Project-II: Comparing Mean Value, Harmonic and Green Coordinates for Mesh Deformation in Computer Animation

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1 Project-II: Comparing Mean Value, Harmonic and Green Coordinates for Mesh Deformation in Computer Animation Chaman Singh Verma April 20, 2011 Figure 1: Barycentric Mapping usage: (a) Straightening the Pisa Tower (b) Shape manipulation 1 Introduction Cage based mesh deformation (CMD) is perhaps the simplest, yet fairly effective approach to manipulate geometric/image objects in computer animation. Two major issues in CMD are (I) How to build a cage? (II) How to define barycentric coordinates for arbitrary shaped cage that produces deformations that are smooth, intuitive and conforming to some of the user s expectations. Barycentric coordinates were first introduced by Mobius ( circa 1830) for convex shapes. Many approaches have been made to generalize these coordinates for concave shapes, and currently some of the widely used techniques are: 1

2 1. Wachspress Coordinates (WC) 2. Mean Value Coordinates (MVC) 3. Harmonic Coordinates (HC) 4. Cauchy Green Coordinates (GC) 5. Maximum Entropy Coordinates (MEC) Many theoretical studies have been performed for these coordinates and true to the engineering adage Nothing is perfect, all these techniques have their pros and cons. Mean values have closed form solution, but fails for concave shapes, Harmonic coordinate have nice and smooth properties but sometimes computationally too expensive unless some sophisticated algorithms such as Multi-grid methods are used. Green coordinates have been introduced very recently, and they have some promise to supersede both MVC and HC. Alternative approaches such as MEC have been proposed, but there are not yet popular. 2 Forward and Backward Interpolation When an object undergoes transformation, there are two approaches by which attributes such as colors are interpolated. Figure 2: Interpolation (A) left image source (B) right image destination Forward Interpolation In this the source image attributes are directly copied on the corresponding barycentric coordinates. Although it is simplest one, it create many artifacts including possible holes in the distored image. Backward Interpolation In this method, for every destination pixel, a search is performed on the sourceimage, and the corresponding pixel values are copied. 2

3 3 Mean Value Coordinates Horman and Floater defined Mean Value Coordinates for arbitrary planar polygons as w i = 2 tan(α i 1/2)+tan(α i /2) r i (1) A modified version avoids calculating the angles is given by tan(α i /2) = r ir i+1 D i 2A i (2) If we denote s i = v i v then D i is the dot product of s i and s i+1. Normalized barycentric coordinates λ i is given by λ i = w i NC i=1 w i Where NC is the number of cage points. A complete C++ source code which handles all the special cases is given below. void mean_value_coordinates( const vector<point2d> &cagecoords, const Point2D &querycoord, vector<double> &barycoords) { int nsize = cagecoords.size(); assert( nsize ); double dx, dy; vector<point2d> s(nsize); for( int i = 0; i < nsize; i++) { dx = cagecoords[i][0] - querycoord[0]; dy = cagecoords[i][1] - querycoord[1]; s[i][0] = dx; s[i][1] = dy; barycoords.resize(nsize); for( int i = 0; i < nsize; i++) barycoords[i] = 0.0; int ip, im; // (i+1) and (i-1) double ri, rp, Ai, Di, dl, mu; // Distance double eps = 10.0*std::numeric_limits<double>::min(); // First check if any coordinates close to the cage point or // lie on the cage boundary. These are special cases. for( int i = 0; i < nsize; i++) { ip = (i+1)%nsize; ri = sqrt( s[i][0]*s[i][0] + s[i][1]*s[i][1] ); Ai = 0.5*(s[i][0]*s[ip][1] - s[ip][0]*s[i][1]); (3) 3

4 Di = s[ip][0]*s[i][0] + s[ip][1]*s[i][1]; if( ri <= eps) { barycoords[i] = 1.0; return; else if( fabs(ai) <= 0 && Di < 0.0) { dx = s[ip][0] - s[i][0]; dy = s[ip][1] - s[i][1]; dl = sqrt(dx*dx + dy*dy); assert(dl > eps); dx = querycoord[0] - s[i][0]; dy = querycoord[1] - s[i][1]; mu = sqrt(dx*dx + dy*dy)/dl; barycoords[i] = 1.0-mu; barycoords[ip] = mu; return; // Page #12, from the paper vector<double> tanalpha(nsize); // tan(alpha/2) for( int i = 0; i < nsize; i++) { ip = (i+1)%nsize; im = (nsize-1+i)%nsize; ri = sqrt( s[i][0]*s[i][0] + s[i][1]*s[i][1] ); rp = sqrt( s[ip][0]*s[ip][0] + s[ip][1]*s[ip][1] ); Ai = 0.5*(s[i][0]*s[ip][1] - s[ip][0]*s[i][1]); Di = s[ip][0]*s[i][0] + s[ip][1]*s[i][1]; tanalpha[i] = (ri*rp - Di)/(2.0*Ai); // Equation #11, from the paper double wi, wsum = 0.0; for( int i = 0; i < nsize; i++) { ip = (i+1)%nsize; im = (nsize-1+i)%nsize; ri = sqrt( s[i][0]*s[i][0] + s[i][1]*s[i][1] ); wi = 2.0*( tanalpha[im] + tanalpha[ip] )/ri; wsum += wi; barycoords[i] = wi; // Normalize for( int i = 0; i < nsize; i++) barycoords[i] /= wsum; 4

5 4 Harmonic Coordinates A smooth harmonic field is generated by solving Laplace equation. 2 φ = 0 (4) Figure 3: For each cage point (28 in this example), we need to solve Laplace equation In the interior the discretized form of the Laplacian is φ i 1,j 2φ i,j +φ i+1,j dx 2 + φ i,j 1 2φ i,j +φ i,j+1 dy 2 = 0 (5) A simple matrix free iterative solver is given by φ n+1 i,j = 0.25(φ n i 1,j +φn i+1,j +φn i,j 1 +φn i,j+1 ) (6) This system of linear equation will have a unique solution, if at least one boundary condition is applied. In HC evaluation, for every cage vertices, we set φ(c i ) = 1,φ(C j ) = 0.0 j i and solve this system. In general, HC evaluation is expensive because 5

6 The five point stencil is a high pass filter and its convergence rate is poor. Moreover, when the mesh size increases, the convergence become worse. For every cage vertices we have to solve the system after changing the boundary conditions. Since the Laplacian operator produces Symmetric and positive definite sparse system, we use preconditioned conjugate gradient ( PCG) method. For small system, direct solvers such as SuperLU could also be used. Since the Harmonic field can be generated only within the region, Harmonic Coordinates are also generated only for the internal vertices. Therefore, a given domain must be classified into three components i.e. External, Boundary and Internal. For images, this is done by region growing algorithm as follows 1. Tag all the cells ( pixels ) as UNIDENTIFIED 2. Scan-convert boundary pixels and tag them as BOUNDARY. 3. Start from one corner pixel in flood fill the region till it reaches BOUND- ARY pixels. All these pixels are tagged as EXTERNAL 4. All remaining cells are tagged INTERNAL The above method will work only for the waterright models, therefore, it is essential that the cage is simply conneted and closed. For structured mesh, this means that every boundary cells has atleast one neighboring cell ( total 8 neihboring cells ) is also tagged as BOUNDARY Figure 4: Multiple cage boundaries can be easily handle with Harmomic Coordinates 6

7 5 Green Coordinates Unlike MVC and HC, Green coordinates utilizes both the position of the cage nodes and normals to the edges. Although the derivation of the GC is quite involved, the pseudo code for 2D provided by Lipman etc is as follows. Figure 5: The pseudo code for 2D Green Coordinates given by Horman and Floater s paper A complete C++ code to calculate the GC is given below. int green_coordinates( const vector<point2d> &cagecoords, const Point2D &querycoord, vector<double> &phi, vector<double> &psi) { int nsegments = cagecoords.size()-1; Point2D a, b, nj; double Q, S, R, BA, SRT, L0, L1, A0, A1, A10, L10; psi.resize( nsegments ); phi.resize( nsegments + 1); for( int j = 0; j < nsegments; j++) { const Point2D &vj1 = cagecoords[j]; const Point2D &vj2 = cagecoords[(j+1)]; 7

8 a[0] = vj2[0] - vj1[0]; a[1] = vj2[1] - vj1[1]; nj[0] = a[1]; nj[1] = -a[0]; b[0] = vj1[0] - querycoord[0]; b[1] = vj1[1] - querycoord[1]; Q = a[0]*a[0] + a[1]*a[1]; S = b[0]*b[0] + b[1]*b[1]; R = 2.0*( a[0]*b[0] + a[1]*b[1]); BA = b[0]*nj[0] + b[1]*nj[1]; SRT = sqrt( 4.0*S*Q - R*R); L0 = log(s); L1 = log( S + Q + R ); A0 = atan( R/SRT)/SRT; A1 = atan((2*q+r)/srt)/srt; A10 = A1-A0; L10 = L1-L0; psi[j] = sqrt(q)/(4.0*m_pi)*(( 4.0*S-(R*R/Q))*A10 + (R/(2.0*Q))*L10 + L1-2); phi[j+1] += - (BA/(2.0*M_PI))*( (L10/(2.0*Q)) - A10*R/Q ); phi[j] += + (BA/(2.0*M_PI))*( (L10/(2.0*Q)) - A10*(2.0 + R/Q) ); //////////////////////////////////////////////////////////////////////////////////// References [1] Mean Value Coordinates Michael S. Floater [2] Harmonic Coordinates Tony DeRose Mark Meyer Pixar Technical Memo #06-02 Pixar Animation Studios [3] Harmonic Coordinates for Character Articulation Pushkar Joshi Mark Meyer Tony DeRose Brian Green Tom Sanocki Pixar Animation Studios [4] Green Coordinates Yaron Lipman, David Levin, Daniel Cohen-Or 8