David Levin Tel-Aviv University Afrigraph 2009 Shape Preserving Deformation David Levin Tel-Aviv University Afrigraph 2009 Based on joint works with Yaron Lipman andDaniel Cohen-Or Yaron Lipman and Daniel Cohen-Or Tel-Aviv University

Table of Contents Representation and manipulation of discrete surfaces.  Rigid motion invariant discrete surface representation.  Moving Frames for Surface Deformation.  Volume representations and Green coordinates.

Representation and manipulation of discrete surfaces. deformation In this part we address the deformation problem: surface representations. Approach: use special surface representations.  Rotation invariant surface representation.  Moving frames.  Volume representation.

Linear Rotation-invariant Coordinates for Meshes SIGGRAPH 2005, special issue of ACM Transactions on Graphics. Shape is invariant to rigid-motions: shape representation for discrete surfaces (meshes) We wish to find a rigid-motion invariant shape representation for discrete surfaces (meshes):  Local support.  Easy to transform from other representations.  Easy to reconstruct.

Moving Frames on surfaces Moving frames on surfaces, generalization of Frenet-frame:  Now we have two parameters.

Rotation invariant Surface representation On Discrete Surfaces (Meshes): place a frame at each vertex (In a rigid-motion invariant manner). Tangential form:

Rotation invariant Surface representation On Discrete Surfaces (Meshes)

Rotation invariant Surface representation Discrete surface (linear!) equations

Rotation invariant Surface representation 1. Solve for the frames (Linear Least Squares) Rigid motion invariantrepresentation. 2. Solve for the vertices position. (Integration – Linear Least Squares) Theorem 1 Theorem 1: The set of coefficients define a unique discrete surface up to rigid motion.

Applications Deformation Deformation (prescribing frames on the surface):

Applications Morphing Morphing (linear blending of the coordinates) – can handle large rotations. Linear blending of rotation-invariant Coords. Linear blending of world Coords.

Applications Morphing Morphing (linear blending of the coordinates) – can handle large rotations. can handle large rotations

Applications

Moving Frames for Surface Deformation ACM Transactions on Graphics (2007). In previous method we look for new orientations of local frames. The deformation energy can be quantified in a more precise manner. Geometric distance We define: Geometric distance which reflects resemblance between curves and as such should ignore rigid motions. Curvature torsion Use a classical rigid motion invariant representation: Curvature and torsion to define:

Geometric Deformation of Curves Reduction: Instead for looking for f, we search for R: Theorem:

Geometric Deformation of Curves Reduction: Instead for looking for f, we search for R: Dirichlet energy

Geometric Deformation of Curves harmonic mappings Critical solutions of Dirichlet integral are called harmonic mappings. We have (still for curves): Shape is preserved if: harmonic quantity  Frenet Frames where rotated by harmonic quantity. geodesic curve  The Frenet Frames rotation map is geodesic curve on SO(3). But we know what are geodesics on SO(3): one parameter subgroup of SO(3)

Geometric Deformation of Curves Algorithm:

Geometric Distance and Optimal Rotation Field [Lipman et al. 2007] Let us seek optimal deformation. geometric distance  Maintain the first fundamental form (isometries) and minimize the second fundamental form. Define a geometric distance between isometric surfaces: H the matrix of (Gauss map) in local frame,

Geometric Distance and Optimal Rotation Field [Lipman et al. 2007] Similarly to the curve case we look for mapping R. Analogous to the curve case we have and therefore shapeharmonic The shape is preserved if the rotation map is harmonic

Algorithm To Minimize we choose a parameterization of SO(3):  Orthogonal  Orthogonal parameterization  Conformal  Conformal parameterization: Rotation angle Rotation axis And when one rotation axis is involved we can assume 0

Discretization Choose your favorite Laplace-Beltrami discretization (E.g. cot weight). Solve for the rotations Solve for the rotations (with constraints). Two cases:  One rotation axis: exist linear solution– the angle of rotation should be harmonic function of the mesh.  More than one rotation angle: A non-linear problem. However linear approximation is enough. Integrate Integrate: use the first discrete surface equation.

Results Discrete surface equations Rotations solved In SO(3)

Results

Green Coordinates Yaron Lipman, David Levin, Daniel Cohen-Or Tel Aviv University

Related work Free-form (space) deformations. Free-form (space) deformations.  Lattice-based (Sederberg & Parry 86, Coquillart 90, …)  Curve-/handle-based (Singh & Fiume 98, Botsch et al. 05, …)  Cage-based (Mean Value Coordinates) (Floater 03, Ju et al. 05, Langer et al. 06, Joshi et al. 07, Lipman et al. 07, Langer et al.08,… )  Vector Field constructions (Angelidis 04, Von Funck 06,…)

Generalized Barycentric Coordinates barycentric coordinates Recent generalization to barycentric coordinates allow defining space deformations using a flexible control polyhedra [Floater 03, Ju et al 05, Joshi et al. 07]: Advantages: Advantages: Fast, simple, robust, not limited to surfaces. Disadvantage Disadvantage (in our context): Do not preserve shape. Joshi et.al. 2007

Shape Preserving Free-Form deformation look What is the class of space mappings we look for? (preserve shape)  Conformal mappings  Conformal mappings are shape preserving: Angle preserving. Conformal mappings induce local similarity transformations :

Shape Preserving Free-Form deformation Two problems:  Conformal mappings  Conformal mappings are hard to compute. Schwarz-Christoffel mapping – computationally hard [ T.A.Driscoll L.N.Trefethen 02].  Conformal mappings  Conformal mappings exist only in 2D.

Shape Preserving Free-Form deformation can be produced What class of space mappings can be produced using the standard Free-from deformation formulation? Each axis is treated independently. Affine invariance. Scalar affine weights (coordinates) Constant points (w.r.t. p) No hope for conformal mappings in this formulation of free-form deformations

Result break Conformal space deformation Harmonic coordinates

Shape Preserving Free-Form deformation Idea Idea: use the normals to blend between the coordinates…

Shape Preserving Free-Form deformation Then the deformation is defined by:

Green Coordinates Question Question: what are and ? Green’s third identity Green’s third identity encodes a similar relation:  A harmonic function can be written as

Green Coordinates Use the coordinate functions for u:

Green Coordinates Use the coordinate functions for u:

Green Coordinates Theorem 1: Theorem 1:  The mapping for d=2 is conformal for all. Harmonic Coordinates Green Coordinates Might go out of the cage

Green Coordinates Theorem 2: Theorem 2: closed form expressions  The coordinate functions have closed form expressions for d=2,3. 2D Green Coordinates 3D Green Coordinates

Results Green CoordinatesHarmonic coordinates

Results Green Coordinates Mean-Value Coordinates Harmonic Coordinates Distortion histogram Observation (Conjecture): Observation (Conjecture): The mapping is quasi- conformal for d=3.

Results Green Coordinates Mean Value Coordinates

Results Green Coordinates Mean Value coordinates

Results Green Coordinates Mean Value coordinates

Results Green Coordinates Mean Value Coordinates

Green Coordinates Theorem 3: Theorem 3: unique analytic continuationreal-analytic continuation  The extension through every simplicial face is unique in the sense of analytic continuation in 2D and real-analytic continuation in 3D.  The formula for the unique extension is simply affine map where is an affine map.

Employment of partial cages Green Coordinates Mean-Value Coordinates

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