meshlaplacian.c

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/*
 * $Id: meshlaplacian.c 36276 2011-04-21 15:53:30Z campbellbarton $
 *
 * ***** BEGIN GPL LICENSE BLOCK *****
 *
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 *
 * The Original Code is Copyright (C) 2001-2002 by NaN Holding BV.
 * All rights reserved.
 *
 * The Original Code is: all of this file.
 *
 * Contributor(s): none yet.
 *
 * ***** END GPL LICENSE BLOCK *****
 * meshlaplacian.c: Algorithms using the mesh laplacian.
 */

#include <math.h>
#include <string.h>

#include "MEM_guardedalloc.h"

#include "DNA_object_types.h"
#include "DNA_mesh_types.h"
#include "DNA_meshdata_types.h"
#include "DNA_scene_types.h"

#include "BLI_math.h"
#include "BLI_edgehash.h"
#include "BLI_memarena.h"
#include "BLI_utildefines.h"

#include "BKE_DerivedMesh.h"
#include "BKE_modifier.h"


#ifdef RIGID_DEFORM
#include "BLI_editVert.h"
#include "BLI_polardecomp.h"
#endif

#include "ONL_opennl.h"

#include "BLO_sys_types.h" // for intptr_t support

#include "ED_mesh.h"
#include "ED_armature.h"

#include "meshlaplacian.h"


/* ************* XXX *************** */
static void waitcursor(int UNUSED(val)) {}
static void progress_bar(int UNUSED(dummy_val), const char *UNUSED(dummy)) {}
static void start_progress_bar(void) {}
static void end_progress_bar(void) {}
static void error(const char *str) { printf("error: %s\n", str); }
/* ************* XXX *************** */


/************************** Laplacian System *****************************/

struct LaplacianSystem {
        NLContext context;      /* opennl context */

        int totvert, totface;

        float **verts;                  /* vertex coordinates */
        float *varea;                   /* vertex weights for laplacian computation */
        char *vpinned;                  /* vertex pinning */
        int (*faces)[3];                /* face vertex indices */
        float (*fweights)[3];   /* cotangent weights per face */

        int areaweights;                /* use area in cotangent weights? */
        int storeweights;               /* store cotangent weights in fweights */
        int nlbegun;                    /* nlBegin(NL_SYSTEM/NL_MATRIX) done */

        EdgeHash *edgehash;             /* edge hash for construction */

        struct HeatWeighting {
                MFace *mface;
                int totvert;
                int totface;
                float (*verts)[3];      /* vertex coordinates */
                float (*vnors)[3];      /* vertex normals */

                float (*root)[3];       /* bone root */
                float (*tip)[3];        /* bone tip */
                float (*source)[3]; /* vertex source */
                int numsource;

                float *H;                       /* diagonal H matrix */
                float *p;                       /* values from all p vectors */
                float *mindist;         /* minimum distance to a bone for all vertices */
                
                BVHTree   *bvhtree;     /* ray tracing acceleration structure */
                MFace     **vface;      /* a face that the vertex belongs to */
        } heat;

#ifdef RIGID_DEFORM
        struct RigidDeformation {
                EditMesh *mesh;

                float (*R)[3][3];
                float (*rhs)[3];
                float (*origco)[3];
                int thrownerror;
        } rigid;
#endif
};

/* Laplacian matrix construction */

/* Computation of these weights for the laplacian is based on:
   "Discrete Differential-Geometry Operators for Triangulated 2-Manifolds",
   Meyer et al, 2002. Section 3.5, formula (8).
   
   We do it a bit different by going over faces instead of going over each
   vertex and adjacent faces, since we don't store this adjacency. Also, the
   formulas are tweaked a bit to work for non-manifold meshes. */

static void laplacian_increase_edge_count(EdgeHash *edgehash, int v1, int v2)
{
        void **p = BLI_edgehash_lookup_p(edgehash, v1, v2);

        if(p)
                *p = (void*)((intptr_t)*p + (intptr_t)1);
        else
                BLI_edgehash_insert(edgehash, v1, v2, (void*)(intptr_t)1);
}

static int laplacian_edge_count(EdgeHash *edgehash, int v1, int v2)
{
        return (int)(intptr_t)BLI_edgehash_lookup(edgehash, v1, v2);
}

static float cotan_weight(float *v1, float *v2, float *v3)
{
        float a[3], b[3], c[3], clen;

        sub_v3_v3v3(a, v2, v1);
        sub_v3_v3v3(b, v3, v1);
        cross_v3_v3v3(c, a, b);

        clen = len_v3(c);

        if (clen == 0.0f)
                return 0.0f;
        
        return dot_v3v3(a, b)/clen;
}

static void laplacian_triangle_area(LaplacianSystem *sys, int i1, int i2, int i3)
{
        float t1, t2, t3, len1, len2, len3, area;
        float *varea= sys->varea, *v1, *v2, *v3;
        int obtuse = 0;

        v1= sys->verts[i1];
        v2= sys->verts[i2];
        v3= sys->verts[i3];

        t1= cotan_weight(v1, v2, v3);
        t2= cotan_weight(v2, v3, v1);
        t3= cotan_weight(v3, v1, v2);

        if(RAD2DEGF(angle_v3v3v3(v2, v1, v3)) > 90) obtuse= 1;
        else if(RAD2DEGF(angle_v3v3v3(v1, v2, v3)) > 90) obtuse= 2;
        else if(RAD2DEGF(angle_v3v3v3(v1, v3, v2)) > 90) obtuse= 3;

        if (obtuse > 0) {
                area= area_tri_v3(v1, v2, v3);

                varea[i1] += (obtuse == 1)? area: area*0.5f;
                varea[i2] += (obtuse == 2)? area: area*0.5f;
                varea[i3] += (obtuse == 3)? area: area*0.5f;
        }
        else {
                len1= len_v3v3(v2, v3);
                len2= len_v3v3(v1, v3);
                len3= len_v3v3(v1, v2);

                t1 *= len1*len1;
                t2 *= len2*len2;
                t3 *= len3*len3;

                varea[i1] += (t2 + t3)*0.25f;
                varea[i2] += (t1 + t3)*0.25f;
                varea[i3] += (t1 + t2)*0.25f;
        }
}

static void laplacian_triangle_weights(LaplacianSystem *sys, int f, int i1, int i2, int i3)
{
        float t1, t2, t3;
        float *varea= sys->varea, *v1, *v2, *v3;

        v1= sys->verts[i1];
        v2= sys->verts[i2];
        v3= sys->verts[i3];

        /* instead of *0.5 we divided by the number of faces of the edge, it still
           needs to be verified that this is indeed the correct thing to do! */
        t1= cotan_weight(v1, v2, v3)/laplacian_edge_count(sys->edgehash, i2, i3);
        t2= cotan_weight(v2, v3, v1)/laplacian_edge_count(sys->edgehash, i3, i1);
        t3= cotan_weight(v3, v1, v2)/laplacian_edge_count(sys->edgehash, i1, i2);

        nlMatrixAdd(i1, i1, (t2+t3)*varea[i1]);
        nlMatrixAdd(i2, i2, (t1+t3)*varea[i2]);
        nlMatrixAdd(i3, i3, (t1+t2)*varea[i3]);

        nlMatrixAdd(i1, i2, -t3*varea[i1]);
        nlMatrixAdd(i2, i1, -t3*varea[i2]);

        nlMatrixAdd(i2, i3, -t1*varea[i2]);
        nlMatrixAdd(i3, i2, -t1*varea[i3]);

        nlMatrixAdd(i3, i1, -t2*varea[i3]);
        nlMatrixAdd(i1, i3, -t2*varea[i1]);

        if(sys->storeweights) {
                sys->fweights[f][0]= t1*varea[i1];
                sys->fweights[f][1]= t2*varea[i2];
                sys->fweights[f][2]= t3*varea[i3];
        }
}

static LaplacianSystem *laplacian_system_construct_begin(int totvert, int totface, int lsq)
{
        LaplacianSystem *sys;

        sys= MEM_callocN(sizeof(LaplacianSystem), "LaplacianSystem");

        sys->verts= MEM_callocN(sizeof(float*)*totvert, "LaplacianSystemVerts");
        sys->vpinned= MEM_callocN(sizeof(char)*totvert, "LaplacianSystemVpinned");
        sys->faces= MEM_callocN(sizeof(int)*3*totface, "LaplacianSystemFaces");

        sys->totvert= 0;
        sys->totface= 0;

        sys->areaweights= 1;
        sys->storeweights= 0;

        /* create opennl context */
        nlNewContext();
        nlSolverParameteri(NL_NB_VARIABLES, totvert);
        if(lsq)
                nlSolverParameteri(NL_LEAST_SQUARES, NL_TRUE);

        sys->context= nlGetCurrent();

        return sys;
}

void laplacian_add_vertex(LaplacianSystem *sys, float *co, int pinned)
{
        sys->verts[sys->totvert]= co;
        sys->vpinned[sys->totvert]= pinned;
        sys->totvert++;
}

void laplacian_add_triangle(LaplacianSystem *sys, int v1, int v2, int v3)
{
        sys->faces[sys->totface][0]= v1;
        sys->faces[sys->totface][1]= v2;
        sys->faces[sys->totface][2]= v3;
        sys->totface++;
}

static void laplacian_system_construct_end(LaplacianSystem *sys)
{
        int (*face)[3];
        int a, totvert=sys->totvert, totface=sys->totface;

        laplacian_begin_solve(sys, 0);

        sys->varea= MEM_callocN(sizeof(float)*totvert, "LaplacianSystemVarea");

        sys->edgehash= BLI_edgehash_new();
        for(a=0, face=sys->faces; a<sys->totface; a++, face++) {
                laplacian_increase_edge_count(sys->edgehash, (*face)[0], (*face)[1]);
                laplacian_increase_edge_count(sys->edgehash, (*face)[1], (*face)[2]);
                laplacian_increase_edge_count(sys->edgehash, (*face)[2], (*face)[0]);
        }

        if(sys->areaweights)
                for(a=0, face=sys->faces; a<sys->totface; a++, face++)
                        laplacian_triangle_area(sys, (*face)[0], (*face)[1], (*face)[2]);
        
        for(a=0; a<totvert; a++) {
                if(sys->areaweights) {
                        if(sys->varea[a] != 0.0f)
                                sys->varea[a]= 0.5f/sys->varea[a];
                }
                else
                        sys->varea[a]= 1.0f;

                /* for heat weighting */
                if(sys->heat.H)
                        nlMatrixAdd(a, a, sys->heat.H[a]);
        }

        if(sys->storeweights)
                sys->fweights= MEM_callocN(sizeof(float)*3*totface, "LaplacianFWeight");
        
        for(a=0, face=sys->faces; a<totface; a++, face++)
                laplacian_triangle_weights(sys, a, (*face)[0], (*face)[1], (*face)[2]);

        MEM_freeN(sys->faces);
        sys->faces= NULL;

        if(sys->varea) {
                MEM_freeN(sys->varea);
                sys->varea= NULL;
        }

        BLI_edgehash_free(sys->edgehash, NULL);
        sys->edgehash= NULL;
}

static void laplacian_system_delete(LaplacianSystem *sys)
{
        if(sys->verts) MEM_freeN(sys->verts);
        if(sys->varea) MEM_freeN(sys->varea);
        if(sys->vpinned) MEM_freeN(sys->vpinned);
        if(sys->faces) MEM_freeN(sys->faces);
        if(sys->fweights) MEM_freeN(sys->fweights);

        nlDeleteContext(sys->context);
        MEM_freeN(sys);
}

void laplacian_begin_solve(LaplacianSystem *sys, int index)
{
        int a;

        if (!sys->nlbegun) {
                nlBegin(NL_SYSTEM);

                if(index >= 0) {
                        for(a=0; a<sys->totvert; a++) {
                                if(sys->vpinned[a]) {
                                        nlSetVariable(0, a, sys->verts[a][index]);
                                        nlLockVariable(a);
                                }
                        }
                }

                nlBegin(NL_MATRIX);
                sys->nlbegun = 1;
        }
}

void laplacian_add_right_hand_side(LaplacianSystem *UNUSED(sys), int v, float value)
{
        nlRightHandSideAdd(0, v, value);
}

int laplacian_system_solve(LaplacianSystem *sys)
{
        nlEnd(NL_MATRIX);
        nlEnd(NL_SYSTEM);
        sys->nlbegun = 0;

        //nlPrintMatrix();

        return nlSolveAdvanced(NULL, NL_TRUE);
}

float laplacian_system_get_solution(int v)
{
        return nlGetVariable(0, v);
}

/************************* Heat Bone Weighting ******************************/
/* From "Automatic Rigging and Animation of 3D Characters"
                 Ilya Baran and Jovan Popovic, SIGGRAPH 2007 */

#define C_WEIGHT                        1.0f
#define WEIGHT_LIMIT_START      0.05f
#define WEIGHT_LIMIT_END        0.025f
#define DISTANCE_EPSILON        1e-4f

typedef struct BVHCallbackUserData {
        float start[3];
        float vec[3];
        LaplacianSystem *sys;
} BVHCallbackUserData;

static void bvh_callback(void *userdata, int index, const BVHTreeRay *UNUSED(ray), BVHTreeRayHit *hit)
{
        BVHCallbackUserData *data = (struct BVHCallbackUserData*)userdata;
        MFace *mf = data->sys->heat.mface + index;
        float (*verts)[3] = data->sys->heat.verts;
        float lambda, uv[2], n[3], dir[3];

        mul_v3_v3fl(dir, data->vec, hit->dist);

        if(isect_ray_tri_v3(data->start, dir, verts[mf->v1], verts[mf->v2], verts[mf->v3], &lambda, uv)) {
                normal_tri_v3(n, verts[mf->v1], verts[mf->v2], verts[mf->v3]);
                if(lambda < 1.0f && dot_v3v3(n, data->vec) < -1e-5f) {
                        hit->index = index;
                        hit->dist *= lambda;
                }
        }

        mul_v3_v3fl(dir, data->vec, hit->dist);

        if(isect_ray_tri_v3(data->start, dir, verts[mf->v1], verts[mf->v3], verts[mf->v4], &lambda, uv)) {
                normal_tri_v3(n, verts[mf->v1], verts[mf->v3], verts[mf->v4]);
                if(lambda < 1.0f && dot_v3v3(n, data->vec) < -1e-5f) {
                        hit->index = index;
                        hit->dist *= lambda;
                }
        }
}

/* Raytracing for vertex to bone/vertex visibility */
static void heat_ray_tree_create(LaplacianSystem *sys)
{
        MFace *mface = sys->heat.mface;
        float (*verts)[3] = sys->heat.verts;
        int totface = sys->heat.totface;
        int totvert = sys->heat.totvert;
        int a;

        sys->heat.bvhtree = BLI_bvhtree_new(totface, 0.0f, 4, 6);
        sys->heat.vface = MEM_callocN(sizeof(MFace*)*totvert, "HeatVFaces");

        for(a=0; a<totface; a++) {
                MFace *mf = mface+a;
                float bb[6];

                INIT_MINMAX(bb, bb+3);
                DO_MINMAX(verts[mf->v1], bb, bb+3);
                DO_MINMAX(verts[mf->v2], bb, bb+3);
                DO_MINMAX(verts[mf->v3], bb, bb+3);
                if(mf->v4) {
                        DO_MINMAX(verts[mf->v4], bb, bb+3);
                }

                BLI_bvhtree_insert(sys->heat.bvhtree, a, bb, 2);
                
                //Setup inverse pointers to use on isect.orig
                sys->heat.vface[mf->v1]= mf;
                sys->heat.vface[mf->v2]= mf;
                sys->heat.vface[mf->v3]= mf;
                if(mf->v4) sys->heat.vface[mf->v4]= mf;
        }

        BLI_bvhtree_balance(sys->heat.bvhtree); 
}

static int heat_ray_source_visible(LaplacianSystem *sys, int vertex, int source)
{
        BVHTreeRayHit hit;
        BVHCallbackUserData data;
        MFace *mface;
        float end[3];
        int visible;

        mface= sys->heat.vface[vertex];
        if(!mface)
                return 1;

        data.sys= sys;
        copy_v3_v3(data.start, sys->heat.verts[vertex]);

        if(sys->heat.root) /* bone */
                closest_to_line_segment_v3(end, data.start,
                        sys->heat.root[source], sys->heat.tip[source]);
        else /* vertex */
                copy_v3_v3(end, sys->heat.source[source]);

        sub_v3_v3v3(data.vec, end, data.start);
        madd_v3_v3v3fl(data.start, data.start, data.vec, 1e-5);
        mul_v3_fl(data.vec, 1.0f - 2e-5f);

        /* pass normalized vec + distance to bvh */
        hit.index = -1;
        hit.dist = normalize_v3(data.vec);

        visible= BLI_bvhtree_ray_cast(sys->heat.bvhtree, data.start, data.vec, 0.0f, &hit, bvh_callback, (void*)&data) == -1;

        return visible;
}

static float heat_source_distance(LaplacianSystem *sys, int vertex, int source)
{
        float closest[3], d[3], dist, cosine;
        
        /* compute euclidian distance */
        if(sys->heat.root) /* bone */
                closest_to_line_segment_v3(closest, sys->heat.verts[vertex],
                        sys->heat.root[source], sys->heat.tip[source]);
        else /* vertex */
                copy_v3_v3(closest, sys->heat.source[source]);

        sub_v3_v3v3(d, sys->heat.verts[vertex], closest);
        dist= normalize_v3(d);

        /* if the vertex normal does not point along the bone, increase distance */
        cosine= INPR(d, sys->heat.vnors[vertex]);

        return dist/(0.5f*(cosine + 1.001f));
}

static int heat_source_closest(LaplacianSystem *sys, int vertex, int source)
{
        float dist;

        dist= heat_source_distance(sys, vertex, source);

        if(dist <= sys->heat.mindist[vertex]*(1.0f + DISTANCE_EPSILON))
                if(heat_ray_source_visible(sys, vertex, source))
                        return 1;
                
        return 0;
}

static void heat_set_H(LaplacianSystem *sys, int vertex)
{
        float dist, mindist, h;
        int j, numclosest = 0;

        mindist= 1e10;

        /* compute minimum distance */
        for(j=0; j<sys->heat.numsource; j++) {
                dist= heat_source_distance(sys, vertex, j);

                if(dist < mindist)
                        mindist= dist;
        }

        sys->heat.mindist[vertex]= mindist;

        /* count number of sources with approximately this minimum distance */
        for(j=0; j<sys->heat.numsource; j++)
                if(heat_source_closest(sys, vertex, j))
                        numclosest++;

        sys->heat.p[vertex]= (numclosest > 0)? 1.0f/numclosest: 0.0f;

        /* compute H entry */
        if(numclosest > 0) {
                mindist= maxf(mindist, 1e-4f);
                h= numclosest*C_WEIGHT/(mindist*mindist);
        }
        else
                h= 0.0f;
        
        sys->heat.H[vertex]= h;
}

static void heat_calc_vnormals(LaplacianSystem *sys)
{
        float fnor[3];
        int a, v1, v2, v3, (*face)[3];

        sys->heat.vnors= MEM_callocN(sizeof(float)*3*sys->totvert, "HeatVNors");

        for(a=0, face=sys->faces; a<sys->totface; a++, face++) {
                v1= (*face)[0];
                v2= (*face)[1];
                v3= (*face)[2];

                normal_tri_v3( fnor,sys->verts[v1], sys->verts[v2], sys->verts[v3]);
                
                add_v3_v3(sys->heat.vnors[v1], fnor);
                add_v3_v3(sys->heat.vnors[v2], fnor);
                add_v3_v3(sys->heat.vnors[v3], fnor);
        }

        for(a=0; a<sys->totvert; a++)
                normalize_v3(sys->heat.vnors[a]);
}

static void heat_laplacian_create(LaplacianSystem *sys)
{
        MFace *mface = sys->heat.mface, *mf;
        int totface= sys->heat.totface;
        int totvert= sys->heat.totvert;
        int a;

        /* heat specific definitions */
        sys->heat.mindist= MEM_callocN(sizeof(float)*totvert, "HeatMinDist");
        sys->heat.H= MEM_callocN(sizeof(float)*totvert, "HeatH");
        sys->heat.p= MEM_callocN(sizeof(float)*totvert, "HeatP");

        /* add verts and faces to laplacian */
        for(a=0; a<totvert; a++)
                laplacian_add_vertex(sys, sys->heat.verts[a], 0);

        for(a=0, mf=mface; a<totface; a++, mf++) {
                laplacian_add_triangle(sys, mf->v1, mf->v2, mf->v3);
                if(mf->v4)
                        laplacian_add_triangle(sys, mf->v1, mf->v3, mf->v4);
        }

        /* for distance computation in set_H */
        heat_calc_vnormals(sys);

        for(a=0; a<totvert; a++)
                heat_set_H(sys, a);
}

static void heat_system_free(LaplacianSystem *sys)
{
        BLI_bvhtree_free(sys->heat.bvhtree);
        MEM_freeN(sys->heat.vface);

        MEM_freeN(sys->heat.mindist);
        MEM_freeN(sys->heat.H);
        MEM_freeN(sys->heat.p);
        MEM_freeN(sys->heat.vnors);
}

static float heat_limit_weight(float weight)
{
        float t;

        if(weight < WEIGHT_LIMIT_END) {
                return 0.0f;
        }
        else if(weight < WEIGHT_LIMIT_START) {
                t= (weight - WEIGHT_LIMIT_END)/(WEIGHT_LIMIT_START - WEIGHT_LIMIT_END);
                return t*WEIGHT_LIMIT_START;
        }
        else
                return weight;
}

void heat_bone_weighting(Object *ob, Mesh *me, float (*verts)[3], int numsource, bDeformGroup **dgrouplist, bDeformGroup **dgroupflip, float (*root)[3], float (*tip)[3], int *selected, const char **err_str)
{
        LaplacianSystem *sys;
        MFace *mface;
        float solution, weight;
        int *vertsflipped = NULL, *mask= NULL;
        int a, totface, j, bbone, firstsegment, lastsegment;

        *err_str= NULL;

        /* count triangles and create mask */
        if(me->editflag & ME_EDIT_PAINT_MASK)
                mask= MEM_callocN(sizeof(int)*me->totvert, "heat_bone_weighting mask");

        for(totface=0, a=0, mface=me->mface; a<me->totface; a++, mface++) {
                totface++;
                if(mface->v4) totface++;

                if(mask && (mface->flag & ME_FACE_SEL)) {
                        mask[mface->v1]= 1;
                        mask[mface->v2]= 1;
                        mask[mface->v3]= 1;
                        if(mface->v4)
                                mask[mface->v4]= 1;
                }
        }

        /* create laplacian */
        sys = laplacian_system_construct_begin(me->totvert, totface, 1);

        sys->heat.mface= me->mface;
        sys->heat.totface= me->totface;
        sys->heat.totvert= me->totvert;
        sys->heat.verts= verts;
        sys->heat.root= root;
        sys->heat.tip= tip;
        sys->heat.numsource= numsource;

        heat_ray_tree_create(sys);
        heat_laplacian_create(sys);

        laplacian_system_construct_end(sys);

        if(dgroupflip) {
                vertsflipped = MEM_callocN(sizeof(int)*me->totvert, "vertsflipped");
                for(a=0; a<me->totvert; a++)
                        vertsflipped[a] = mesh_get_x_mirror_vert(ob, a);
        }

        /* compute weights per bone */
        for(j=0; j<numsource; j++) {
                if(!selected[j])
                        continue;

                firstsegment= (j == 0 || dgrouplist[j-1] != dgrouplist[j]);
                lastsegment= (j == numsource-1 || dgrouplist[j] != dgrouplist[j+1]);
                bbone= !(firstsegment && lastsegment);

                /* clear weights */
                if(bbone && firstsegment) {
                        for(a=0; a<me->totvert; a++) {
                                if(mask && !mask[a])
                                        continue;

                                ED_vgroup_vert_remove(ob, dgrouplist[j], a);
                                if(vertsflipped && dgroupflip[j] && vertsflipped[a] >= 0)
                                        ED_vgroup_vert_remove(ob, dgroupflip[j], vertsflipped[a]);
                        }
                }

                /* fill right hand side */
                laplacian_begin_solve(sys, -1);

                for(a=0; a<me->totvert; a++)
                        if(heat_source_closest(sys, a, j))
                                laplacian_add_right_hand_side(sys, a,
                                        sys->heat.H[a]*sys->heat.p[a]);

                /* solve */
                if(laplacian_system_solve(sys)) {
                        /* load solution into vertex groups */
                        for(a=0; a<me->totvert; a++) {
                                if(mask && !mask[a])
                                        continue;

                                solution= laplacian_system_get_solution(a);
                                
                                if(bbone) {
                                        if(solution > 0.0f)
                                                ED_vgroup_vert_add(ob, dgrouplist[j], a, solution,
                                                        WEIGHT_ADD);
                                }
                                else {
                                        weight= heat_limit_weight(solution);
                                        if(weight > 0.0f)
                                                ED_vgroup_vert_add(ob, dgrouplist[j], a, weight,
                                                        WEIGHT_REPLACE);
                                        else
                                                ED_vgroup_vert_remove(ob, dgrouplist[j], a);
                                }

                                /* do same for mirror */
                                if(vertsflipped && dgroupflip[j] && vertsflipped[a] >= 0) {
                                        if(bbone) {
                                                if(solution > 0.0f)
                                                        ED_vgroup_vert_add(ob, dgroupflip[j], vertsflipped[a],
                                                                solution, WEIGHT_ADD);
                                        }
                                        else {
                                                weight= heat_limit_weight(solution);
                                                if(weight > 0.0f)
                                                        ED_vgroup_vert_add(ob, dgroupflip[j], vertsflipped[a],
                                                                weight, WEIGHT_REPLACE);
                                                else
                                                        ED_vgroup_vert_remove(ob, dgroupflip[j], vertsflipped[a]);
                                        }
                                }
                        }
                }
                else if(*err_str == NULL) {
                        *err_str= "Bone Heat Weighting: failed to find solution for one or more bones";
                        break;
                }

                /* remove too small vertex weights */
                if(bbone && lastsegment) {
                        for(a=0; a<me->totvert; a++) {
                                if(mask && !mask[a])
                                        continue;

                                weight= ED_vgroup_vert_weight(ob, dgrouplist[j], a);
                                weight= heat_limit_weight(weight);
                                if(weight <= 0.0f)
                                        ED_vgroup_vert_remove(ob, dgrouplist[j], a);

                                if(vertsflipped && dgroupflip[j] && vertsflipped[a] >= 0) {
                                        weight= ED_vgroup_vert_weight(ob, dgroupflip[j], vertsflipped[a]);
                                        weight= heat_limit_weight(weight);
                                        if(weight <= 0.0f)
                                                ED_vgroup_vert_remove(ob, dgroupflip[j], vertsflipped[a]);
                                }
                        }
                }
        }

        /* free */
        if(vertsflipped) MEM_freeN(vertsflipped);
        if(mask) MEM_freeN(mask);

        heat_system_free(sys);

        laplacian_system_delete(sys);
}

#ifdef RIGID_DEFORM
/********************** As-Rigid-As-Possible Deformation ******************/
/* From "As-Rigid-As-Possible Surface Modeling",
                Olga Sorkine and Marc Alexa, ESGP 2007. */

/* investigate:
   - transpose R in orthogonal
   - flipped normals and per face adding
   - move cancelling to transform, make origco pointer
*/

static LaplacianSystem *RigidDeformSystem = NULL;

static void rigid_add_half_edge_to_R(LaplacianSystem *sys, EditVert *v1, EditVert *v2, float w)
{
        float e[3], e_[3];
        int i;

        sub_v3_v3v3(e, sys->rigid.origco[v1->tmp.l], sys->rigid.origco[v2->tmp.l]);
        sub_v3_v3v3(e_, v1->co, v2->co);

        /* formula (5) */
        for (i=0; i<3; i++) {
                sys->rigid.R[v1->tmp.l][i][0] += w*e[0]*e_[i];
                sys->rigid.R[v1->tmp.l][i][1] += w*e[1]*e_[i];
                sys->rigid.R[v1->tmp.l][i][2] += w*e[2]*e_[i];
        }
}

static void rigid_add_edge_to_R(LaplacianSystem *sys, EditVert *v1, EditVert *v2, float w)
{
        rigid_add_half_edge_to_R(sys, v1, v2, w);
        rigid_add_half_edge_to_R(sys, v2, v1, w);
}

static void rigid_orthogonalize_R(float R[][3])
{
        HMatrix M, Q, S;

        copy_m4_m3(M, R);
        polar_decomp(M, Q, S);
        copy_m3_m4(R, Q);
}

static void rigid_add_half_edge_to_rhs(LaplacianSystem *sys, EditVert *v1, EditVert *v2, float w)
{
        /* formula (8) */
        float Rsum[3][3], rhs[3];

        if (sys->vpinned[v1->tmp.l])
                return;

        add_m3_m3m3(Rsum, sys->rigid.R[v1->tmp.l], sys->rigid.R[v2->tmp.l]);
        transpose_m3(Rsum);

        sub_v3_v3v3(rhs, sys->rigid.origco[v1->tmp.l], sys->rigid.origco[v2->tmp.l]);
        mul_m3_v3(Rsum, rhs);
        mul_v3_fl(rhs, 0.5f);
        mul_v3_fl(rhs, w);

        add_v3_v3(sys->rigid.rhs[v1->tmp.l], rhs);
}

static void rigid_add_edge_to_rhs(LaplacianSystem *sys, EditVert *v1, EditVert *v2, float w)
{
        rigid_add_half_edge_to_rhs(sys, v1, v2, w);
        rigid_add_half_edge_to_rhs(sys, v2, v1, w);
}

void rigid_deform_iteration()
{
        LaplacianSystem *sys= RigidDeformSystem;
        EditMesh *em;
        EditVert *eve;
        EditFace *efa;
        int a, i;

        if(!sys)
                return;
        
        nlMakeCurrent(sys->context);
        em= sys->rigid.mesh;

        /* compute R */
        memset(sys->rigid.R, 0, sizeof(float)*3*3*sys->totvert);
        memset(sys->rigid.rhs, 0, sizeof(float)*3*sys->totvert);

        for(a=0, efa=em->faces.first; efa; efa=efa->next, a++) {
                rigid_add_edge_to_R(sys, efa->v1, efa->v2, sys->fweights[a][2]);
                rigid_add_edge_to_R(sys, efa->v2, efa->v3, sys->fweights[a][0]);
                rigid_add_edge_to_R(sys, efa->v3, efa->v1, sys->fweights[a][1]);

                if(efa->v4) {
                        a++;
                        rigid_add_edge_to_R(sys, efa->v1, efa->v3, sys->fweights[a][2]);
                        rigid_add_edge_to_R(sys, efa->v3, efa->v4, sys->fweights[a][0]);
                        rigid_add_edge_to_R(sys, efa->v4, efa->v1, sys->fweights[a][1]);
                }
        }

        for(a=0, eve=em->verts.first; eve; eve=eve->next, a++) {
                rigid_orthogonalize_R(sys->rigid.R[a]);
                eve->tmp.l= a;
        }
        
        /* compute right hand sides for solving */
        for(a=0, efa=em->faces.first; efa; efa=efa->next, a++) {
                rigid_add_edge_to_rhs(sys, efa->v1, efa->v2, sys->fweights[a][2]);
                rigid_add_edge_to_rhs(sys, efa->v2, efa->v3, sys->fweights[a][0]);
                rigid_add_edge_to_rhs(sys, efa->v3, efa->v1, sys->fweights[a][1]);

                if(efa->v4) {
                        a++;
                        rigid_add_edge_to_rhs(sys, efa->v1, efa->v3, sys->fweights[a][2]);
                        rigid_add_edge_to_rhs(sys, efa->v3, efa->v4, sys->fweights[a][0]);
                        rigid_add_edge_to_rhs(sys, efa->v4, efa->v1, sys->fweights[a][1]);
                }
        }

        /* solve for positions, for X,Y and Z separately */
        for(i=0; i<3; i++) {
                laplacian_begin_solve(sys, i);

                for(a=0; a<sys->totvert; a++)
                        if(!sys->vpinned[a])
                                laplacian_add_right_hand_side(sys, a, sys->rigid.rhs[a][i]);

                if(laplacian_system_solve(sys)) {
                        for(a=0, eve=em->verts.first; eve; eve=eve->next, a++)
                                eve->co[i]= laplacian_system_get_solution(a);
                }
                else {
                        if(!sys->rigid.thrownerror) {
                                error("RigidDeform: failed to find solution.");
                                sys->rigid.thrownerror= 1;
                        }
                        break;
                }
        }
}

static void rigid_laplacian_create(LaplacianSystem *sys)
{
        EditMesh *em = sys->rigid.mesh;
        EditVert *eve;
        EditFace *efa;
        int a;

        /* add verts and faces to laplacian */
        for(a=0, eve=em->verts.first; eve; eve=eve->next, a++) {
                laplacian_add_vertex(sys, eve->co, eve->pinned);
                eve->tmp.l= a;
        }

        for(efa=em->faces.first; efa; efa=efa->next) {
                laplacian_add_triangle(sys,
                        efa->v1->tmp.l, efa->v2->tmp.l, efa->v3->tmp.l);
                if(efa->v4)
                        laplacian_add_triangle(sys,
                                efa->v1->tmp.l, efa->v3->tmp.l, efa->v4->tmp.l);
        }
}

void rigid_deform_begin(EditMesh *em)
{
        LaplacianSystem *sys;
        EditVert *eve;
        EditFace *efa;
        int a, totvert, totface;

        /* count vertices, triangles */
        for(totvert=0, eve=em->verts.first; eve; eve=eve->next)
                totvert++;

        for(totface=0, efa=em->faces.first; efa; efa=efa->next) {
                totface++;
                if(efa->v4) totface++;
        }

        /* create laplacian */
        sys = laplacian_system_construct_begin(totvert, totface, 0);

        sys->rigid.mesh= em;
        sys->rigid.R = MEM_callocN(sizeof(float)*3*3*totvert, "RigidDeformR");
        sys->rigid.rhs = MEM_callocN(sizeof(float)*3*totvert, "RigidDeformRHS");
        sys->rigid.origco = MEM_callocN(sizeof(float)*3*totvert, "RigidDeformCo");

        for(a=0, eve=em->verts.first; eve; eve=eve->next, a++)
                copy_v3_v3(sys->rigid.origco[a], eve->co);

        sys->areaweights= 0;
        sys->storeweights= 1;

        rigid_laplacian_create(sys);

        laplacian_system_construct_end(sys);

        RigidDeformSystem = sys;
}

void rigid_deform_end(int cancel)
{
        LaplacianSystem *sys = RigidDeformSystem;

        if(sys) {
                EditMesh *em = sys->rigid.mesh;
                EditVert *eve;
                int a;

                if(cancel)
                        for(a=0, eve=em->verts.first; eve; eve=eve->next, a++)
                                if(!eve->pinned)
                                        copy_v3_v3(eve->co, sys->rigid.origco[a]);

                if(sys->rigid.R) MEM_freeN(sys->rigid.R);
                if(sys->rigid.rhs) MEM_freeN(sys->rigid.rhs);
                if(sys->rigid.origco) MEM_freeN(sys->rigid.origco);

                /* free */
                laplacian_system_delete(sys);
        }

        RigidDeformSystem = NULL;
}
#endif

/************************** Harmonic Coordinates ****************************/
/* From "Harmonic Coordinates for Character Articulation",
        Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green and Tom Sanocki,
        SIGGRAPH 2007. */

#define EPSILON 0.0001f

#define MESHDEFORM_TAG_UNTYPED  0
#define MESHDEFORM_TAG_BOUNDARY 1
#define MESHDEFORM_TAG_INTERIOR 2
#define MESHDEFORM_TAG_EXTERIOR 3

#define MESHDEFORM_LEN_THRESHOLD 1e-6f

#define MESHDEFORM_MIN_INFLUENCE 0.0005f

static int MESHDEFORM_OFFSET[7][3] =
                {{0,0,0}, {1,0,0}, {-1,0,0}, {0,1,0}, {0,-1,0}, {0,0,1}, {0,0,-1}};

typedef struct MDefBoundIsect {
        float co[3], uvw[4];
        int nvert, v[4], facing;
        float len;
} MDefBoundIsect;

typedef struct MDefBindInfluence {
        struct MDefBindInfluence *next;
        float weight;
        int vertex;
} MDefBindInfluence;

typedef struct MeshDeformBind {
        /* grid dimensions */
        float min[3], max[3];
        float width[3], halfwidth[3];
        int size, size3;

        /* meshes */
        DerivedMesh *cagedm;
        float (*cagecos)[3];
        float (*vertexcos)[3];
        int totvert, totcagevert;

        /* grids */
        MemArena *memarena;
        MDefBoundIsect *(*boundisect)[6];
        int *semibound;
        int *tag;
        float *phi, *totalphi;

        /* mesh stuff */
        int *inside;
        float *weights;
        MDefBindInfluence **dyngrid;
        float cagemat[4][4];

        /* direct solver */
        int *varidx;
} MeshDeformBind;

typedef struct MeshDeformIsect {
        float start[3];
        float vec[3];
        float labda;

        void *face;
        int isect;
        float u, v;
        
} MeshDeformIsect;

/* ray intersection */

/* our own triangle intersection, so we can fully control the epsilons and
 * prevent corner case from going wrong*/
static int meshdeform_tri_intersect(float orig[3], float end[3], float vert0[3],
        float vert1[3], float vert2[3], float *isectco, float *uvw)
{
        float edge1[3], edge2[3], tvec[3], pvec[3], qvec[3];
        float det,inv_det, u, v, dir[3], isectdir[3];

        sub_v3_v3v3(dir, end, orig);

        /* find vectors for two edges sharing vert0 */
        sub_v3_v3v3(edge1, vert1, vert0);
        sub_v3_v3v3(edge2, vert2, vert0);

        /* begin calculating determinant - also used to calculate U parameter */
        cross_v3_v3v3(pvec, dir, edge2);

        /* if determinant is near zero, ray lies in plane of triangle */
        det = INPR(edge1, pvec);

        if (det == 0.0f)
                return 0;
        inv_det = 1.0f / det;

        /* calculate distance from vert0 to ray origin */
        sub_v3_v3v3(tvec, orig, vert0);

        /* calculate U parameter and test bounds */
        u = INPR(tvec, pvec) * inv_det;
        if (u < -EPSILON || u > 1.0f+EPSILON)
                return 0;

        /* prepare to test V parameter */
        cross_v3_v3v3(qvec, tvec, edge1);

        /* calculate V parameter and test bounds */
        v = INPR(dir, qvec) * inv_det;
        if (v < -EPSILON || u + v > 1.0f+EPSILON)
                return 0;

        isectco[0]= (1.0f - u - v)*vert0[0] + u*vert1[0] + v*vert2[0];
        isectco[1]= (1.0f - u - v)*vert0[1] + u*vert1[1] + v*vert2[1];
        isectco[2]= (1.0f - u - v)*vert0[2] + u*vert1[2] + v*vert2[2];

        uvw[0]= 1.0f - u - v;
        uvw[1]= u;
        uvw[2]= v;

        /* check if it is within the length of the line segment */
        sub_v3_v3v3(isectdir, isectco, orig);

        if(INPR(dir, isectdir) < -EPSILON)
                return 0;
        
        if(INPR(dir, dir) + EPSILON < INPR(isectdir, isectdir))
                return 0;

        return 1;
}

static int meshdeform_intersect(MeshDeformBind *mdb, MeshDeformIsect *isec)
{
        MFace *mface;
        float face[4][3], co[3], uvw[3], len, nor[3], end[3];
        int f, hit, is= 0, totface;

        isec->labda= 1e10;

        mface= mdb->cagedm->getFaceArray(mdb->cagedm);
        totface= mdb->cagedm->getNumFaces(mdb->cagedm);

        add_v3_v3v3(end, isec->start, isec->vec);

        for(f=0; f<totface; f++, mface++) {
                copy_v3_v3(face[0], mdb->cagecos[mface->v1]);
                copy_v3_v3(face[1], mdb->cagecos[mface->v2]);
                copy_v3_v3(face[2], mdb->cagecos[mface->v3]);

                if(mface->v4) {
                        copy_v3_v3(face[3], mdb->cagecos[mface->v4]);
                        hit = meshdeform_tri_intersect(isec->start, end, face[0], face[1], face[2], co, uvw);

                        if(hit) {
                                normal_tri_v3( nor,face[0], face[1], face[2]);
                        }
                        else {
                                hit= meshdeform_tri_intersect(isec->start, end, face[0], face[2], face[3], co, uvw);
                                normal_tri_v3( nor,face[0], face[2], face[3]);
                        }
                }
                else {
                        hit= meshdeform_tri_intersect(isec->start, end, face[0], face[1], face[2], co, uvw);
                        normal_tri_v3( nor,face[0], face[1], face[2]);
                }

                if(hit) {
                        len= len_v3v3(isec->start, co)/len_v3v3(isec->start, end);
                        if(len < isec->labda) {
                                isec->labda= len;
                                isec->face = mface;
                                isec->isect= (INPR(isec->vec, nor) <= 0.0f);
                                is= 1;
                        }
                }
        }

        return is;
}

static MDefBoundIsect *meshdeform_ray_tree_intersect(MeshDeformBind *mdb, float *co1, float *co2)
{
        MDefBoundIsect *isect;
        MeshDeformIsect isec;
        float (*cagecos)[3];
        MFace *mface;
        float vert[4][3], len, end[3];
        static float epsilon[3]= {0, 0, 0}; //1e-4, 1e-4, 1e-4};

        /* setup isec */
        memset(&isec, 0, sizeof(isec));
        isec.labda= 1e10f;

        VECADD(isec.start, co1, epsilon);
        VECADD(end, co2, epsilon);
        sub_v3_v3v3(isec.vec, end, isec.start);

        if(meshdeform_intersect(mdb, &isec)) {
                len= isec.labda;
                mface=(MFace*)isec.face;

                /* create MDefBoundIsect */
                isect= BLI_memarena_alloc(mdb->memarena, sizeof(*isect));

                /* compute intersection coordinate */
                isect->co[0]= co1[0] + isec.vec[0]*len;
                isect->co[1]= co1[1] + isec.vec[1]*len;
                isect->co[2]= co1[2] + isec.vec[2]*len;

                isect->len= len_v3v3(co1, isect->co);
                if(isect->len < MESHDEFORM_LEN_THRESHOLD)
                        isect->len= MESHDEFORM_LEN_THRESHOLD;

                isect->v[0]= mface->v1;
                isect->v[1]= mface->v2;
                isect->v[2]= mface->v3;
                isect->v[3]= mface->v4;
                isect->nvert= (mface->v4)? 4: 3;

                isect->facing= isec.isect;

                /* compute mean value coordinates for interpolation */
                cagecos= mdb->cagecos;
                copy_v3_v3(vert[0], cagecos[mface->v1]);
                copy_v3_v3(vert[1], cagecos[mface->v2]);
                copy_v3_v3(vert[2], cagecos[mface->v3]);
                if(mface->v4) copy_v3_v3(vert[3], cagecos[mface->v4]);
                interp_weights_poly_v3( isect->uvw,vert, isect->nvert, isect->co);

                return isect;
        }

        return NULL;
}

static int meshdeform_inside_cage(MeshDeformBind *mdb, float *co)
{
        MDefBoundIsect *isect;
        float outside[3], start[3], dir[3];
        int i;

        for(i=1; i<=6; i++) {
                outside[0] = co[0] + (mdb->max[0] - mdb->min[0] + 1.0f)*MESHDEFORM_OFFSET[i][0];
                outside[1] = co[1] + (mdb->max[1] - mdb->min[1] + 1.0f)*MESHDEFORM_OFFSET[i][1];
                outside[2] = co[2] + (mdb->max[2] - mdb->min[2] + 1.0f)*MESHDEFORM_OFFSET[i][2];

                copy_v3_v3(start, co);
                sub_v3_v3v3(dir, outside, start);
                normalize_v3(dir);
                
                isect = meshdeform_ray_tree_intersect(mdb, start, outside);
                if(isect && !isect->facing)
                        return 1;
        }

        return 0;
}

/* solving */

static int meshdeform_index(MeshDeformBind *mdb, int x, int y, int z, int n)
{
        int size= mdb->size;
        
        x += MESHDEFORM_OFFSET[n][0];
        y += MESHDEFORM_OFFSET[n][1];
        z += MESHDEFORM_OFFSET[n][2];

        if(x < 0 || x >= mdb->size)
                return -1;
        if(y < 0 || y >= mdb->size)
                return -1;
        if(z < 0 || z >= mdb->size)
                return -1;

        return x + y*size + z*size*size;
}

static void meshdeform_cell_center(MeshDeformBind *mdb, int x, int y, int z, int n, float *center)
{
        x += MESHDEFORM_OFFSET[n][0];
        y += MESHDEFORM_OFFSET[n][1];
        z += MESHDEFORM_OFFSET[n][2];

        center[0]= mdb->min[0] + x*mdb->width[0] + mdb->halfwidth[0];
        center[1]= mdb->min[1] + y*mdb->width[1] + mdb->halfwidth[1];
        center[2]= mdb->min[2] + z*mdb->width[2] + mdb->halfwidth[2];
}

static void meshdeform_add_intersections(MeshDeformBind *mdb, int x, int y, int z)
{
        MDefBoundIsect *isect;
        float center[3], ncenter[3];
        int i, a;

        a= meshdeform_index(mdb, x, y, z, 0);
        meshdeform_cell_center(mdb, x, y, z, 0, center);

        /* check each outgoing edge for intersection */
        for(i=1; i<=6; i++) {
                if(meshdeform_index(mdb, x, y, z, i) == -1)
                        continue;

                meshdeform_cell_center(mdb, x, y, z, i, ncenter);

                isect= meshdeform_ray_tree_intersect(mdb, center, ncenter);
                if(isect) {
                        mdb->boundisect[a][i-1]= isect;
                        mdb->tag[a]= MESHDEFORM_TAG_BOUNDARY;
                }
        }
}

static void meshdeform_bind_floodfill(MeshDeformBind *mdb)
{
        int *stack, *tag= mdb->tag;
        int a, b, i, xyz[3], stacksize, size= mdb->size;

        stack= MEM_callocN(sizeof(int)*mdb->size3, "MeshDeformBindStack");

        /* we know lower left corner is EXTERIOR because of padding */
        tag[0]= MESHDEFORM_TAG_EXTERIOR;
        stack[0]= 0;
        stacksize= 1;

        /* floodfill exterior tag */
        while(stacksize > 0) {
                a= stack[--stacksize];

                xyz[2]= a/(size*size);
                xyz[1]= (a - xyz[2]*size*size)/size;
                xyz[0]= a - xyz[1]*size - xyz[2]*size*size;

                for(i=1; i<=6; i++) {
                        b= meshdeform_index(mdb, xyz[0], xyz[1], xyz[2], i);

                        if(b != -1) {
                                if(tag[b] == MESHDEFORM_TAG_UNTYPED ||
                                   (tag[b] == MESHDEFORM_TAG_BOUNDARY && !mdb->boundisect[a][i-1])) {
                                        tag[b]= MESHDEFORM_TAG_EXTERIOR;
                                        stack[stacksize++]= b;
                                }
                        }
                }
        }

        /* other cells are interior */
        for(a=0; a<size*size*size; a++)
                if(tag[a]==MESHDEFORM_TAG_UNTYPED)
                        tag[a]= MESHDEFORM_TAG_INTERIOR;

#if 0
        {
                int tb, ti, te, ts;
                tb= ti= te= ts= 0;
                for(a=0; a<size*size*size; a++)
                        if(tag[a]==MESHDEFORM_TAG_BOUNDARY)
                                tb++;
                        else if(tag[a]==MESHDEFORM_TAG_INTERIOR)
                                ti++;
                        else if(tag[a]==MESHDEFORM_TAG_EXTERIOR) {
                                te++;

                                if(mdb->semibound[a])
                                        ts++;
                        }
                
                printf("interior %d exterior %d boundary %d semi-boundary %d\n", ti, te, tb, ts);
        }
#endif

        MEM_freeN(stack);
}

static float meshdeform_boundary_phi(MeshDeformBind *UNUSED(mdb), MDefBoundIsect *isect, int cagevert)
{
        int a;

        for(a=0; a<isect->nvert; a++)
                if(isect->v[a] == cagevert)
                        return isect->uvw[a];
        
        return 0.0f;
}

static float meshdeform_interp_w(MeshDeformBind *mdb, float *gridvec, float *UNUSED(vec), int UNUSED(cagevert))
{
        float dvec[3], ivec[3], wx, wy, wz, result=0.0f;
        float weight, totweight= 0.0f;
        int i, a, x, y, z;

        for(i=0; i<3; i++) {
                ivec[i]= (int)gridvec[i];
                dvec[i]= gridvec[i] - ivec[i];
        }

        for(i=0; i<8; i++) {
                if(i & 1) { x= ivec[0]+1; wx= dvec[0]; }
                else { x= ivec[0]; wx= 1.0f-dvec[0]; } 

                if(i & 2) { y= ivec[1]+1; wy= dvec[1]; }
                else { y= ivec[1]; wy= 1.0f-dvec[1]; } 

                if(i & 4) { z= ivec[2]+1; wz= dvec[2]; }
                else { z= ivec[2]; wz= 1.0f-dvec[2]; } 

                CLAMP(x, 0, mdb->size-1);
                CLAMP(y, 0, mdb->size-1);
                CLAMP(z, 0, mdb->size-1);

                a= meshdeform_index(mdb, x, y, z, 0);
                weight= wx*wy*wz;
                result += weight*mdb->phi[a];
                totweight += weight;
        }

        if(totweight > 0.0f)
                result /= totweight;

        return result;
}

static void meshdeform_check_semibound(MeshDeformBind *mdb, int x, int y, int z)
{
        int i, a;

        a= meshdeform_index(mdb, x, y, z, 0);
        if(mdb->tag[a] != MESHDEFORM_TAG_EXTERIOR)
                return;

        for(i=1; i<=6; i++)
                if(mdb->boundisect[a][i-1]) 
                        mdb->semibound[a]= 1;
}

static float meshdeform_boundary_total_weight(MeshDeformBind *mdb, int x, int y, int z)
{
        float weight, totweight= 0.0f;
        int i, a;

        a= meshdeform_index(mdb, x, y, z, 0);

        /* count weight for neighbour cells */
        for(i=1; i<=6; i++) {
                if(meshdeform_index(mdb, x, y, z, i) == -1)
                        continue;

                if(mdb->boundisect[a][i-1])
                        weight= 1.0f/mdb->boundisect[a][i-1]->len;
                else if(!mdb->semibound[a])
                        weight= 1.0f/mdb->width[0];
                else
                        weight= 0.0f;

                totweight += weight;
        }

        return totweight;
}

static void meshdeform_matrix_add_cell(MeshDeformBind *mdb, int x, int y, int z)
{
        MDefBoundIsect *isect;
        float weight, totweight;
        int i, a, acenter;

        acenter= meshdeform_index(mdb, x, y, z, 0);
        if(mdb->tag[acenter] == MESHDEFORM_TAG_EXTERIOR)
                return;

        nlMatrixAdd(mdb->varidx[acenter], mdb->varidx[acenter], 1.0f);
        
        totweight= meshdeform_boundary_total_weight(mdb, x, y, z);
        for(i=1; i<=6; i++) {
                a= meshdeform_index(mdb, x, y, z, i);
                if(a == -1 || mdb->tag[a] == MESHDEFORM_TAG_EXTERIOR)
                        continue;

                isect= mdb->boundisect[acenter][i-1];
                if (!isect) {
                        weight= (1.0f/mdb->width[0])/totweight;
                        nlMatrixAdd(mdb->varidx[acenter], mdb->varidx[a], -weight);
                }
        }
}

static void meshdeform_matrix_add_rhs(MeshDeformBind *mdb, int x, int y, int z, int cagevert)
{
        MDefBoundIsect *isect;
        float rhs, weight, totweight;
        int i, a, acenter;

        acenter= meshdeform_index(mdb, x, y, z, 0);
        if(mdb->tag[acenter] == MESHDEFORM_TAG_EXTERIOR)
                return;

        totweight= meshdeform_boundary_total_weight(mdb, x, y, z);
        for(i=1; i<=6; i++) {
                a= meshdeform_index(mdb, x, y, z, i);
                if(a == -1)
                        continue;

                isect= mdb->boundisect[acenter][i-1];

                if (isect) {
                        weight= (1.0f/isect->len)/totweight;
                        rhs= weight*meshdeform_boundary_phi(mdb, isect, cagevert);
                        nlRightHandSideAdd(0, mdb->varidx[acenter], rhs);
                }
        }
}

static void meshdeform_matrix_add_semibound_phi(MeshDeformBind *mdb, int x, int y, int z, int cagevert)
{
        MDefBoundIsect *isect;
        float rhs, weight, totweight;
        int i, a;

        a= meshdeform_index(mdb, x, y, z, 0);
        if(!mdb->semibound[a])
                return;
        
        mdb->phi[a]= 0.0f;

        totweight= meshdeform_boundary_total_weight(mdb, x, y, z);
        for(i=1; i<=6; i++) {
                isect= mdb->boundisect[a][i-1];

                if (isect) {
                        weight= (1.0f/isect->len)/totweight;
                        rhs= weight*meshdeform_boundary_phi(mdb, isect, cagevert);
                        mdb->phi[a] += rhs;
                }
        }
}

static void meshdeform_matrix_add_exterior_phi(MeshDeformBind *mdb, int x, int y, int z, int UNUSED(cagevert))
{
        float phi, totweight;
        int i, a, acenter;

        acenter= meshdeform_index(mdb, x, y, z, 0);
        if(mdb->tag[acenter] != MESHDEFORM_TAG_EXTERIOR || mdb->semibound[acenter])
                return;

        phi= 0.0f;
        totweight= 0.0f;
        for(i=1; i<=6; i++) {
                a= meshdeform_index(mdb, x, y, z, i);

                if(a != -1 && mdb->semibound[a]) {
                        phi += mdb->phi[a];
                        totweight += 1.0f;
                }
        }

        if(totweight != 0.0f)
                mdb->phi[acenter]= phi/totweight;
}

static void meshdeform_matrix_solve(MeshDeformBind *mdb)
{
        NLContext *context;
        float vec[3], gridvec[3];
        int a, b, x, y, z, totvar;
        char message[1024];

        /* setup variable indices */
        mdb->varidx= MEM_callocN(sizeof(int)*mdb->size3, "MeshDeformDSvaridx");
        for(a=0, totvar=0; a<mdb->size3; a++)
                mdb->varidx[a]= (mdb->tag[a] == MESHDEFORM_TAG_EXTERIOR)? -1: totvar++;

        if(totvar == 0) {
                MEM_freeN(mdb->varidx);
                return;
        }

        progress_bar(0, "Starting mesh deform solve");

        /* setup opennl solver */
        nlNewContext();
        context= nlGetCurrent();

        nlSolverParameteri(NL_NB_VARIABLES, totvar);
        nlSolverParameteri(NL_NB_ROWS, totvar);
        nlSolverParameteri(NL_NB_RIGHT_HAND_SIDES, 1);

        nlBegin(NL_SYSTEM);
        nlBegin(NL_MATRIX);

        /* build matrix */
        for(z=0; z<mdb->size; z++)
                for(y=0; y<mdb->size; y++)
                        for(x=0; x<mdb->size; x++)
                                meshdeform_matrix_add_cell(mdb, x, y, z);

        /* solve for each cage vert */
        for(a=0; a<mdb->totcagevert; a++) {
                if(a != 0) {
                        nlBegin(NL_SYSTEM);
                        nlBegin(NL_MATRIX);
                }

                /* fill in right hand side and solve */
                for(z=0; z<mdb->size; z++)
                        for(y=0; y<mdb->size; y++)
                                for(x=0; x<mdb->size; x++)
                                        meshdeform_matrix_add_rhs(mdb, x, y, z, a);

                nlEnd(NL_MATRIX);
                nlEnd(NL_SYSTEM);

#if 0
                nlPrintMatrix();
#endif

                if(nlSolveAdvanced(NULL, NL_TRUE)) {
                        for(z=0; z<mdb->size; z++)
                                for(y=0; y<mdb->size; y++)
                                        for(x=0; x<mdb->size; x++)
                                                meshdeform_matrix_add_semibound_phi(mdb, x, y, z, a);

                        for(z=0; z<mdb->size; z++)
                                for(y=0; y<mdb->size; y++)
                                        for(x=0; x<mdb->size; x++)
                                                meshdeform_matrix_add_exterior_phi(mdb, x, y, z, a);

                        for(b=0; b<mdb->size3; b++) {
                                if(mdb->tag[b] != MESHDEFORM_TAG_EXTERIOR)
                                        mdb->phi[b]= nlGetVariable(0, mdb->varidx[b]);
                                mdb->totalphi[b] += mdb->phi[b];
                        }

                        if(mdb->weights) {
                                /* static bind : compute weights for each vertex */
                                for(b=0; b<mdb->totvert; b++) {
                                        if(mdb->inside[b]) {
                                                copy_v3_v3(vec, mdb->vertexcos[b]);
                                                gridvec[0]= (vec[0] - mdb->min[0] - mdb->halfwidth[0])/mdb->width[0];
                                                gridvec[1]= (vec[1] - mdb->min[1] - mdb->halfwidth[1])/mdb->width[1];
                                                gridvec[2]= (vec[2] - mdb->min[2] - mdb->halfwidth[2])/mdb->width[2];

                                                mdb->weights[b*mdb->totcagevert + a]= meshdeform_interp_w(mdb, gridvec, vec, a);
                                        }
                                }
                        }
                        else {
                                MDefBindInfluence *inf;

                                /* dynamic bind */
                                for(b=0; b<mdb->size3; b++) {
                                        if(mdb->phi[b] >= MESHDEFORM_MIN_INFLUENCE) {
                                                inf= BLI_memarena_alloc(mdb->memarena, sizeof(*inf));
                                                inf->vertex= a;
                                                inf->weight= mdb->phi[b];
                                                inf->next= mdb->dyngrid[b];
                                                mdb->dyngrid[b]= inf;
                                        }
                                }
                        }
                }
                else {
                        error("Mesh Deform: failed to find solution.");
                        break;
                }

                sprintf(message, "Mesh deform solve %d / %d       |||", a+1, mdb->totcagevert);
                progress_bar((float)(a+1)/(float)(mdb->totcagevert), message);
        }

#if 0
        /* sanity check */
        for(b=0; b<mdb->size3; b++)
                if(mdb->tag[b] != MESHDEFORM_TAG_EXTERIOR)
                        if(fabs(mdb->totalphi[b] - 1.0f) > 1e-4)
                                printf("totalphi deficiency [%s|%d] %d: %.10f\n",
                                        (mdb->tag[b] == MESHDEFORM_TAG_INTERIOR)? "interior": "boundary", mdb->semibound[b], mdb->varidx[b], mdb->totalphi[b]);
#endif
        
        /* free */
        MEM_freeN(mdb->varidx);

        nlDeleteContext(context);
}

static void harmonic_coordinates_bind(Scene *UNUSED(scene), MeshDeformModifierData *mmd, MeshDeformBind *mdb)
{
        MDefBindInfluence *inf;
        MDefInfluence *mdinf;
        MDefCell *cell;
        float center[3], vec[3], maxwidth, totweight;
        int a, b, x, y, z, totinside, offset;

        /* compute bounding box of the cage mesh */
        INIT_MINMAX(mdb->min, mdb->max);

        for(a=0; a<mdb->totcagevert; a++)
                DO_MINMAX(mdb->cagecos[a], mdb->min, mdb->max);

        /* allocate memory */
        mdb->size= (2<<(mmd->gridsize-1)) + 2;
        mdb->size3= mdb->size*mdb->size*mdb->size;
        mdb->tag= MEM_callocN(sizeof(int)*mdb->size3, "MeshDeformBindTag");
        mdb->phi= MEM_callocN(sizeof(float)*mdb->size3, "MeshDeformBindPhi");
        mdb->totalphi= MEM_callocN(sizeof(float)*mdb->size3, "MeshDeformBindTotalPhi");
        mdb->boundisect= MEM_callocN(sizeof(*mdb->boundisect)*mdb->size3, "MDefBoundIsect");
        mdb->semibound= MEM_callocN(sizeof(int)*mdb->size3, "MDefSemiBound");

        mdb->inside= MEM_callocN(sizeof(int)*mdb->totvert, "MDefInside");

        if(mmd->flag & MOD_MDEF_DYNAMIC_BIND)
                mdb->dyngrid= MEM_callocN(sizeof(MDefBindInfluence*)*mdb->size3, "MDefDynGrid");
        else
                mdb->weights= MEM_callocN(sizeof(float)*mdb->totvert*mdb->totcagevert, "MDefWeights");

        mdb->memarena= BLI_memarena_new(BLI_MEMARENA_STD_BUFSIZE, "harmonic coords arena");
        BLI_memarena_use_calloc(mdb->memarena);

        /* make bounding box equal size in all directions, add padding, and compute
         * width of the cells */
        maxwidth = -1.0f;
        for(a=0; a<3; a++)
                if(mdb->max[a]-mdb->min[a] > maxwidth)
                        maxwidth= mdb->max[a]-mdb->min[a];

        for(a=0; a<3; a++) {
                center[a]= (mdb->min[a]+mdb->max[a])*0.5f;
                mdb->min[a]= center[a] - maxwidth*0.5f;
                mdb->max[a]= center[a] + maxwidth*0.5f;

                mdb->width[a]= (mdb->max[a]-mdb->min[a])/(mdb->size-4);
                mdb->min[a] -= 2.1f*mdb->width[a];
                mdb->max[a] += 2.1f*mdb->width[a];

                mdb->width[a]= (mdb->max[a]-mdb->min[a])/mdb->size;
                mdb->halfwidth[a]= mdb->width[a]*0.5f;
        }

        progress_bar(0, "Setting up mesh deform system");

        totinside= 0;
        for(a=0; a<mdb->totvert; a++) {
                copy_v3_v3(vec, mdb->vertexcos[a]);
                mdb->inside[a]= meshdeform_inside_cage(mdb, vec);
                if(mdb->inside[a])
                        totinside++;
        }

        /* free temporary MDefBoundIsects */
        BLI_memarena_free(mdb->memarena);
        mdb->memarena= BLI_memarena_new(BLI_MEMARENA_STD_BUFSIZE, "harmonic coords arena");

        /* start with all cells untyped */
        for(a=0; a<mdb->size3; a++)
                mdb->tag[a]= MESHDEFORM_TAG_UNTYPED;
        
        /* detect intersections and tag boundary cells */
        for(z=0; z<mdb->size; z++)
                for(y=0; y<mdb->size; y++)
                        for(x=0; x<mdb->size; x++)
                                meshdeform_add_intersections(mdb, x, y, z);

        /* compute exterior and interior tags */
        meshdeform_bind_floodfill(mdb);

        for(z=0; z<mdb->size; z++)
                for(y=0; y<mdb->size; y++)
                        for(x=0; x<mdb->size; x++)
                                meshdeform_check_semibound(mdb, x, y, z);

        /* solve */
        meshdeform_matrix_solve(mdb);

        /* assign results */
        if(mmd->flag & MOD_MDEF_DYNAMIC_BIND) {
                mmd->totinfluence= 0;
                for(a=0; a<mdb->size3; a++)
                        for(inf=mdb->dyngrid[a]; inf; inf=inf->next)
                                mmd->totinfluence++;

                /* convert MDefBindInfluences to smaller MDefInfluences */
                mmd->dyngrid= MEM_callocN(sizeof(MDefCell)*mdb->size3, "MDefDynGrid");
                mmd->dyninfluences= MEM_callocN(sizeof(MDefInfluence)*mmd->totinfluence, "MDefInfluence");
                offset= 0;
                for(a=0; a<mdb->size3; a++) {
                        cell= &mmd->dyngrid[a];
                        cell->offset= offset;

                        totweight= 0.0f;
                        mdinf= mmd->dyninfluences + cell->offset;
                        for(inf=mdb->dyngrid[a]; inf; inf=inf->next, mdinf++) {
                                mdinf->weight= inf->weight;
                                mdinf->vertex= inf->vertex;
                                totweight += mdinf->weight;
                                cell->totinfluence++;
                        }

                        if(totweight > 0.0f) {
                                mdinf= mmd->dyninfluences + cell->offset;
                                for(b=0; b<cell->totinfluence; b++, mdinf++)
                                        mdinf->weight /= totweight;
                        }

                        offset += cell->totinfluence;
                }

                mmd->dynverts= mdb->inside;
                mmd->dyngridsize= mdb->size;
                copy_v3_v3(mmd->dyncellmin, mdb->min);
                mmd->dyncellwidth= mdb->width[0];
                MEM_freeN(mdb->dyngrid);
        }
        else {
                mmd->bindweights= mdb->weights;
                MEM_freeN(mdb->inside);
        }

        MEM_freeN(mdb->tag);
        MEM_freeN(mdb->phi);
        MEM_freeN(mdb->totalphi);
        MEM_freeN(mdb->boundisect);
        MEM_freeN(mdb->semibound);
        BLI_memarena_free(mdb->memarena);
}

#if 0
static void heat_weighting_bind(Scene *scene, DerivedMesh *dm, MeshDeformModifierData *mmd, MeshDeformBind *mdb)
{
        LaplacianSystem *sys;
        MFace *mface= dm->getFaceArray(dm), *mf;
        int totvert= dm->getNumVerts(dm);
        int totface= dm->getNumFaces(dm);
        float solution, weight;
        int a, tottri, j, thrownerror = 0;

        mdb->weights= MEM_callocN(sizeof(float)*mdb->totvert*mdb->totcagevert, "MDefWeights");

        /* count triangles */
        for(tottri=0, a=0, mf=mface; a<totface; a++, mf++) {
                tottri++;
                if(mf->v4) tottri++;
        }

        /* create laplacian */
        sys = laplacian_system_construct_begin(totvert, tottri, 1);

        sys->heat.mface= mface;
        sys->heat.totface= totface;
        sys->heat.totvert= totvert;
        sys->heat.verts= mdb->vertexcos;
        sys->heat.source = mdb->cagecos;
        sys->heat.numsource= mdb->totcagevert;

        heat_ray_tree_create(sys);
        heat_laplacian_create(sys);

        laplacian_system_construct_end(sys);

        /* compute weights per bone */
        for(j=0; j<mdb->totcagevert; j++) {
                /* fill right hand side */
                laplacian_begin_solve(sys, -1);

                for(a=0; a<totvert; a++)
                        if(heat_source_closest(sys, a, j))
                                laplacian_add_right_hand_side(sys, a,
                                        sys->heat.H[a]*sys->heat.p[a]);

                /* solve */
                if(laplacian_system_solve(sys)) {
                        /* load solution into vertex groups */
                        for(a=0; a<totvert; a++) {
                                solution= laplacian_system_get_solution(a);
                                
                                weight= heat_limit_weight(solution);
                                if(weight > 0.0f)
                                        mdb->weights[a*mdb->totcagevert + j] = weight;
                        }
                }
                else if(!thrownerror) {
                        error("Mesh Deform Heat Weighting:"
                                " failed to find solution for one or more vertices");
                        thrownerror= 1;
                        break;
                }
        }

        /* free */
        heat_system_free(sys);
        laplacian_system_delete(sys);

        mmd->bindweights= mdb->weights;
}
#endif

void mesh_deform_bind(Scene *scene, MeshDeformModifierData *mmd, float *vertexcos, int totvert, float cagemat[][4])
{
        MeshDeformBind mdb;
        MVert *mvert;
        int a;

        waitcursor(1);
        start_progress_bar();

        memset(&mdb, 0, sizeof(MeshDeformBind));

        /* get mesh and cage mesh */
        mdb.vertexcos= MEM_callocN(sizeof(float)*3*totvert, "MeshDeformCos");
        mdb.totvert= totvert;
        
        mdb.cagedm= mesh_create_derived_no_deform(scene, mmd->object, NULL, CD_MASK_BAREMESH);
        mdb.totcagevert= mdb.cagedm->getNumVerts(mdb.cagedm);
        mdb.cagecos= MEM_callocN(sizeof(*mdb.cagecos)*mdb.totcagevert, "MeshDeformBindCos");
        copy_m4_m4(mdb.cagemat, cagemat);

        mvert= mdb.cagedm->getVertArray(mdb.cagedm);
        for(a=0; a<mdb.totcagevert; a++)
                copy_v3_v3(mdb.cagecos[a], mvert[a].co);
        for(a=0; a<mdb.totvert; a++)
                mul_v3_m4v3(mdb.vertexcos[a], mdb.cagemat, vertexcos + a*3);

        /* solve */
#if 0
        if(mmd->mode == MOD_MDEF_VOLUME)
                harmonic_coordinates_bind(scene, mmd, &mdb);
        else
                heat_weighting_bind(scene, dm, mmd, &mdb);
#else
        harmonic_coordinates_bind(scene, mmd, &mdb);
#endif

        /* assign bind variables */
        mmd->bindcagecos= (float*)mdb.cagecos;
        mmd->totvert= mdb.totvert;
        mmd->totcagevert= mdb.totcagevert;
        copy_m4_m4(mmd->bindmat, mmd->object->obmat);

        /* transform bindcagecos to world space */
        for(a=0; a<mdb.totcagevert; a++)
                mul_m4_v3(mmd->object->obmat, mmd->bindcagecos+a*3);

        /* free */
        mdb.cagedm->release(mdb.cagedm);
        MEM_freeN(mdb.vertexcos);

        /* compact weights */
        modifier_mdef_compact_influences((ModifierData*)mmd);

        end_progress_bar();
        waitcursor(0);
}