21
❭ ❪ ❳ Curve construction ❨❬❩ Case 3: the curve must have at least one pivot vertex VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.22
❭ ❳ ❪ ❭ ❪ ❳ ❪ ❩ ❴ ❩ ❴ ❪ ❩ ❩ ❴ ❴ ❳ ❴ ❩ ❪ ❳ ❴ ❳ ❪ ❩ ❪ ❳ ❪ ❪ ❭ ❳ Curve construction ❨❥❩ ❨✐❩ ❨✐❩ and/or will be pivot vertices of the new curve VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.23
❙ ❋ 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 on is a ❨❬❦ constriction: constriction = straight between two successive pivot vertices + angle at each pivot vertex first condition OK, but what about the second one ?? VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.24
Plan 1. Definition and characterization of constrictions 2. Algorithm 3. Results and comments VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.25
Results on synthetic surfaces VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.26
Results on synthetic surfaces VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.27
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 VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.28
Results on scanned data VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.29
Results on scanned data VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.30
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 VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.31
✂ Perspectives “Loosening” algorithm to slide the curve to a constriction done, other paper VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.32
✂ ✂ 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 ? VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.33
✂ Perspectives Results highly depend on the simplification algorithm must preserve as much as possible the shape of the object theoretical study, comparison between existing algorithms VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.34
Thank you VisSym2003 Detection of constrictions on closed polyhedral surfaces – p.35
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