source: src/ellipsoid.cpp@ 02da9e

/*
 * ellipsoid.cpp
 *
 *      Created on: Jan 20, 2009
 *                      Author: heber
 */

#include "ellipsoid.hpp"

/** Determines squared distance for a given point \a x to surface of ellipsoid.
 * \param x given point
 * \param EllipsoidCenter center of ellipsoid
 * \param EllipsoidLength[3] three lengths of half axis of ellipsoid
 * \param EllipsoidAngle[3] three rotation angles of ellipsoid
 * \return squared distance from point to surface
 */
double SquaredDistanceToEllipsoid(Vector &x, Vector &EllipsoidCenter, double *EllipsoidLength, double *EllipsoidAngle)
{
        Vector helper, RefPoint;
        double distance = -1.;
        double Matrix[NDIM*NDIM];
        double InverseLength[3];
        double psi,theta,phi; // euler angles in ZX'Z'' convention

        //cout << Verbose(3) << "Begin of SquaredDistanceToEllipsoid" << endl;

        for(int i=0;i<3;i++)
                InverseLength[i] = 1./EllipsoidLength[i];

        // 1. translate coordinate system so that ellipsoid center is in origin
        helper.CopyVector(&x);
        helper.SubtractVector(&EllipsoidCenter);
        RefPoint.CopyVector(&helper);
        //cout << Verbose(4) << "Translated given point is at " << RefPoint << "." << endl;

        // 2. transform coordinate system by inverse of rotation matrix and of diagonal matrix
        psi = EllipsoidAngle[0];
        theta = EllipsoidAngle[1];
        phi = EllipsoidAngle[2];
        Matrix[0] = cos(psi)*cos(phi) - sin(psi)*cos(theta)*sin(phi);
        Matrix[1] = -cos(psi)*sin(phi) - sin(psi)*cos(theta)*cos(phi);
        Matrix[2] = sin(psi)*sin(theta);
        Matrix[3] = sin(psi)*cos(phi) + cos(psi)*cos(theta)*sin(phi);
        Matrix[4] = cos(psi)*cos(theta)*cos(phi) - sin(psi)*sin(phi);
        Matrix[5] = -cos(psi)*sin(theta);
        Matrix[6] = sin(theta)*sin(phi);
        Matrix[7] = sin(theta)*cos(phi);
        Matrix[8] = cos(theta);
        helper.MatrixMultiplication(Matrix);
        helper.Scale(InverseLength);
        //cout << Verbose(4) << "Transformed RefPoint is at " << helper << "." << endl;

        // 3. construct intersection point with unit sphere and ray between origin and x
        helper.Normalize(); // is simply normalizes vector in distance direction
        //cout << Verbose(4) << "Transformed intersection is at " << helper << "." << endl;

        // 4. transform back the constructed intersection point
        psi = -EllipsoidAngle[0];
        theta = -EllipsoidAngle[1];
        phi = -EllipsoidAngle[2];
        helper.Scale(EllipsoidLength);
        Matrix[0] = cos(psi)*cos(phi) - sin(psi)*cos(theta)*sin(phi);
        Matrix[1] = -cos(psi)*sin(phi) - sin(psi)*cos(theta)*cos(phi);
        Matrix[2] = sin(psi)*sin(theta);
        Matrix[3] = sin(psi)*cos(phi) + cos(psi)*cos(theta)*sin(phi);
        Matrix[4] = cos(psi)*cos(theta)*cos(phi) - sin(psi)*sin(phi);
        Matrix[5] = -cos(psi)*sin(theta);
        Matrix[6] = sin(theta)*sin(phi);
        Matrix[7] = sin(theta)*cos(phi);
        Matrix[8] = cos(theta);
        helper.MatrixMultiplication(Matrix);
        //cout << Verbose(4) << "Intersection is at " << helper << "." << endl;

        // 5. determine distance between backtransformed point and x
        distance = RefPoint.DistanceSquared(&helper);
        //cout << Verbose(4) << "Squared distance between intersection and RefPoint is " << distance << "." << endl;

        return distance;
        //cout << Verbose(3) << "End of SquaredDistanceToEllipsoid" << endl;
};

/** structure for ellipsoid minimisation containing points to fit to.
 */
struct EllipsoidMinimisation {
        int N;                  //!< dimension of vector set
        Vector *x;      //!< array of vectors
};

/** Sum of squared distance to ellipsoid to be minimised.
 * \param *x parameters for the ellipsoid
 * \param *params EllipsoidMinimisation with set of data points to minimise distance to and dimension
 * \return sum of squared distance, \sa SquaredDistanceToEllipsoid()
 */
double SumSquaredDistance (const gsl_vector * x, void * params)
{
        Vector *set= ((struct EllipsoidMinimisation *)params)->x;
        int N = ((struct EllipsoidMinimisation *)params)->N;
        double SumDistance = 0.;
        double distance;
        Vector Center;
        double EllipsoidLength[3], EllipsoidAngle[3];

        // put parameters into suitable ellipsoid form
        for (int i=0;i<3;i++) {
                Center.x[i] = gsl_vector_get(x, i+0);
                EllipsoidLength[i] = gsl_vector_get(x, i+3);
                EllipsoidAngle[i] = gsl_vector_get(x, i+6);
        }

        // go through all points and sum distance
        for (int i=0;i<N;i++) {
                distance = SquaredDistanceToEllipsoid(set[i], Center, EllipsoidLength, EllipsoidAngle);
                if (!isnan(distance)) {
                        SumDistance += distance;
                } else {
                        SumDistance = GSL_NAN;
                        break;
                }
        }

        //cout << "Current summed distance is " << SumDistance << "." << endl;
        return SumDistance;
};

/** Finds best fitting ellipsoid parameter set in Least square sense for a given point set.
 * \param *out output stream for debugging
 * \param *set given point set
 * \param N number of points in set
 * \param EllipsoidParamter[3] three parameters in ellipsoid equation
 * \return true - fit successful, false - fit impossible
 */
bool FitPointSetToEllipsoid(ofstream *out, Vector *set, int N, Vector *EllipsoidCenter, double *EllipsoidLength, double *EllipsoidAngle)
{
        int status = GSL_SUCCESS;
        *out << Verbose(2) << "Begin of FitPointSetToEllipsoid " << endl;
        if (N >= 3) { // check that enough points are given (9 d.o.f.)
                struct EllipsoidMinimisation par;
                const gsl_multimin_fminimizer_type *T = gsl_multimin_fminimizer_nmsimplex;
                gsl_multimin_fminimizer *s = NULL;
                gsl_vector *ss, *x;
                gsl_multimin_function minex_func;

                size_t iter = 0;
                double size;

                /* Starting point */
                x = gsl_vector_alloc (9);
                for (int i=0;i<3;i++) {
                        gsl_vector_set (x, i+0, EllipsoidCenter->x[i]);
                        gsl_vector_set (x, i+3, EllipsoidLength[i]);
                        gsl_vector_set (x, i+6, EllipsoidAngle[i]);
                }
                par.x = set;
                par.N = N;

                /* Set initial step sizes */
                ss = gsl_vector_alloc (9);
                for (int i=0;i<3;i++) {
                        gsl_vector_set (ss, i+0, 0.1);
                        gsl_vector_set (ss, i+3, 1.0);
                        gsl_vector_set (ss, i+6, M_PI/20.);
                }

                /* Initialize method and iterate */
                minex_func.n = 9;
                minex_func.f = &SumSquaredDistance;
                minex_func.params = (void *)&par;

                s = gsl_multimin_fminimizer_alloc (T, 9);
                gsl_multimin_fminimizer_set (s, &minex_func, x, ss);

                do {
                        iter++;
                        status = gsl_multimin_fminimizer_iterate(s);

                        if (status)
                                break;

                        size = gsl_multimin_fminimizer_size (s);
                        status = gsl_multimin_test_size (size, 1e-2);

                        if (status == GSL_SUCCESS) {
                                for (int i=0;i<3;i++) {
                                        EllipsoidCenter->x[i] = gsl_vector_get (s->x,i+0);
                                        EllipsoidLength[i] = gsl_vector_get (s->x, i+3);
                                        EllipsoidAngle[i] = gsl_vector_get (s->x, i+6);
                                }
                                *out << setprecision(3) << Verbose(4) << "Converged fit at: " << *EllipsoidCenter << ", lengths " << EllipsoidLength[0] << ", " << EllipsoidLength[1] << ", " << EllipsoidLength[2] << ", angles " << EllipsoidAngle[0] << ", " << EllipsoidAngle[1] << ", " << EllipsoidAngle[2] << " with summed distance " << s->fval << "." << endl;
                        }

                } while (status == GSL_CONTINUE && iter < 1000);

                gsl_vector_free(x);
                gsl_vector_free(ss);
                gsl_multimin_fminimizer_free (s);

        } else {
                *out << Verbose(3) << "Not enough points provided for fit to ellipsoid." << endl;
                return false;
        }
        *out << Verbose(2) << "End of FitPointSetToEllipsoid" << endl;
        if (status == GSL_SUCCESS)
                return true;
        else
                return false;
};

/** Picks a number of random points from a LC neighbourhood as a fitting set.
 * \param *out output stream for debugging
 * \param *T Tesselation containing boundary points
 * \param *LC linked cell list of all atoms
 * \param *&x random point set on return (not allocated!)
 * \param PointsToPick number of points in set to pick
 */
void PickRandomNeighbouredPointSet(ofstream *out, class Tesselation *T, class LinkedCell *LC, Vector *&x, int PointsToPick)
{
        int PointsLeft = 0;
        int PointsPicked = 0;
        double value, threshold;
        int Nlower[NDIM], Nupper[NDIM];
        set<int> PickedAtomNrs;  // ordered list of picked atoms
        set<int>::iterator current;
        int index;
        atom *Candidate = NULL;
        LinkedAtoms *List = NULL;
        *out << Verbose(2) << "Begin of PickRandomPointSet" << endl;

        // allocate array
        if (x == NULL) {
                x = new Vector[PointsToPick];
        } else {
                *out << "WARNING: Given pointer to vector array seems already allocated." << endl;
        }

        do {
                for(int i=0;i<NDIM;i++) // pick three random indices
                        LC->n[i] = (rand() % LC->N[i]);
                *out << Verbose(2) << "INFO: Center cell is " << LC->n[0] << ", " << LC->n[1] << ", " << LC->n[2] << " ... ";
                // get random cell
                List = LC->GetCurrentCell();
                if (List == NULL) {     // set index to it
                        continue;
                }
                *out << "with No. " << LC->index << "." << endl;

                *out << Verbose(2) << "LC Intervals:";
                for (int i=0;i<NDIM;i++) {
                        Nlower[i] = ((LC->n[i]-1) >= 0) ? LC->n[i]-1 : 0;
                        Nupper[i] = ((LC->n[i]+1) < LC->N[i]) ? LC->n[i]+1 : LC->N[i]-1;
                        *out << " [" << Nlower[i] << "," << Nupper[i] << "] ";
                }
                *out << endl;

                // count whether there are sufficient atoms in this cell+neighbors
                PointsLeft=0;
                for (LC->n[0] = Nlower[0]; LC->n[0] <= Nupper[0]; LC->n[0]++)
                        for (LC->n[1] = Nlower[1]; LC->n[1] <= Nupper[1]; LC->n[1]++)
                                for (LC->n[2] = Nlower[2]; LC->n[2] <= Nupper[2]; LC->n[2]++) {
                                        List = LC->GetCurrentCell();
                                        PointsLeft += List->size();
                                }
                *out << Verbose(2) << "There are " << PointsLeft << " atoms in this neighbourhood." << endl;
                if (PointsLeft < PointsToPick) {        // ensure that we can pick enough points in its neighbourhood at all.
                        continue;
                }

                // pre-pick a fixed number of atoms
                PickedAtomNrs.clear();
                do {
                        index = (rand() % PointsLeft);
                        current = PickedAtomNrs.find(index);    // not present?
                        if (current == PickedAtomNrs.end()) {
                                //*out << Verbose(2) << "Picking atom nr. " << index << "." << endl;
                                PickedAtomNrs.insert(index);
                        }
                } while (PickedAtomNrs.size() < PointsToPick);

                index = 0; // now go through all and pick those whose from PickedAtomsNr
                PointsPicked=0;
                current = PickedAtomNrs.begin();
                for (LC->n[0] = Nlower[0]; LC->n[0] <= Nupper[0]; LC->n[0]++)
                        for (LC->n[1] = Nlower[1]; LC->n[1] <= Nupper[1]; LC->n[1]++)
                                for (LC->n[2] = Nlower[2]; LC->n[2] <= Nupper[2]; LC->n[2]++) {
                                        List = LC->GetCurrentCell();
//                                      *out << Verbose(2) << "Current cell is " << LC->n[0] << ", " << LC->n[1] << ", " << LC->n[2] << " with No. " << LC->index << " containing " << List->size() << " points." << endl;
                                        if (List != NULL) {
//                                              if (List->begin() != List->end())
//                                                      *out << Verbose(2) << "Going through candidates ... " << endl;
//                                              else
//                                                      *out << Verbose(2) << "Cell is empty ... " << endl;
                                                for (LinkedAtoms::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
                                                        if ((current != PickedAtomNrs.end()) && (*current == index)) {
                                                                Candidate = (*Runner);
                                                                *out << Verbose(2) << "Current picked node is " << **Runner << " with index " << index << "." << endl;
                                                                x[PointsPicked++].CopyVector(&(Candidate->x));          // we have one more atom picked
                                                                current++;              // next pre-picked atom
                                                        }
                                                        index++;        // next atom nr.
                                                }
//                                      } else {
//                                              *out << Verbose(2) << "List for this index not allocated!" << endl;
                                        }
                                }
                *out << Verbose(2) << "The following points were picked: " << endl;
                for (int i=0;i<PointsPicked;i++)
                        *out << Verbose(2) << x[i] << endl;
                if (PointsPicked == PointsToPick)       // break out of loop if we have all
                        break;
        } while(1);

        *out << Verbose(2) << "End of PickRandomPointSet" << endl;
};

/** Picks a number of random points from a set of boundary points as a fitting set.
 * \param *out output stream for debugging
 * \param *T Tesselation containing boundary points
 * \param *&x random point set on return (not allocated!)
 * \param PointsToPick number of points in set to pick
 */
void PickRandomPointSet(ofstream *out, class Tesselation *T, Vector *&x, int PointsToPick)
{
        int PointsLeft = T->PointsOnBoundaryCount;
        int PointsPicked = 0;
        double value, threshold;
        PointMap *List = &T->PointsOnBoundary;
        *out << Verbose(2) << "Begin of PickRandomPointSet" << endl;

        // allocate array
        if (x == NULL) {
                x = new Vector[PointsToPick];
        } else {
                *out << "WARNING: Given pointer to vector array seems already allocated." << endl;
        }

        if (List != NULL)
                for (PointMap::iterator Runner = List->begin(); Runner != List->end(); Runner++) {
                        threshold = 1. - (double)(PointsToPick - PointsPicked)/(double)PointsLeft;
                        value = (double)rand()/(double)RAND_MAX;
                        //*out << Verbose(3) << "Current node is " << *Runner->second->node << " with " << value << " ... " << threshold << ": ";
                        if (value > threshold) {
                                x[PointsPicked].CopyVector(&(Runner->second->node->x));
                                PointsPicked++;
                                //*out << "IN." << endl;
                        } else {
                                //*out << "OUT." << endl;
                        }
                        PointsLeft--;
                }
        *out << Verbose(2) << "The following points were picked: " << endl;
        for (int i=0;i<PointsPicked;i++)
                *out << Verbose(3) << x[i] << endl;

        *out << Verbose(2) << "End of PickRandomPointSet" << endl;
};

/** Finds best fitting ellipsoid parameter set in least square sense for a given point set.
 * \param *out output stream for debugging
 * \param *T Tesselation containing boundary points
 * \param *LCList linked cell list of all atoms
 * \param N number of unique points in ellipsoid fit, must be greater equal 6
 * \param number of fits (i.e. parameter sets in output file)
 * \param *filename name for output file
 */
void FindDistributionOfEllipsoids(ofstream *out, class Tesselation *T, class LinkedCell *LCList, int N, int number, const char *filename)
{
        ofstream output;
        Vector *x = NULL;
        Vector Center;
        Vector EllipsoidCenter;
        double EllipsoidLength[3];
        double EllipsoidAngle[3];
        double distance, MaxDistance, MinDistance;
        *out << Verbose(0) << "Begin of FindDistributionOfEllipsoids" << endl;

        // construct center of gravity of boundary point set for initial ellipsoid center
        Center.Zero();
        for (PointMap::iterator Runner = T->PointsOnBoundary.begin(); Runner != T->PointsOnBoundary.end(); Runner++)
                Center.AddVector(&Runner->second->node->x);
        Center.Scale(1./T->PointsOnBoundaryCount);
        *out << Verbose(1) << "Center is at " << Center << "." << endl;

        // Output header
        output.open(filename, ios::trunc);
        output << "# Nr.\tCenterX\tCenterY\tCenterZ\ta\tb\tc\tpsi\ttheta\tphi" << endl;

        // loop over desired number of parameter sets
        for (;number >0;number--) {
                *out << Verbose(1) << "Determining data set " << number << " ... " << endl;
                // pick the point set
                x = NULL;
                //PickRandomPointSet(out, T, LCList, x, N);
                PickRandomNeighbouredPointSet(out, T, LCList, x, N);

                // calculate some sensible starting values for parameter fit
                MaxDistance = 0.;
                MinDistance = x[0].ScalarProduct(&x[0]);
                for (int i=0;i<N;i++) {
                        distance = x[i].ScalarProduct(&x[i]);
                        if (distance > MaxDistance)
                                MaxDistance = distance;
                        if (distance < MinDistance)
                                MinDistance = distance;
                }
                //*out << Verbose(2) << "MinDistance " << MinDistance << ", MaxDistance " << MaxDistance << "." << endl;
                EllipsoidCenter.CopyVector(&Center);    // use Center of Gravity as initial center of ellipsoid
                for (int i=0;i<3;i++)
                        EllipsoidAngle[i] = 0.;
                EllipsoidLength[0] = sqrt(MaxDistance);
                EllipsoidLength[1] = sqrt((MaxDistance+MinDistance)/2.);
                EllipsoidLength[2] = sqrt(MinDistance);

                // fit the parameters
                if (FitPointSetToEllipsoid(out, x, N, &EllipsoidCenter, &EllipsoidLength[0], &EllipsoidAngle[0])) {
                        *out << Verbose(1) << "Picking succeeded!" << endl;
                        // output obtained parameter set
                        output << number << "\t";
                        for (int i=0;i<3;i++)
                                output << setprecision(9) << EllipsoidCenter.x[i] << "\t";
                        for (int i=0;i<3;i++)
                                output << setprecision(9) << EllipsoidLength[i] << "\t";
                        for (int i=0;i<3;i++)
                                output << setprecision(9) << EllipsoidAngle[i] << "\t";
                        output << endl;
                } else { // increase N to pick one more
                        *out << Verbose(1) << "Picking failed!" << endl;
                        number++;
                }
                delete[](x);    // free allocated memory for point set
        }
        // close output and finish
        output.close();

        *out << Verbose(0) << "End of FindDistributionOfEllipsoids" << endl;
};