Laplacian surface editing

先讲下各个公式

公式(2)是laplacian coordinates的定义

公式(5)第一部分是保证laplacian coordinate坐标的一致性, 后面是handle点的约束

公式(8)是T的构造

公式(12)是求T中的系数

2D MATLAB 版在它的project里面自带, 这里将其拓展成3D版本, 源代码如下:

function U = laplacian_surface_editing_3D(vertex,faces,BI,BC)%The file is an implementation of 'Laplacian surface editing' in 3D. The
%original accompany code with this paper is in 2D. Based on this code, I
%build the 3D version.%Inputs: vertex %   vertex  #vertex by 3 list of rest domain positions%   faces   #faces by 3 list of triangle indices into vertex%   b       #b list of indices of constraint (boundary) vertices%   bc      #b by 3 list of constraint positions for b%Output: %   U       #V by dim list of new positions% By seamanj @ NCCAn = length(vertex);options.symmetrize=0;
options.normalize=1;
L = compute_mesh_laplacian(vertex, faces, 'combinatorial', options );delta = L * vertex; %delta为拉普拉斯坐标
L_prime = [   L     zeros(n) zeros(n)   % the x-partzeros(n)    L     zeros(n)       zeros(n) zeros(n)    L    ];
neighbors = compute_vertex_ring(faces);for i = 1:nring = [i neighbors{i}];V = vertex(ring,:)';V = [Vones(1,length(ring))];%这里为1, 乘上v'就变成公式(10)中的A了,这里这么写就是把v'合并了%V第一行为x,第二行为y,第三行为z,第四行为1C = zeros(length(ring) * 3, 7);% ... Fill C infor r=1:length(ring)C(r,:) =                [V(1,r) 0 V(3,r) (-1)*V(2,r) V(4,r) 0 0];%V第一行为x，第二行为y,第三行为z,第四行为1C(length(ring)+r,:) =   [V(2,r) (-1)*V(3,r) 0 V(1,r) 0 V(4,r) 0];C(2*length(ring)+r,:) = [V(3,r) V(2,r) (-1)*V(1,r) 0 0 0 V(4,r)];end;  Cinv = pinv(C);s =   Cinv(1,:);h1 =  Cinv(2,:);h2 =  Cinv(3,:);h3 =  Cinv(4,:);delta_i = delta(i,:)';delta_ix = delta_i(1);delta_iy = delta_i(2);delta_iz = delta_i(3);% T*delta gives us an array of coefficients  % T*delta*V'等于公式(5)的T(V')*delta, 注意这里我们要求的是V'Tdelta = [delta_ix*s       + delta_iy*(-1)*h3 + delta_iz*h2delta_ix*h3      + delta_iy*s       + delta_iz*(-1)*h1delta_ix*(-1)*h2 + delta_iy*h1      + delta_iz*s];% updating the weights in Lx_prime, Ly_prime, Lw_prime%L_prime里面原本有L,这里就是公式(5)中的T(V')*delta - L(V')L_prime(i,[ring (ring + n) (ring + 2*n)]) = ...L_prime(i,[ring (ring + n) (ring + 2*n)]) + (-1)*Tdelta(1,:);L_prime(i+n,[ring (ring + n) (ring + 2*n)]) = ...L_prime(i+n,[ring (ring + n) (ring + 2*n)]) + (-1)*Tdelta(2,:);     L_prime(i+n*2,[ring (ring + n) (ring + 2*n)]) = ...L_prime(i+n*2,[ring (ring + n) (ring + 2*n)]) + (-1)*Tdelta(3,:); end% weight for the constraints
w=1;% building the least-squares system matrix
A_prime = L_prime;
rhs = zeros(3*n,1);for j=1:length(BI)A_prime = [A_primew*((1:(3*n))==BI(j))w*((1:(3*n))==(BI(j)+n))w*((1:(3*n))==(BI(j)+2*n))];rhs = [rhsw*BC(j,1)w*BC(j,2)w*BC(j,3)];
end;% solving for v-primes
xyz_col = A_prime\rhs;
U = [xyz_col(1:n) xyz_col((n+1):(2*n)) xyz_col((2*n+1):(3*n))];

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