├── .gitmodules ├── LICENSE ├── README.md ├── cpp ├── CMakeLists.txt ├── armadillo-598-init.vtk ├── armadillo-598-rest.vtk ├── david-A-input.obj ├── untangle2d.cpp └── untangle3d.cpp ├── octopus.mp4 ├── python ├── laplace.py ├── mesh.py ├── untangle.py └── winslow.py ├── slides.pdf └── title-page.png /.gitmodules: -------------------------------------------------------------------------------- 1 | [submodule "cpp/ultimaille"] 2 | path = cpp/ultimaille 3 | url = https://github.com/ssloy/ultimaille.git 4 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU AFFERO GENERAL PUBLIC LICENSE 2 | Version 3, 19 November 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU Affero General Public License is a free, copyleft license for 11 | software and other kinds of works, specifically designed to ensure 12 | cooperation with the community in the case of network server software. 13 | 14 | The licenses for most software and other practical works are designed 15 | to take away your freedom to share and change the works. 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It is safest 628 | to attach them to the start of each source file to most effectively 629 | state the exclusion of warranty; and each file should have at least 630 | the "copyright" line and a pointer to where the full notice is found. 631 | 632 | 633 | Copyright (C) 634 | 635 | This program is free software: you can redistribute it and/or modify 636 | it under the terms of the GNU Affero General Public License as published 637 | by the Free Software Foundation, either version 3 of the License, or 638 | (at your option) any later version. 639 | 640 | This program is distributed in the hope that it will be useful, 641 | but WITHOUT ANY WARRANTY; without even the implied warranty of 642 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 643 | GNU Affero General Public License for more details. 644 | 645 | You should have received a copy of the GNU Affero General Public License 646 | along with this program. If not, see . 647 | 648 | Also add information on how to contact you by electronic and paper mail. 649 | 650 | If your software can interact with users remotely through a computer 651 | network, you should also make sure that it provides a way for users to 652 | get its source. For example, if your program is a web application, its 653 | interface could display a "Source" link that leads users to an archive 654 | of the code. There are many ways you could offer source, and different 655 | solutions will be better for different programs; see section 13 for the 656 | specific requirements. 657 | 658 | You should also get your employer (if you work as a programmer) or school, 659 | if any, to sign a "copyright disclaimer" for the program, if necessary. 660 | For more information on this, and how to apply and follow the GNU AGPL, see 661 | . 662 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # How to compute locally invertible maps 2 | 3 | This repository contains source code and slides shown during MIT Vision and Graphics Seminar (March 2, 2021) 4 | 5 | [![](https://raw.githubusercontent.com/ssloy/invertible-maps/6aa25ed6f6741e63eefc8b41969beabe9d6fec66/title-page.png)](https://github.com/ssloy/invertible-maps/raw/main/slides.pdf) 6 | 7 | 8 | # C++ code for mesh smoothing/untangling 9 | 10 | This repository also contains the source code for 2d/3d constrained boundary mesh untangling. 11 | 12 | This code successfully passes the entire Locally Injective Mappings Benchmark [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.3827969.svg)](https://doi.org/10.5281/zenodo.3827969) 13 | 14 | For the initial testing purposes, we provide a copy of two example problems (`cpp/david-A-input.obj` and `cpp/armadillo-598-init.vtk`) taken from the benchmark. 15 | Challenge the code with your data! 16 | 17 | # Compile and run: 18 | ```sh 19 | git clone --recurse-submodules https://github.com/ssloy/invertible-maps && 20 | cd invertible-maps/cpp && 21 | mkdir build && 22 | cd build && 23 | cmake .. && 24 | make -j && 25 | ./untangle2d ../david-A-input.obj result2d.obj && 26 | ./untangle3d ../armadillo-598-init.vtk ../armadillo-598-rest.vtk result3d.vtk 27 | ``` 28 | 29 | -------------------------------------------------------------------------------- /cpp/CMakeLists.txt: -------------------------------------------------------------------------------- 1 | cmake_minimum_required(VERSION 2.8) 2 | project(untangle) 3 | 4 | if(NOT CMAKE_BUILD_TYPE) 5 | set(CMAKE_BUILD_TYPE Release) 6 | endif() 7 | 8 | set(CMAKE_CXX_STANDARD 17) 9 | set(CMAKE_CXX_STANDARD_REQUIRED ON) 10 | find_package(OpenMP) 11 | 12 | if (NOT WIN32) 13 | if(OPENMP_FOUND) 14 | set(CMAKE_C_FLAGS "${CMAKE_C_FLAGS} ${OpenMP_C_FLAGS}") 15 | set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${OpenMP_CXX_FLAGS}") 16 | set(CMAKE_EXE_LINKER_FLAGS "${CMAKE_EXE_LINKER_FLAGS} ${OpenMP_EXE_LINKER_FLAGS}") 17 | endif() 18 | endif() 19 | 20 | add_subdirectory(ultimaille) 21 | include_directories(ultimaille) 22 | include_directories(ultimaille/ext) 23 | 24 | if (MSVC) 25 | # warning level 4 26 | add_compile_options(/W4) 27 | else() 28 | # lots of warnings and all warnings as errors 29 | add_compile_options(-Wall -Wextra -pedantic) 30 | endif() 31 | 32 | add_executable(untangle2d untangle2d.cpp) 33 | target_link_libraries(untangle2d ultimaille) 34 | 35 | add_executable(untangle3d untangle3d.cpp) 36 | target_link_libraries(untangle3d ultimaille) 37 | 38 | -------------------------------------------------------------------------------- /cpp/untangle2d.cpp: -------------------------------------------------------------------------------- 1 | #include 2 | #include 3 | #include 4 | #include 5 | #include 6 | 7 | #include 8 | 9 | using namespace UM; 10 | 11 | double triangle_area_2d(vec2 a, vec2 b, vec2 c) { 12 | return .5*((b.y-a.y)*(b.x+a.x) + (c.y-b.y)*(c.x+b.x) + (a.y-c.y)*(a.x+c.x)); 13 | } 14 | 15 | double triangle_aspect_ratio_2d(vec2 a, vec2 b, vec2 c) { 16 | double l1 = (b-a).norm(); 17 | double l2 = (c-b).norm(); 18 | double l3 = (a-c).norm(); 19 | double lmax = std::max(l1, std::max(l2, l3)); 20 | return lmax*(l1+l2+l3)/(4.*std::sqrt(3.)*triangle_area_2d(a, b, c)); 21 | } 22 | 23 | inline double chi(double eps, double det) { 24 | if (det>0) 25 | return (det + std::sqrt(eps*eps + det*det))*.5; 26 | return .5*eps*eps / (std::sqrt(eps*eps + det*det) - det); 27 | } 28 | 29 | inline double chi_deriv(double eps, double det) { 30 | return .5+det/(2.*std::sqrt(eps*eps + det*det)); 31 | } 32 | 33 | struct Untangle2D { 34 | Untangle2D(Triangles &mesh) : m(mesh), X(m.nverts()*2), lock(m.points), ref_tri(m), J(m), K(m), det(m), area(m) { 35 | for (int t : facet_iter(m)) { 36 | area[t] = m.util.unsigned_area(t); 37 | vec2 A,B,C; 38 | m.util.project(t, A, B, C); 39 | 40 | double ar = triangle_aspect_ratio_2d(A, B, C); 41 | if (ar>10) { // if the aspect ratio is bad, assign an equilateral reference triangle 42 | double a = ((B-A).norm() + (C-B).norm() + (A-C).norm())/3.; // edge length is the average of the original triangle 43 | area[t] = sqrt(3.)/4.*a*a; 44 | A = {0., 0.}; 45 | B = {a, 0.}; 46 | C = {a/2., std::sqrt(3.)/2.*a}; 47 | } 48 | 49 | mat<2,2> ST = {{B-A, C-A}}; 50 | ref_tri[t] = mat<3,2>{{ {-1,-1},{1,0},{0,1} }}*ST.invert_transpose(); 51 | } 52 | } 53 | 54 | void lock_boundary_verts() { 55 | SurfaceConnectivity fec(m); 56 | for (int v : vert_iter(m)) 57 | lock[v] = fec.is_boundary_vert(v); 58 | } 59 | 60 | void evaluate_jacobian(const std::vector &X) { 61 | detmin = std::numeric_limits::max(); 62 | ninverted = 0; 63 | #pragma omp parallel for reduction(min:detmin) reduction(+:ninverted) 64 | for (int t=0; t &J = this->J[t]; 66 | J = {}; 67 | for (int i=0; i<3; i++) 68 | for (int d : range(2)) 69 | J[d] += ref_tri[t][i]*X[2*m.vert(t,i) + d]; 70 | this->K[t] = { {{ +J[1].y, -J[1].x }, { -J[0].y, +J[0].x }} }; // dual basis 71 | det[t] = J.det(); 72 | detmin = std::min(detmin, det[t]); 73 | ninverted += (det[t]<=0); 74 | } 75 | } 76 | 77 | bool go() { 78 | std::vector spin_locks(X.size()); 79 | eps = 1; 80 | evaluate_jacobian(X); 81 | if (debug>0) std::cerr << "number of inverted elements: " << ninverted << std::endl; 82 | for (int iter=0; iter0) std::cerr << "iteration #" << iter << std::endl; 84 | const LBFGS_Optimizer::func_grad_eval func = [&](const std::vector& X, double& F, std::vector& G) { 85 | std::fill(G.begin(), G.end(), 0); 86 | F = 0; 87 | evaluate_jacobian(X); 88 | //#pragma omp parallel for reduction(vec_double_plus:G) reduction(+:F) 89 | #pragma omp parallel for reduction(+:F) 90 | for (int t=0; t trash(X.size()); 119 | func(X, E_prev, trash); 120 | 121 | LBFGS_Optimizer opt(func); 122 | opt.gtol = bfgs_threshold; 123 | opt.maxiter = bfgs_maxiter; 124 | opt.run(X); 125 | 126 | func(X, E, trash); 127 | if (debug>0) std::cerr << "E: " << E << " eps: " << eps << " detmin: " << detmin << " ninv: " << ninverted << std::endl; 128 | 129 | #if 0 130 | double sigma = std::max(1.-E/E_prev, 1e-1); 131 | if (detmin>=0) 132 | eps *= (1-sigma); 133 | else 134 | eps *= 1 - (sigma*std::sqrt(detmin*detmin + eps*eps))/(std::abs(detmin) + std::sqrt(detmin*detmin + eps*eps)); 135 | 136 | #else 137 | double sigma = std::max(1.-E/E_prev, 1e-1); 138 | double mu = (1-sigma)*chi(eps, detmin); 139 | if (detmin0 && std::abs(E_prev - E)/E<1e-5) break; 145 | } 146 | return !ninverted; 147 | } 148 | 149 | //////////////////////////////// 150 | // Untangle2D state variables // 151 | //////////////////////////////// 152 | 153 | // optimization input parameters 154 | Triangles &m; // the mesh to optimize 155 | double theta = 1./128.; // the energy is (1-theta)*(shape energy) + theta*(area energy) 156 | int maxiter = 10000; // max number of outer iterations 157 | double bfgs_threshold = 1e-4; 158 | int bfgs_maxiter = 30000; // max number of inner iterations 159 | int debug = 1; // verbose level 160 | 161 | // optimization state variables 162 | 163 | std::vector X; // current geometry 164 | PointAttribute lock; // currently lock = boundary vertices 165 | FacetAttribute> ref_tri; 166 | FacetAttribute> J; // per-tet Jacobian matrix = [[JX.x JX.y, JX.z], [JY.x, JY.y, JY.z], [JZ.x, JZ.y, JZ.z]] 167 | FacetAttribute> K; // per-tet dual basis: det J = dot J[i] * K[i] 168 | FacetAttribute det; // per-tet determinant of the Jacobian matrix 169 | FacetAttribute area; // reference area 170 | double eps; // regularization parameter, depends on min(jacobian) 171 | 172 | double detmin; // min(jacobian) over all tetrahedra 173 | int ninverted; // number of inverted tetrahedra 174 | }; 175 | 176 | int main(int argc, char** argv) { 177 | if (2>argc) { 178 | std::cerr << "Usage: " << argv[0] << " model.mesh [result.mesh]" << std::endl; 179 | return 1; 180 | } 181 | 182 | std::string res_filename = "result.mesh"; 183 | if (3<=argc) { 184 | res_filename = std::string(argv[2]); 185 | } 186 | 187 | Triangles m; 188 | SurfaceAttributes attr = read_by_extension(argv[1], m); 189 | std::cerr << "Untangling " << argv[1] << "," << m.nverts() << "," << std::endl; 190 | PointAttribute tex_coord("tex_coord", attr, m); 191 | 192 | vec2 bbmin, bbmax; // these are used to undo the scaling we apply to the model 193 | const double boxsize = 10.; 194 | { // scale the target domain for better numerical stability 195 | bbmin = bbmax = tex_coord[0]; 196 | for (int v : vert_iter(m)) { 197 | for (int d : range(2)) { 198 | bbmin[d] = std::min(bbmin[d], tex_coord[v][d]); 199 | bbmax[d] = std::max(bbmax[d], tex_coord[v][d]); 200 | } 201 | } 202 | double maxside = std::max(bbmax.x-bbmin.x, bbmax.y-bbmin.y); 203 | for (int v : vert_iter(m)) 204 | tex_coord[v] = (tex_coord[v] - (bbmax+bbmin)/2.)*boxsize/maxside + vec2(1,1)*boxsize/2.; 205 | } 206 | 207 | // { // scale the input geometry to have the same area as the target domain 208 | double target_area = 0; 209 | for (int t : facet_iter(m)) { 210 | vec2 a = tex_coord[m.vert(t, 0)]; 211 | vec2 b = tex_coord[m.vert(t, 1)]; 212 | vec2 c = tex_coord[m.vert(t, 2)]; 213 | target_area += triangle_area_2d(a, b, c); 214 | } 215 | um_assert(target_area>0); // ascertain mesh requirements 216 | double source_area = 0; 217 | for (int t : facet_iter(m)) 218 | source_area += m.util.unsigned_area(t); 219 | for (vec3 &p : m.points) 220 | p *= std::sqrt(target_area/source_area); 221 | // } 222 | 223 | Untangle2D opt(m); 224 | 225 | #if 0 226 | for (int t : facet_iter(m)) { 227 | opt.area[t] = target_area/m.nfacets(); 228 | double a = sqrt(opt.area[t]*4./sqrt(3.)); 229 | vec2 A = {0., 0.}; 230 | vec2 B = {a, 0.}; 231 | vec2 C = {a/2., std::sqrt(3.)/2.*a}; 232 | mat<2,2> ST = {{B-A, C-A}}; 233 | opt.ref_tri[t] = mat<3,2>{{ {-1,-1},{1,0},{0,1} }}*ST.invert_transpose(); 234 | } 235 | #endif 236 | 237 | for (int v : vert_iter(m)) 238 | for (int d : range(2)) 239 | opt.X[2*v+d] = tex_coord[v][d]; 240 | 241 | opt.lock_boundary_verts(); 242 | 243 | auto t1 = std::chrono::high_resolution_clock::now(); 244 | bool success = opt.go(); 245 | auto t2 = std::chrono::high_resolution_clock::now(); 246 | std::chrono::duration time = t2 - t1; 247 | 248 | if (success) 249 | std::cerr << "SUCCESS; running time: " << time.count() << " s; min det J = " << opt.detmin << std::endl; 250 | else 251 | std::cerr << "FAIL TO UNTANGLE!" << std::endl; 252 | 253 | for (int v : vert_iter(m)) { 254 | for (int d : range(2)) 255 | m.points[v][d] = opt.X[2*v+d]; 256 | m.points[v].z = 0; 257 | } 258 | 259 | { // restore scale 260 | double maxside = std::max(bbmax.x-bbmin.x, bbmax.y-bbmin.y); 261 | for (vec3 &p : m.points) 262 | p = (p - vec3(1,1,1)*boxsize/2)/boxsize*maxside + (vec3(bbmax.x, bbmax.y, 0)+vec3(bbmin.x, bbmin.y, 0))/2.; 263 | } 264 | 265 | write_by_extension(res_filename, m, SurfaceAttributes{ { {"selection", opt.lock.ptr} }, { {"det", opt.det.ptr} }, {} }); 266 | return 0; 267 | } 268 | 269 | -------------------------------------------------------------------------------- /cpp/untangle3d.cpp: -------------------------------------------------------------------------------- 1 | #include 2 | #include 3 | #include 4 | #include 5 | #include 6 | 7 | #include 8 | 9 | using namespace UM; 10 | 11 | inline double chi(double eps, double det) { 12 | if (det>0) 13 | return (det + std::sqrt(eps*eps + det*det))*.5; 14 | return .5*eps*eps / (std::sqrt(eps*eps + det*det) - det); 15 | } 16 | 17 | inline double chi_deriv(double eps, double det) { 18 | return .5+det/(2.*std::sqrt(eps*eps + det*det)); 19 | } 20 | 21 | struct Untangle3D { 22 | Untangle3D(Tetrahedra &mesh) : m(mesh), X(m.nverts()*3), lock(m.points, false), J(m), K(m), det(m), ref_tet(m), volume(m) { 23 | for (int t : cell_iter(m)) { 24 | volume[t] = m.util.cell_volume(t); 25 | #if 1 26 | mat<3,3> ST = {{ 27 | m.points[m.vert(t, 1)] - m.points[m.vert(t, 0)], 28 | m.points[m.vert(t, 2)] - m.points[m.vert(t, 0)], 29 | m.points[m.vert(t, 3)] - m.points[m.vert(t, 0)] 30 | }}; 31 | #else 32 | Tetrahedra R; // regular tetrahedron with unit edge length, centered at the origin (sqrt(2)/12 volume) 33 | R.cells = {0,1,2,3}; 34 | *R.points.data = { 35 | { .5, 0, -1./(2.*std::sqrt(2.))}, 36 | {-.5, 0, -1./(2.*std::sqrt(2.))}, 37 | { 0, -.5, 1./(2.*std::sqrt(2.))}, 38 | { 0, .5, 1./(2.*std::sqrt(2.))} 39 | }; 40 | double a = std::cbrt(volume[t]*6.*std::sqrt(2.)); 41 | for (vec3 &p : R.points) // scale the tet 42 | p = p*a; 43 | mat<3,3> ST = {{ 44 | R.points[1] - R.points[0], 45 | R.points[2] - R.points[0], 46 | R.points[3] - R.points[0] 47 | }}; 48 | #endif 49 | ref_tet[t] = mat<4,3>{{ {-1,-1,-1},{1,0,0},{0,1,0},{0,0,1} }}*ST.invert_transpose(); 50 | } 51 | } 52 | 53 | void lock_boundary_verts() { 54 | VolumeConnectivity vec(m); 55 | for (int c : cell_iter(m)) 56 | for (int lf : range(4)) 57 | if (vec.adjacent[m.facet(c, lf)]<0) 58 | for (int lv : range(3)) 59 | lock[m.facet_vert(c, lf, lv)] = true; 60 | } 61 | 62 | void evaluate_jacobian(const std::vector &X) { 63 | detmin = std::numeric_limits::max(); 64 | ninverted = 0; 65 | #pragma omp parallel for reduction(min:detmin) reduction(+:ninverted) 66 | for (int c=0; c &J = this->J[c]; 68 | J = {}; 69 | for (int i=0; i<4; i++) 70 | for (int d : range(3)) 71 | J[d] += ref_tet[c][i]*X[3*m.vert(c,i) + d]; 72 | det[c] = J.det(); 73 | detmin = std::min(detmin, det[c]); 74 | ninverted += (det[c]<=0); 75 | 76 | this->K[c] = { // dual basis 77 | {{ 78 | J[1].y*J[2].z - J[1].z*J[2].y, 79 | J[1].z*J[2].x - J[1].x*J[2].z, 80 | J[1].x*J[2].y - J[1].y*J[2].x 81 | }, 82 | { 83 | J[0].z*J[2].y - J[0].y*J[2].z, 84 | J[0].x*J[2].z - J[0].z*J[2].x, 85 | J[0].y*J[2].x - J[0].x*J[2].y 86 | }, 87 | { 88 | J[0].y*J[1].z - J[0].z*J[1].y, 89 | J[0].z*J[1].x - J[0].x*J[1].z, 90 | J[0].x*J[1].y - J[0].y*J[1].x 91 | }} 92 | }; 93 | } 94 | } 95 | 96 | bool go() { 97 | std::vector spin_locks(X.size()); 98 | eps = 1.; 99 | evaluate_jacobian(X); 100 | if (debug>0) std::cerr << "number of inverted elements: " << ninverted << std::endl; 101 | for (int iter=0; iter0) std::cerr << "iteration #" << iter << std::endl; 103 | 104 | const LBFGS_Optimizer::func_grad_eval func = [&](const std::vector& X, double& F, std::vector& G) { 105 | std::fill(G.begin(), G.end(), 0); 106 | F = 0; 107 | evaluate_jacobian(X); 108 | #pragma omp parallel for reduction(+:F) 109 | for (int t=0; t &a = this->J[t]; // tangent basis 111 | mat<3,3> &b = this->K[t]; // dual basis 112 | double c1 = chi(eps, det[t]); 113 | double c2 = pow(c1, 2./3.); 114 | double c3 = chi_deriv(eps, det[t]); 115 | 116 | double f = (a[0]*a[0] + a[1]*a[1] + a[2]*a[2])/c2; 117 | double g = (1+det[t]*det[t])/c1; 118 | F += ((1-theta)*f + theta*g)*volume[t]; 119 | 120 | for (int dim : range(3)) { 121 | vec3 dfda = a[dim]*(2./c2) - b[dim]*((2.*f*c3)/(3.*c1)); 122 | vec3 dgda = b[dim]*((2*det[t]-g*c3)/c1); 123 | 124 | for (int i=0; i<4; i++) { 125 | int v = m.vert(t,i); 126 | if (lock[v]) continue; 127 | spin_locks[v*3+dim].lock(); 128 | G[v*3+dim] += ((dfda*(1.-theta) + dgda*theta)*ref_tet[t][i])*volume[t]; 129 | spin_locks[v*3+dim].unlock(); 130 | } 131 | } 132 | } 133 | }; 134 | 135 | double E_prev, E; 136 | std::vector trash(X.size()); 137 | func(X, E_prev, trash); 138 | 139 | LBFGS_Optimizer opt(func); 140 | opt.gtol = bfgs_threshold; 141 | opt.maxiter = bfgs_maxiter; 142 | opt.run(X); 143 | 144 | func(X, E, trash); 145 | if (debug>0) std::cerr << "E: " << E << " eps: " << eps << " detmin: " << detmin << " ninv: " << ninverted << std::endl; 146 | 147 | double sigma = std::max(1.-E/E_prev, 1e-1); 148 | double mu = (1-sigma)*chi(eps, detmin); 149 | if (detmin0 && std::abs(E_prev - E)/E<1e-5) break; 154 | } 155 | return !ninverted; 156 | } 157 | 158 | //////////////////////////////// 159 | // Untangle3D state variables // 160 | //////////////////////////////// 161 | 162 | // optimization input parameters 163 | Tetrahedra &m; // the mesh to optimize 164 | double theta = 1./2.; // the energy is (1-theta)*(shape energy) + theta*(area energy) 165 | int maxiter = 10000; // max number of outer iterations 166 | int bfgs_maxiter = 3000; // max number of inner iterations 167 | double bfgs_threshold = 1e-4; 168 | 169 | int debug = 1; // verbose level 170 | 171 | // optimization state variables 172 | 173 | std::vector X; // current geometry 174 | PointAttribute lock; // currently lock = boundary vertices 175 | CellAttribute> J; // per-tet Jacobian matrix = [[JX.x JX.y, JX.z], [JY.x, JY.y, JY.z], [JZ.x, JZ.y, JZ.z]] 176 | CellAttribute> K; // per-tet dual basis: det J = dot J[i] * K[i] 177 | CellAttribute det; // per-tet determinant of the Jacobian matrix 178 | CellAttribute> ref_tet; // reference tetrahedron: array of 4 normal vectors to compute the gradients 179 | CellAttribute volume; // reference volume 180 | double eps; // regularization parameter, depends on min(jacobian) 181 | 182 | double detmin; // min(jacobian) over all tetrahedra 183 | int ninverted; // number of inverted tetrahedra 184 | }; 185 | 186 | int main(int argc, char** argv) { 187 | if (3>argc) { 188 | std::cerr << "Usage: " << argv[0] << " init.mesh reference.mesh [result.mesh]" << std::endl; 189 | return 1; 190 | } 191 | 192 | std::string res_filename = "result.mesh"; 193 | if (4<=argc) { 194 | res_filename = std::string(argv[3]); 195 | } 196 | 197 | Tetrahedra ini, ref; 198 | read_by_extension(argv[1], ini); 199 | read_by_extension(argv[2], ref); 200 | std::cerr << "Untangling " << argv[1] << "," << ini.nverts() << "," << std::endl; 201 | 202 | if (ini.nverts()!=ref.nverts() || ini.ncells()!=ref.ncells()) { 203 | std::cerr << "Error: " << argv[1] << " and " << argv[2] << " must have the same number of vertices and tetrahedra, aborting" << std::endl; 204 | return -1; 205 | } 206 | 207 | /* 208 | std::vector tokill(ref.ncells(), false); 209 | std::vector> new_cells; 210 | VolumeConnectivity vec(ref); 211 | for (int c : cell_iter(ref)) { 212 | int cf2 = -1; 213 | int cf1 = -1; 214 | for (int cf : range(4)) if (vec.adjacent[4*c + cf] == -1) { 215 | if (cf1 == -1) cf1 = cf; 216 | else cf2 = cf; 217 | } 218 | if (cf2<0) continue; 219 | 220 | int he = vec.halfedge(c, cf1, 0); 221 | for (int i : range(2)) if (vec.cell_facet(vec.opposite_f(he)) != cf2) he = vec.next(he); 222 | 223 | he = vec.prev(vec.opposite_f(vec.next(he))); 224 | 225 | int mid = ref.nverts(); 226 | ref.points.push_back(0.5 * (ref.points[vec.from(he)] + ref.points[vec.to(he)])); 227 | ini.points.push_back(0.5 * (ini.points[vec.from(he)] + ini.points[vec.to(he)])); 228 | 229 | for (int he2split : vec.halfedges_around_edge(he)) { 230 | tokill[vec.cell(he2split)] = true; 231 | new_cells.push_back({ vec.from(he2split), vec.to(vec.next(he2split)), mid, vec.to(vec.next(vec.opposite_f(he2split))) }); 232 | he2split = vec.opposite_f(he2split); 233 | new_cells.push_back({ vec.from(he2split), vec.to(vec.next(he2split)), mid, vec.to(vec.next(vec.opposite_f(he2split))) }); 234 | } 235 | } 236 | ref.delete_cells(tokill); 237 | ini.delete_cells(tokill); 238 | 239 | { 240 | int off = ref.create_cells(new_cells.size()); 241 | for (int i : range(new_cells.size())) for (int lv : range(4)) ref.vert(off + i, lv) = new_cells[i][lv]; 242 | } 243 | { 244 | int off = ini.create_cells(new_cells.size()); 245 | for (int i : range(new_cells.size())) for (int lv : range(4)) ini.vert(off + i, lv) = new_cells[i][lv]; 246 | } 247 | 248 | write_by_extension("split-rest.mesh", ref); 249 | write_by_extension("split-init.mesh", ini); 250 | // return 0; 251 | */ 252 | 253 | 254 | 255 | #if 0 256 | Permutation perm(ref.nverts()); 257 | Permutation perm2(ref.nverts()); 258 | HilbertSort hs(*ref.points.data); 259 | hs.apply(perm.ind); 260 | perm.apply(*ref.points.data); 261 | perm.apply(*ini.points.data); 262 | perm.apply_reverse(perm2.ind); 263 | for (int t : cell_iter(ref)) 264 | for (int lv : range(4)) 265 | ini.vert(t, lv) = ref.vert(t, lv) = perm2[ref.vert(t, lv)]; 266 | write_geogram("gna.geogram", ref); 267 | #endif 268 | 269 | 270 | bool inverted = false; 271 | { // ascertain the mesh requirements 272 | double ref_volume = 0, ini_volume = 0; 273 | for (int c : cell_iter(ref)) { 274 | ref_volume += ref.util.cell_volume(c); 275 | ini_volume += ini.util.cell_volume(c); 276 | } 277 | 278 | if ( 279 | (ref_volume<0 && ini_volume>0) || 280 | (ref_volume>0 && ini_volume<0) 281 | ) { 282 | std::cerr << "Error: " << argv[1] << " and " << argv[2] << " must have the orientation, aborting" << std::endl; 283 | return -1; 284 | } 285 | 286 | inverted = (ini_volume<=0); 287 | if (inverted) { 288 | std::cerr << "Warning: the input has negative volume, inverting" << std::endl; 289 | for (vec3 &p : ini.points) 290 | p.x *= -1; 291 | for (vec3 &p : ref.points) 292 | p.x *= -1; 293 | } 294 | } 295 | 296 | vec3 bbmin, bbmax; // these are used to undo the scaling we apply to the model 297 | const double boxsize = 10.; 298 | 299 | { // scale 300 | ref.points.util.bbox(bbmin, bbmax); 301 | double maxside = std::max(bbmax.x-bbmin.x, bbmax.y-bbmin.y); 302 | for (vec3 &p : ref.points) 303 | p = (p - (bbmax+bbmin)/2.)*boxsize/maxside + vec3(1,1,1)*boxsize/2; 304 | for (vec3 &p : ini.points) 305 | p = (p - (bbmax+bbmin)/2.)*boxsize/maxside + vec3(1,1,1)*boxsize/2; 306 | } 307 | 308 | Untangle3D opt(ref); 309 | 310 | for (int v : vert_iter(ref)) 311 | for (int d : range(3)) 312 | opt.X[3*v+d] = ini.points[v][d]; 313 | 314 | opt.lock_boundary_verts(); 315 | 316 | auto t1 = std::chrono::high_resolution_clock::now(); 317 | bool success = opt.go(); 318 | auto t2 = std::chrono::high_resolution_clock::now(); 319 | std::chrono::duration time = t2 - t1; 320 | 321 | if (success) 322 | std::cerr << "SUCCESS; running time: " << time.count() << " s; min det J = " << opt.detmin << std::endl; 323 | else 324 | std::cerr << "FAIL TO UNTANGLE!" << std::endl; 325 | 326 | for (int v : vert_iter(ref)) 327 | for (int d : range(3)) 328 | ref.points[v][d] = opt.X[3*v+d]; 329 | 330 | { // restore scale 331 | double maxside = std::max(bbmax.x-bbmin.x, bbmax.y-bbmin.y); 332 | for (vec3 &p : ref.points) 333 | p = (p - vec3(1,1,1)*boxsize/2)/boxsize*maxside + (bbmax+bbmin)/2.; 334 | } 335 | 336 | if (inverted) 337 | for (vec3 &p : ref.points) 338 | p.x *= -1; 339 | 340 | write_by_extension(res_filename, ref, VolumeAttributes{ { {"selection", opt.lock.ptr} }, { {"det", opt.det.ptr} }, {}, {} }); 341 | return 0; 342 | } 343 | 344 | -------------------------------------------------------------------------------- /octopus.mp4: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ssloy/invertible-maps/b3f2048f63a37b6853ae15b8fb677c122696480c/octopus.mp4 -------------------------------------------------------------------------------- /python/laplace.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/python3 2 | 3 | from mesh import Mesh 4 | mesh = Mesh("z") # a quad mesh with regular grid connectivity 5 | n = mesh.size 6 | u,v = mesh.x[:n*n], mesh.x[n*n:] # the grid is made of n*n verts 7 | 8 | for _ in range(128): # Gauss-Seidel iterations solving for zero Laplacian 9 | for j in range(1, n-1): # the boundary is fixed, so we iterate 10 | for i in range(1, n-1): # through interior vertices only 11 | idx = i+j*n 12 | u[idx] = (u[idx-1] + u[idx+1] + u[idx-n] + u[idx+n])/4. 13 | v[idx] = (v[idx-1] + v[idx+1] + v[idx-n] + v[idx+n])/4. 14 | print(mesh) 15 | mesh.show() 16 | 17 | -------------------------------------------------------------------------------- /python/mesh.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | from PIL import Image, ImageDraw 3 | 4 | class Mesh(): 5 | def __init__(self, test_case): # generate the test problem 6 | n = self.size 7 | self.quads = [ [i+j*n, i+1+j*n, i+1+(j+1)*n, i+(j+1)*n] for j in range(n-1) for i in range(n-1) ] # connectivity 8 | self.boundary = [ i==0 or i==n-1 or j==0 or j==n-1 for j in range(n) for i in range(n) ] # vertex boundary flags 9 | 10 | self.x = np.array([ i/n for j in range(n) for i in range(n) ] + \ 11 | [ j/n for j in range(n) for i in range(n) ]) # regular grid 12 | 13 | if (test_case=="z"): # Belinsky Z test case 14 | self.x = np.array([ i/n + int(j>=n//2)*3/5 for j in range(n) for i in range(n) ] + \ 15 | [ 2*j/n - int(j>=n//2)*3/5 for j in range(n) for i in range(n) ]) # 2D geometry 16 | elif (test_case=="chicane"): # chicane test case 17 | self.x = np.array([ i/n + int(j>=n//2)*3/5 for j in range(n) for i in range(n) ] + \ 18 | [ 2*j/n for j in range(n) for i in range(n) ]) # 2D geometry 19 | elif (test_case=="disc"): # disc test case 20 | blist = [i for i in range(n)] + [(i+2)*n-1 for i in range(n-2)] + [n*n - 1 - i for i in range(n)] + [n*(n-1) - (i+1)*n for i in range(n-2)] 21 | for i,v in enumerate(blist): 22 | self.x[v ] = np.cos(i/len(blist)*2.*np.pi+np.pi/4.) 23 | self.x[v+n*n] = np.sin(i/len(blist)*2.*np.pi+np.pi/4.) 24 | 25 | @property 26 | def size(self): 27 | return 8 28 | 29 | @property 30 | def nverts(self): 31 | return self.size*self.size 32 | 33 | def __str__(self): # wavefront .obj output 34 | ret = "" 35 | for v in range(self.nverts): 36 | ret = ret + ("v %f %f 0\n" % (self.x[v], self.x[v+self.nverts])) 37 | for f in self.quads: 38 | ret = ret + ("f %d %d %d %d\n" % (f[0]+1, f[1]+1, f[2]+1, f[3]+1)) 39 | return ret 40 | def show(self): 41 | res = 1000 42 | off = 100 43 | image = Image.new(mode='L', size=(res, res), color=255) 44 | draw = ImageDraw.Draw(image) 45 | 46 | for quad in self.quads: 47 | for e in range(4): 48 | i = quad[e] 49 | j = quad[(e+1)%4] 50 | 51 | line = ((off+self.x[i]*res/2, off+self.x[i+self.nverts]*res/2), (off+self.x[j]*res/2, off+self.x[j+self.nverts]*res/2)) 52 | draw.line(line, fill=128) 53 | del draw 54 | # image.save("winslow.png") 55 | image.show() 56 | -------------------------------------------------------------------------------- /python/untangle.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/python3 2 | 3 | from mesh import Mesh 4 | import numpy as np 5 | from scipy.optimize import fmin_l_bfgs_b 6 | 7 | mesh = Mesh("z") # a quad mesh with regular grid connectivity 8 | n = mesh.nverts 9 | 10 | Q = [ np.matrix('-1,-1;1,0;0,0;0,1'), np.matrix('-1,0;1,-1;0,1;0,0'), # quadratures for 11 | np.matrix('0,0;0,-1;1,1;-1,0'), np.matrix('0,-1;0,0;1,0;-1,1') ] # every quad corner 12 | 13 | def jacobian(U, qc, quad): # evaluate the Jacobian matrix at the given quadrature point 14 | return np.matrix([[U[quad[0] ], U[quad[1] ], U[quad[2] ], U[quad[3] ]], 15 | [U[quad[0]+n], U[quad[1]+n], U[quad[2]+n], U[quad[3]+n]]]) * Q[qc] 16 | 17 | for iter in range(10): # outer L-BFGS loop 18 | mindet = min( [ np.linalg.det( jacobian(mesh.x, qc, quad) ) for quad in mesh.quads for qc in range(4) ] ) 19 | eps = np.sqrt(1e-6**2 + .04*min(mindet, 0)**2) # the regularization parameter e 20 | def energy(U): # compute the energy and its gradient for the map u 21 | F,G = 0, np.zeros(2*n) 22 | for quad in mesh.quads: # sum over all quads 23 | for qc in range(4): # evaluate the Jacobian matrix for every quad corner 24 | J = jacobian(U, qc, quad) 25 | det = np.linalg.det(J) 26 | chi = det/2 + np.sqrt(eps**2 + det**2)/2 # the penalty function 27 | chip = .5 + det/(2*np.sqrt(eps**2 + det**2)) # its derivative 28 | f = np.trace(np.transpose(J)*J)/chi # quad corner shape quality 29 | F += f 30 | dfdj = (2*J - np.matrix([[J[1,1],-J[1,0]],[-J[0,1],J[0,0]]])*f*chip)/chi 31 | dfdu = Q[qc] * np.transpose(dfdj) # chain rule for the actual variables 32 | for i,v in enumerate(quad): 33 | if (mesh.boundary[v]): continue # the boundary verts are locked 34 | G[v ] += dfdu[i,0] 35 | G[v+n] += dfdu[i,1] 36 | return F,G 37 | mesh.x = fmin_l_bfgs_b(energy, mesh.x, factr=1e12)[0] # inner L-BFGS loop 38 | print(mesh) 39 | mesh.show() 40 | 41 | -------------------------------------------------------------------------------- /python/winslow.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/python3 2 | 3 | from mesh import Mesh 4 | 5 | mesh = Mesh("z") # a quad mesh with regular grid connectivity 6 | n = mesh.size 7 | 8 | u,v = mesh.x[:n*n], mesh.x[n*n:] # the grid is made of n*n verts 9 | def g11(i,j): # metric tensor estimation via finite differences 10 | return (u[i+1+j*n]-u[i-1+j*n])**2/4. + (v[i+1+j*n]-v[i-1+j*n])**2/4. 11 | def g22(i,j): 12 | return (u[i+j*n+n]-u[i+j*n-n])**2/4. + (v[i+j*n+n]-v[i+j*n-n])**2/4. 13 | def g12(i,j): 14 | return (u[i+1+j*n]-u[i-1+j*n])*(u[i+j*n+n]-u[i+j*n-n])/4. + \ 15 | (v[i+1+j*n]-v[i-1+j*n])*(v[i+j*n+n]-v[i+j*n-n])/4. 16 | for _ in range(128): # Gauss-Seidel iterations, zero Laplacian of the inverse map 17 | for j in range(1, n-1): # the boundary is fixed, so we iterate 18 | for i in range(1, n-1): # through interior vertices only 19 | a,b,c = g22(i,j), 2*g22(i,j)+2*g11(i,j), g22(i,j) 20 | d = g11(i,j)*(u[i+j*n+n] + u[i+j*n-n]) - 2*g12(i,j)* \ 21 | (u[i+1+j*n+n] + u[i-1+j*n-n] - u[i-1+j*n+n] - u[i+1+j*n-n])/4. 22 | e = g11(i,j)*(v[i+j*n+n] + v[i+j*n-n]) - 2*g12(i,j)* \ 23 | (v[i+1+j*n+n] + v[i-1+j*n-n] - v[i-1+j*n+n] - v[i+1+j*n-n])/4. 24 | u[i+j*n] = (d + a*u[i-1+j*n] + c*u[i+1+j*n])/b # actual Gauss-Seidel 25 | v[i+j*n] = (e + a*v[i-1+j*n] + c*v[i+1+j*n])/b # linear system update 26 | print(mesh) 27 | mesh.show() 28 | 29 | -------------------------------------------------------------------------------- /slides.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ssloy/invertible-maps/b3f2048f63a37b6853ae15b8fb677c122696480c/slides.pdf -------------------------------------------------------------------------------- /title-page.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ssloy/invertible-maps/b3f2048f63a37b6853ae15b8fb677c122696480c/title-page.png --------------------------------------------------------------------------------