Easy modeling generate new shapes by deforming existing ones

Transcription

1 Deformation I

2 Deformation

3 Motivation Easy modeling generate new shapes by deforming existing ones

4 Motivation Easy modeling generate new shapes by deforming existing ones

5 Motivation Character posing for animation

6 Challenges User says as little l as possible algorithm deduces dd the rest

7 Challenges Intuitive deformation global change + local detail preservation

8 Challenges Intuitive deformation global change + local detail preservation

9 Efficient! Challenges

10 Rules of the Game Shape Deformation Algorithm Deformed Shape Constraints Position: This point goes there Orientation/Scale: The environment of this point should rotate/scale Other shape property: Curvature, perimeter, [ Parameterization is also deformation : constraints = curvature 0 everywhere ]

11 Approaches Surface deformation Shape is empty pyshell Curve for 2D deformation Surface for 3D deformation Df Deformation only dfi defined don shape Deformation coupled with shape representation

12 Approaches Space deformation Shape is volumetric Planar domain in 2D Polyhedral domain in3d Deformation defined in neighborhood of shape Can be applied to any shape representation

13 Approaches Surface deformation Find alternative representation which is deformation invariant Space deformation Find a space map which has nice properties

14 Surface Deformation Setup: Choose alternative representation f(s) ) Given S find S such that Constraints(S ) are true f(s ) = f(s) (or close) An optimization problem S R R1 R2 R3

15 Shape Representation How good is the representation? Representation should always be invariant to: Global translation Global rotation Global scale? Depends on application Shapes we want reachable should have similar representations Almost isometric deformation local translation + rotation Almost conformal deformation local translation + rotation + scale

16 Shape Representation Robustness How hard is it to solve the optimization problem? Can we find the global minimum? Small change in constraints similar shape? Efficiency Canit be solvedat interactiverates? rates?

17 Shape Representations Rule of thumb: If representation is a linear function of the coordinates, deformation is: Robust Fast But representation tti is not rotation tti invariant! i (for large rotations)

18 Surface Representations Laplacian coordinates Edge lengths + dihedral angles Pyramid coordinates Local frames.

19 Laplacian Coordinates [Sorkine et al. 04] Control mechanism Handles (vertices) moved by user Region of influence (ROI) Movie

20 Laplacian Coordinates δ = LV = (I-D -1 A)V I = Identity matrix D = Diagonal matrix [d ii = deg(v i )] A = Adjacency matrix V = Vertices in mesh Approximation to normals unique up to translation Reconstruct by solving LV = δ for V, with one constraint Poisson equation

21 Deformation Pose modeling constraints for vertices C V v i = u i i C No exact solution, minimize i i error Laplacian Laplacian User Constraints coordinates of coordinates of original mesh deformed mesh

22 Deformation Laplacian Laplacian User Constraints coordinates of coordinates of original mesh deformed mesh

23 Linear Least Squares Ax = b with m equations, n unknowns Normal equations: (A T A)x = A T b Solution by pseudo inverse: x = A + b= [(A T A) -1 A T ]b If system under determined: x = arg min { x : Ax = b } If system over determined: x = arg min { Ax-b 2 }

24 Laplacian Coordinates Sanity Check Translation invariant? Rotation/scale invariant? δ i δ i δ i

25 Problem

26 input Laplacian coords Rotation invariant coords

27 Rotation Invariant Coords The representation should take into account local rotations + scale δ i = L(v i ) T i δ i = L(v i ) v i v' i δ i δ i T i δ i δ i

28 Rotation Invariant Coords The representation should take into account local rotations + scale δ i = L(v i ) T i δ i = L(v i ) Problem: T i depends on deformed position v i

29 Solution: Implicit Transformations Idea: solve for local transformation and deformed surface simultaneously Transformation of the local frame

30 Similarities Restrict T i to good transformations = rotation + scale similarity transformation Similarity Transformation

31 Similarities Conditions on T i to be a similarity matrix? Linear in 2D: Auxiliary variables ibl Uniform Rotation + scale translation

32 Similarities 2D

33 Similarities 3D case Not linear in 3D: Linearize by dropping the quadratic term Effectively: only small rotations are handled

34 Laplacian Coordinates Realtime? Need to solve a linear system each frame (A T A)x = A T b Precompute sparse Cholesky factorization Only back substitution per frame

35 Some Results

36 Some Results

37 Limitations: Large Rotations

38 How to Find the Rotations? Laplacian coordinates solve for them Problem: not linear Another approach: propagate rotations tti from handles

39 Rotation Propagation Compute handle s deformation gradient Extract rotation and scale/shear components Propagate damped rotations over ROI

40 Deformation Gradient Handle has been transformed affinely Deformation gradient is: Extract rotation R and scale/shear S

41 Smooth Propagation Construct smooth scalar field ld[ [0,1] α(x)=1 Full deformation (handle) α(x)=0 No deformation (fixed part) α(x) [0,1] ( ) [, ] Damp transformation (in between) Linearly damp scale/shear: S(x)= α(x)s(handle) L l d tti Log scale damp rotation: R(x) = exp(α(x)log(r(handle))

42 Limitations Works well for rotations Translations don t change deformation gradient Translation insensitivity

43 The Curse of Rotations Can t solve for them directly using a linear system Can t propagate if the handles don t rotate Some linear methods work for rotations Some work for translations None work for both

44 The Curse of Rotations Non linear methods work for both large rotations and translation only No free lunch: much more expensive