Generalized Barycentric Coordinates for Mesh deformation

Class Project CS777:
Instructor Prof. Mike Gleicher
Date 11th May, 2011

Contents:

  1. Introduction
  2. Barycentric Coordinates:
    1. Mean Value Coordinates
    2. Harmonic Coordinates
    3. Greens Coordinates
  3. Mesh deformation
  4. Image Warping
  5. Results and Analysis
  6. Video Clips
  7. Free C++ Source codes
  8. References

Introduction:

1.   Mean Value Coordinates (MVC)

Assume that the cage vertices are given in counter-clockwise direction. The barycentric coordinate of any point (v) inside the cage is calculated using three point formula as given by Hormann and Floater:

meanvalue

meaneq1

Instead of calculating angles, a faster method is given in the Hormann's paper.

meaneqn2

A robust implementation of the above expression must consider two special cases i.e. (I) when the query vertex is close to a corner (II) when the query vertex is on the edge.


meanval33

MVC are calculated for each point of interest in the domain, and stored in an array. The dimension of barycentric is equal to the number of cage corner points.  Later any movement of the cage corner points results in the calculation of XY coordinates of the query point as a linear combination of cage vertices i.e.

eqn9

Where "Pc" are the XY coordinates of the cage corners, and "Pq" is the XY coordinates of any query point whose position is controlled by cage points.

meanvalue3
 

mean1
 mean2

 Mean Value Coordinates


2. Harmonic Coordinates (HC)

For highly concave polygonal shapes, the MVC could be negative.  Smoother and positive coordinates are generated by solving Laplace equation for each corner of the cage. There are theoretical results based on minmax principles which guarantees that the values inside the domain are at least C2 continuous. 

eq3

Except for very simple geometries, the above equation has no closed form solution, therefore,  it must be solved by some numerical methods. Both direct solvers such as SuperLU or indirect iterative methods could be used. Since this equation is symmetric and positive definite, iterative methods such as Jacobi or Conjugent Gradient (CG) methods have  very nice convergence properties. Jacobi method, is although simplest to implement, has terrible convergence behavior, therefore we have used Preconditioned Conjugate Gradient (PCG) method (with Incomplete LU factorization as preconditioner). 

First of all, in order to solve the above equation, the interior domain must be discretized. In 2D, we used Jonathan Shewchuk's Triangle Software. For example,
delgrid
In order to get smooth results, the cage boundary may also need fine discretization.  Once the domain is appropriately discretized, we have to set proper boundary conditions as follows:

hcbound

As shown in the above picture, if we want to solve the Laplace equation for corner vertex "j", eqn5 is set. The values between (j, j+1) and (j, j-1) are linearly interpolated. Now we are ready for the iterative solver:


eqn6
Which is simple averaging scheme around the neighbor of each vertex. This scheme, although extremely simple as two drawbacks (1) It starts with good convergence towards the final solution, but as the iteration count increases, the convergence starts crawling and it take many iteration before acceptable tolerance value is reached (2) It is grid resolution dependent: when the grid size increases, the convergence becomes slower.Except for some tiny grid size, this method is unacceptable.  Multigrid methods solve this problem, but they need complex data structures which I have not used in this work.  Instead, we solve this problem using Krylov subspace method, which have better convergence properties.

  • Kyrlov Subspace method: 

Standard Conjugent gradient works only for symmetric and positive definite matrices.  With slight modifications, the Laplace equation can be made symmetric. Laplace equation is always positive definite:
eqn12

eqn13


   
I used ITL  library  ( included in MTL4  library to solve the above equations. Some experimental results are shown in the following table ( There are lots of scope for optimization, which is important for models with large number of cage points).

Laplace Solver
 Time(in Seconds)
per cage point
Simple Jacobi
 424
CG ( No Preconditioner)
 6.725
CG( Diagonal Preconditioner)
 6.095
CG( ILU(0) Preconditioner)
 5.450

These results were obtained on Quad-Core, Intel processor with 4GB RAM running Ubuntu 11.04.  With this experiment, we can draw some conclusions:
  • CG methods have far superior ( reduced ) time complexity.
  • For this class of problem ( Laplacian ), the preconditioners did not improve the results, Therefore, standard CG method is preferable because of its lower space complexity.

hc1
 hc2

Harmonic Coordinates

3. Green Coordinates:


Green coordinates have shape preservation properties and like MVC, they have closed formed solution. The exact derivation is lengthy  and matematical, the pseudo code given by Lipamn etc. is pretty straightforward to implement. The following pseudo code is direct cut and paste from the author's paper.

GreenCoord

The only thing, that have to remember is that the above formulation assume that cage coordinates are is in clockwise direction.  Also it should be noted that in the above code only eqn7needs to be normalized ( and it is not done in the code) and not the eqn8.

Once both coordinates have been calculated, the XY coordinate of any point with respect to the deformed cage is given by

eq11

Where S(j) = length of deformed jth cage segment/original length of the jth cage segment and n(tj) is the outward normal to the jth cage segment.  One interesting thing in this equation is that we are adding point ( 1st expression ) with the vector ( 2nd term), which I don't understand, therefore, I have taken these expressions for granted.


gcphi
 gcpsi

Green Coordinates's phivalues at each cage corners.

gc_phi
 gc_psi
Green Coordinate's psivalues at each cage corners.


Comparing Time complexities:

Intuitively, both MVC and GC must be faster than HC because of the closed form expressions and GC must be slower than MVC as it involve many complex mathmatical expressions including exp, atan, and log.  Here is one experimental results which will give some fair idea about time complexities of each method.
For this experiment, sampling was done on 11000 points with 4 node rectangular cage.

Mean Value Coordinates
 ~0.1730 seconds
Green Coordinates
~0.1874 seconds
Harmonic Coordinates
 ~24.0 seconds

These results are  somewhat difficult to compare with DeRose and Meyer's paper, in which they report that HC are faster than MVC. In fact, the timing for HC in their paper for 15362 object points in 30 second and in our case, it is 24 seconds for 11000 points, but their timings for MVC is 113 seconds, and in our case it is 0.1730 seconds, therefore, some more investigations are needed to have fair evaluation of these methods.  Unlike DeRose's paper, I did not perform any sparsification.  (Or may be that MVC are harder for 3D cages, which I did not try). 

Results:


result1_pic1
 result1_pic2

                                                                    Experimental data and triangulation for Harmonic Field Calculation ( 11000 points)



AnimatedGif


Visual evaluation : (A) Mean Value Coordinates  (B) Greens Coordinates (C) Harmonic Coordinates

anim2
Visual evaluation : (A) Mean Value Coordinates  (B) Greens Coordinates (C) Harmonic Coordinates

External Software used:

  • MTL4 library for sparse matrix storage and ITL for conjugate gradient solvers.
  • Jonathan Shewchuk's "Triangle" software for Delaunay triangulation.
  • Yaron Lipman provided Matlab code for the verification purpose.

Source Code C++

Mesh.h
Mesh.cc
GeneralizedBaryCentric.h
GeneralizedBaryCentric.cpp

( At present, the codes are not packeged properly, but fair enough for integration).

References:

  1. Mean Value Coordinates:  Michael Floater
  2. Harmonic Coordinates: Tony DeRose, Mark Meyer
  3. Harmonic Coordinates for Character Articulation:  Pushkar Joshi, Mark Meyer, Toney DeRose, Brian Green, Tom Sonacki
  4. Moving Remy in Harmony: Pixar's Use of Harmonic Functions: David Austin
  5. Green Coordinates: Yaron Lipman, David Levin, Daniel Cohen-Or
  6. Generalized Barycentric Coordinates for Irregular Polygons: Mark Meyer, Haeyoung Lee, Alan Barr, Mathieu Desbrun.