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

2. Introduction 3d objects are usually represented as triangular meshes Every triangle linearly approximates a part of the surface Why are we using linear patches to approximate a smooth manifold?

3. Interpolant C 2 surfaces We want to build a C 2 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

4. 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

5. 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

6. 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

7. Real time fitting of the CC

8. 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

9. 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

10. Fitmaps

11. 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

12. Example

13. 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 p and the input surface along the normal defined on p

14. Displacement Mapping

15. Results – David’s Head

16. Results - Bunny

17. Results - Fertility

18. Results – Gargoyle