discrete distortion for surface meshes mohammed mostefa mesmoudi leila de floriani paola magillo...
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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
C2-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? S
ign of m
ean curvature
+convex/saddle
-concave/saddle
Sign of Gaussian curvature
+convex/concave
-saddle
convex
saddle
0flat
0flat/saddle
flat[imposs.] saddle
concave
saddleridge
valley
Contribution
C2-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
• C2-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) r2
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]:• C2-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
End of the talk
• Thank you!
• Question?
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…