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
• Consider the trihedral angles of tetrahedra defined by each three faces incident in p
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)
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