Automatic Construction of Quad-Based Subdivision Surfaces using Fitmaps

1. Automatic Construction of Quad-Based Subdivision Surfaces using Fitmaps Daniele Panozzo, EnricoPuppo DISI - University of Genova, Italy Marco Tarini DICOM - Dipartimento Informatica e Comunicazione, Universita` dell'Isubria, Varese, Italy NicoPietroni, Paolo Cignoni Visual Computing Group - ISTI-CNR, Pisa, Italy

2. Interpolant C2 surfaces • We want to build a C2 surface that interpolates the input mesh • This representation has several advantages: • The surface can be sampled at arbitrary resolution • We need to store less information • We can compute the fundamental forms for every surface patch and compute differential properties in closed form

3. Uniform B-splines • Uniform B-splines can be used to approximate curves by deciding the number of control points and “fitting” them on the input data • B-splines can be used to define surfaces, the control points must be placed in a regular grid

4. Catmull Clark Subdivision Surfaces • CCs are a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology • CCs are defined recursively • CCs can be evaluated in closed form • CCs can be converted to bi-cubic Bezier patches

5. How can we “fit” a CC? • The limit of the recursive subdivision can be evaluated at the control points • Fitting the CC corresponds to solve a huge linear system

6. Dense Interpolant CC • This solution allows to produce a control grid that has the same number of control points as the number of sampled vertexes • We want to simplify the CC by removing control points, while preserving the initial shape • Intuitively we want to replace a set of small patches with a single one, whenever the substitution does not “alter” the surface

7. Local Topological Editing Operator • The control grid must be composed of quads! • To increase the quality of the surface, the control grid must be: • as regular as possible • aligned with the principal curvatures

8. Fitmaps

9. Local fitting of the CC • After every local operator we re-fit the CC by optimizing: • The quadratic distance between the new patches and the original surface • The shape of the quads of the control mesh

10. Example

11. High frequencies as scalar fields • As the simplification algorithm proceeds, a family of CCs is generated • The high frequency details on the surface are progressively lost • They can be encoded as displacement maps, one for each surface patch • A dispacement map is a scalar, continuous bivariate function that associates, for every point p in a surface patch, the distance between pandthe input surface along the normal defined on p

12. Displacement Mapping

13. Results – David’s Head

14. Results - Bunny

15. Results - Fertility

16. Results – Gargoyle