Discrete Distortion for Surface Meshes Mohammed Mostefa Mesmoudi Leila De Floriani Paola Magillo Dept. of Computer Science, University of Genova, Italy

Outline 1.Context, motivation, contribution… 2.Discrete distortion: idea 3.Definition and properties 4.Experimental results 5.Conclusions and future work

Outline 1.Context, motivation, contribution… 2.Discrete distortion: idea 3.Definition and properties 4.Experimental results 5.Conclusions and future work

C 2 -continuous surface Curvature at any point Discrete surface model: triangle mesh Approximation of curvature at mesh vertices Aim of the work

What is Curvature for? Gaussian curvature Mean curvature  Morphological shape analysis: classify points of the surface...

What is Curvature for? Sign of mean curvature + convex/ saddle - concave/ saddle Sign of Gaussian curvature + convex/concave - saddle convex saddle 0 flat 0 flat/saddle flat [imposs.] saddle concave saddleridge valley

Contribution C 2 -continuous surface Mean curvature at any point Discrete surface model: triangle mesh Approximation of mean curvature at mesh vertices Discrete distortion

Outline 1.Context, motivation, contribution… 2.Discrete distortion: idea 3.Definition and properties 4.Experimental results 5.Conclusions and future work

Distortion: Idea Triangle mesh (with orientation) p (internal) vertex triangles incident in p p

Distortion: Idea Consider a local tetrahedralization extenderd below the surface p

Distortion: Idea Consider a local tetrahedralization extenderd below the surface p

Distortion: Idea Consider the trihedral angles of tetrahedra defined by each three faces incident in p p

Distortion: Idea Solid angle at p = sum of all such angles p

Distortion: Idea If the mesh is flat at p… p

Distortion: Idea Then the solid angle is 2  (equivalent to the area of half a sphere) p

Distortion: Idea Then the solid angle is 2  (equivalent to the area of half a sphere) p

Distortion: Idea If the mesh is not flat at p… p

Distortion: Idea If the mesh is not flat at p… p

Distortion: Idea If the mesh is not flat at p… p

Distortion: Idea Then we measure how much the solid angle is different from 2  p

Outline 1.Context, motivation, contribution… 2.Discrete distortion: idea 3.Definition and properties 4.Experimental results 5.Conclusions and future work

Distortion: Definition Triangle mesh (with orientation) p internal vertex Definition of vertex distortion: D(p) = 2  – (solid angle at p) 

Distortion: Definition Definition of vertex distortion: D(p) = 2  – (solid angle at p)  flat  convex  concave p p

Distortion: Definition Definition of vertex distortion: D(p) = 2  – (solid angle at p) But we compute it in a simpler way…

Distortion: Computation Definition of bond distortion for an edge e: D(e) =  – (dihedral angle at e) Theorem: D(p) =  D(e) over e incident edges in p e

Distortion and Mean Curvature We use the Connolly function to show the relation between : –Mean curvature –Discrete distortion

C 2 -smooth surface (with orientation) p vertex sphere with center in p and radius r r small enough Definition of Connolly function: C(p,r) = (area of sphere part lying under the surface) r 2 Connolly Function (continuous case) p

Connolly Function (discrete case) Triangle mesh (with orientation): p vertex sphere with center in p and radius r r smaller than edges incident in p Connolly function becomes : C(p,r) = solid angle at p  Discrete distortion D(p) = 2  - C(p,r) p

Distortion and Mean Curvature Lemma [from Cazals, Chazals and Lewiner, 2003]: C 2 -smooth surface p internal point H(p) mean curvature at p C(p,r) = 2  +  H(p) r + … other term more fastly tending to 0 with r

Distortion and Mean Curvature C(p,r) Connolly function… Mean curvature C(p,r) ≈ 2  +  H(p) r, for small r Discrete distortion D(p) = 2  - C(p,r)    D(p) ≈ -  H(p) r  For fixed r their behavior is almost the same (up to a constant factor)…

Outline 1.Context, motivation, contribution… 2.Discrete distortion: idea 3.Definition and properties 4.Experimental results 5.Conclusions and future work

Experiments Compare: –Discrete distortion –A commonly used estimator for mean curvature: Mean angle deficit Color scale: from blue (min) to red (max)

Mount Marcy Distortion Mean angle deficit

Kitten Distortion Mean angle deficit

Retinal molecule Distortion Mean angle deficit

Mechanical piece (used piece) Distortion Mean angle deficit

Results Discrete distortion better adapts to surface shape Less sensitive to noise More effective in enhancing convex / concave areas

Outline 1.Context, motivation, contribution… 2.Discrete distortion: idea 3.Definition and properties 4.Experimental results 5.Conclusions and future work

Conclusions Discrete distortion is a good estimate for mean curvature of triangle meshes Discrete distortion provides an easier way to evaluate the Connoly function

Future Work  Many applications fields: physics of particle, chemistry…  Optimization of triangle meshes based on distortion

Acnowledgements This work has been partially supported by: Italian National Science Foundation MIUR-FIRB Project Shalom

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Related Work Analytic methods: approximate the triangle mesh with a smooth function and compute curvature in the continuum (problems: big meshes, which function) Discrete methods: Methods that divide by area Concentrated curvature (for Gaussian curvature) Other discrete methods…