Automatic Construction of Quad-Based Subdivision Surfaces using Fitmaps

Automatic Construction of Quad-Based Subdivision Surfaces using Fitmaps

Daniele Panozzo, Enrico PuppoDISI - University of Genova, Italy

Marco TariniDICOM - Dipartimento Informatica e Comunicazione, Universita` dell'Isubria, Varese, Italy

Nico Pietroni, Paolo CignoniVisual Computing Group - ISTI-CNR, Pisa, Italy

Daniele Panozzo, Enrico PuppoDISI - University of Genova, Italy

Marco TariniDICOM - Dipartimento Informatica e Comunicazione, Universita` dell'Isubria, Varese, Italy

Nico Pietroni, Paolo CignoniVisual Computing Group - ISTI-CNR, Pisa, Italy

Interpolant C2 surfacesInterpolant 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

Uniform B-splinesUniform 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

Catmull Clark Subdivision SurfacesCatmull 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

How can we “fit” a CC?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

Dense Interpolant CCDense 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

Local Topological Editing OperatorLocal 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

FitmapsFitmaps

Local fitting of the CCLocal 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

ExampleExample

High frequencies as scalar fieldsHigh 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

Displacement MappingDisplacement Mapping

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