Research Report Research Report FWF S9206FWF S9206

Helmut PottmannHelmut Pottmann

Geometric Modeling & Industrial GeometryGeometric Modeling & Industrial Geometry

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Map labeling algorithmMap labeling algorithmMap labeling algorithmMap labeling algorithm

• Collision free initial position using digitized images

• Iterative improvement using force directed optimization

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Surface fittingSurface fittingin feature sensitive metricin feature sensitive metricSurface fittingSurface fittingin feature sensitive metricin feature sensitive metric

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Integral invariantsIntegral invariantsIntegral invariantsIntegral invariants

• Integration over local neighborhoods

• „Curvature“ on various scales

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Minkowski sum boundary Minkowski sum boundary surfaces of 3D objectssurfaces of 3D objectsMinkowski sum boundary Minkowski sum boundary surfaces of 3D objectssurfaces of 3D objects

• Minkowski sum A+B of objects A and B is the outer envelope of B with respect to all translations determined by points aA.

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Parametrization of rational Parametrization of rational and convolution surfacesand convolution surfacesParametrization of rational Parametrization of rational and convolution surfacesand convolution surfaces

• The convolution surface A*B is the generalized offset of A w.r.t. B.

• Thm.: The convolution of any rational surface and a polynomial surface with a linear normal vector field is always rational.

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Line element geometryLine element geometryLine element geometryLine element geometry

• Line element = line + point on it

• Surface recognition, reconstruction; segmentation

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Geometric automatic Geometric automatic extraction of sulcal fundiextraction of sulcal fundiGeometric automatic Geometric automatic extraction of sulcal fundiextraction of sulcal fundi

• Fundi = deepest regions of cortex; use a geodesic depth measure w.r.t. to hull surface computed with level set method

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3D shape morphing based on 3D shape morphing based on geodesics in shape spacesgeodesics in shape spaces3D shape morphing based on 3D shape morphing based on geodesics in shape spacesgeodesics in shape spaces

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Scientific talksScientific talksScientific talksScientific talks

• H. Pottmann: Univ. Innsbruck, Tsinghua Univ., Seoul Nat. Univ.: Integral invariants

• M. Peternell, SIAM’05: Envelopes of moving solids

• M. Peternell, SIAM’05: Constrained Surface Approximation from a Geometric Optimization Perspective

• M. Hofer, SIAM’05: 3D Shape Recognition and Reconstruction Based on Line Element Geometry