Surface Reconstruction Level Sets Computer Graphics Hoppe et al, - PDF document

� � � � Surface Reconstruction Level Sets Computer Graphics Hoppe et al, Surface reconstruction from unorganized points, ACM Siggraph’92 Smooth Surface Reconstruction via Natural Neighbour In- terpolation of Distance Functions, ACM SoCG’00 Alexa et al., Point Set Surfaces, IEEE Vis. 2001 Carr et al, Reconstruction and Representation of 3D Ob- jects With Radial Basis Functions, ACM Siggraph’01 1

✕ ✠ ✠ ✒ � ✝ ✞ ✟ ✓ ✠ ✔ ✡ ✔ ✡ ✔ ✠ Barycentric coordinates: exples �☎✆ �✂✁ �☎✄ p rp 1 sp 2 tp 3 area ☛ pp 1 p 3 area ☛ pp 2 p 3 area ☛ pp 1 p 2 ☛✌☞ ☛✏☞ ☛✌☞ r s t ✍✎✍ ✍✎✍ ✍✑✍ area ☛ p 1 p 2 p 3 area ☛ p 1 p 2 p 3 area ☛ p 1 p 2 p 3 ☛✏☞ ☛✌☞ ☛✌☞ ✍✑✍ ✍✎✍ ✍✎✍ No simplices — n 1 ? p ✔✖✕ 2

✁ ✠ Natural coordinates: Sibson’s coordinates Definition. 1 Sibson’s coordinates: area ☛ Vor ☛ p � p i λ i ✍✎✍ area ☛ Vor ☛ p ✍✎✍ ✁✄✂ ☞✍✌✏✎✑✆ ✁✟✞✠✁✒✂✓✡ ☎✝✆ ✁✟✞✠✁✄✂☛✡ Theorem. 1 Barycentric equality: ✠ ∑ λ i p i p i 3

✍ ☞ ✍ ✠ ☛ ✡ ✆ ✍ � ✁ ✠ ✂ ☎ Reconstruction of smooth surfaces ∑ i λ i ∂ ˆ � 1 Defs: h ☛ p ☛ p ✍ h i ☛ p S h ☛ 0 ✂✞✝ ✁✠✟ ✁✄✂ ✌ Observation: h interpolates the points and the h i ✌ Observation: Guarantees. . . 4

✍ ✁ ✄ ☞ ✍ Detecting the bipolar facets �✂✁ ✆✝✆ ✟✡✠ ✆✞✆ �☎✄ Def. A Delaunay triangle is called bipolar if ☛ IF IF ☛ cc 1 ☛ cc 2 0 5

Implicit surface Restricted Delaunay Triangulation 6

� � Implicit versus Modified Implicit Limitations – Natural weights / coordinates – Merits of the 0-level set? Code integrated to CATIA-v5 (March 2001) 7