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

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…