LevMarFitting.h

Go to the documentation of this file.

#ifndef LEVMARFITTING_HEADER
#define LEVMARFITTING_HEADER
#include <algorithm>
#include <iostream>
#include "LevMarFunc.h"
#ifdef DOPARALLEL
#include <omp.h>
#endif
 
template< class ScalarT >
bool Cholesky(ScalarT* a, size_t n, ScalarT p[])
/*Given a positive-definite symmetric matrix a[1..n][1..n], this routine constructs its Cholesky
decomposition, A = L · LT . On input, only the upper triangle of a need be given; it is not
modified. The Cholesky factor L is returned in the lower triangle of a, except for its diagonal
elements which are returned in p[1..n].*/
{
    size_t i, j, k;
    ScalarT sum;
    for (i = 0; i < n; ++i)
    {
        for (j = i; j < n; ++j)
        {
            for (sum = a[i * n + j], k = i - 1; k != -1; --k)
            {
                sum -= a[i * n + k] * a[j * n + k];
            }
            if (i == j)
            {
                if (sum <= ScalarT(0))
                    // a, with rounding errors, is not positive definite.
                {
                    return false;
                }
                p[i] = std::sqrt(sum);
            }
            else
            {
                a[j * n + i] = sum / p[i];
            }
        }
    }
    return true;
}
 
template< class ScalarT, unsigned int N >
bool Cholesky(ScalarT* a, ScalarT p[])
/*Given a positive-definite symmetric matrix a[1..n][1..n], this routine constructs its Cholesky
decomposition, A = L · LT . On input, only the upper triangle of a need be given; it is not
modified. The Cholesky factor L is returned in the lower triangle of a, except for its diagonal
elements which are returned in p[1..n].*/
{
    size_t i, j, k;
    ScalarT sum;
    for (i = 0; i < N; ++i)
    {
        for (j = i; j < N; ++j)
        {
            for (sum = a[i * N + j], k = i - 1; k != -1; --k)
            {
                sum -= a[i * N + k] * a[j * N + k];
            }
            if (i == j)
            {
                if (sum <= ScalarT(0))
                    // a, with rounding errors, is not positive definite.
                {
                    return false;
                }
                p[i] = std::sqrt(sum);
            }
            else
            {
                a[j * N + i] = sum / p[i];
            }
        }
    }
    return true;
}
 
template< class ScalarT >
void CholeskySolve(ScalarT* a, size_t n, ScalarT p[], ScalarT b[], ScalarT x[])
/*Solves the set of n linear equations A · x = b, where a is a positive-definite symmetric matrix.
a[1..n][1..n] and p[1..n] are input as the output of the routine choldc. Only the lower
subdiagonal portion of a is accessed. b[1..n] is input as the right-hand side vector. The
solution vector is returned in x[1..n]. a, n, and p are not modified and can be left in place
for successive calls with different right-hand sides b. b is not modified unless you identify b and
x in the calling sequence, which is allowed.*/
{
    size_t i, k;
    ScalarT sum;
    for (i = 0; i < n; i++)
    {
        // Solve L · y = b, storing y in x.
        for (sum = b[i], k = i - 1; k != -1; --k)
        {
            sum -= a[i * n + k] * x[k];
        }
        x[i] = sum / p[i];
    }
    for (i = n - 1; i != -1; --i)
    {
        // Solve LT · x = y.
        for (sum = x[i], k = i + 1; k < n; ++k)
        {
            sum -= a[k * n + i] * x[k];
        }
        x[i] = sum / p[i];
    }
}
 
template< class ScalarT, unsigned int N >
void CholeskySolve(ScalarT* a, ScalarT p[], ScalarT b[], ScalarT x[])
/*Solves the set of n linear equations A · x = b, where a is a positive-definite symmetric matrix.
a[1..n][1..n] and p[1..n] are input as the output of the routine choldc. Only the lower
subdiagonal portion of a is accessed. b[1..n] is input as the right-hand side vector. The
solution vector is returned in x[1..n]. a, n, and p are not modified and can be left in place
for successive calls with different right-hand sides b. b is not modified unless you identify b and
x in the calling sequence, which is allowed.*/
{
    size_t i, k;
    ScalarT sum;
    for (i = 0; i < N; i++)
    {
        // Solve L · y = b, storing y in x.
        for (sum = b[i], k = i - 1; k != -1; --k)
        {
            sum -= a[i * N + k] * x[k];
        }
        x[i] = sum / p[i];
    }
    for (i = N - 1; i != -1; --i)
    {
        // Solve LT · x = y.
        for (sum = x[i], k = i + 1; k < N; ++k)
        {
            sum -= a[k * N + i] * x[k];
        }
        x[i] = sum / p[i];
    }
}
 
template< class IteratorT, class FuncT >
bool LevMar(IteratorT begin, IteratorT end, FuncT& func,
            typename FuncT::ScalarType* param)
{
    typedef typename FuncT::ScalarType ScalarType;
    enum { paramDim = FuncT::NumParams };
    bool retVal = true;
    unsigned int totalSize = end - begin;
    if (!totalSize)
    {
        return false;
    }
    ScalarType lambda = ScalarType(0.0001);
    ScalarType* F0 = new ScalarType[totalSize * paramDim];
    ScalarType* U = new ScalarType[paramDim * paramDim];
    ScalarType* H = new ScalarType[paramDim * paramDim];
    ScalarType* v = new ScalarType[paramDim];
    ScalarType* d = new ScalarType[totalSize];
    ScalarType* temp = new ScalarType[totalSize];
    ScalarType* x = new ScalarType[paramDim];
    ScalarType* p = new ScalarType[paramDim];
    ScalarType* paramNew = new ScalarType[paramDim];
    size_t nu = 2;
    func.Normalize(param);
    ScalarType paramNorm = 0;
 
    // do fitting in different steps
    unsigned int subsets = std::max(int(std::floor(std::log((float)totalSize) / std::log(2.f))) - 8, 1);
#ifdef PRECISIONLEVMAR
    subsets = 1;
#endif
    MiscLib::Vector< unsigned int > subsetSizes(subsets);
    for (unsigned int i = subsetSizes.size(); i;)
    {
        --i;
        subsetSizes[i] = totalSize;
        if (i)
        {
            subsetSizes[i] = subsetSizes[i] >> 1;
        }
        totalSize -= subsetSizes[i];
    }
    unsigned int curSubset = 0;
    unsigned int size = 0;
 
    // get current error
    ScalarType chi = 0, newChi = 0;
    ScalarType rho = 1;
    unsigned int outerIter = 0,
#ifndef PRECISIONLEVMAR
                 maxOuterIter = 200 / subsetSizes.size(),
#else
                 maxOuterIter = 500,
#endif
                 usefulIter = 0, totalIter = 0;;
    do
    {
        // get current error
        size += subsetSizes[curSubset];
        newChi = func.Chi(param, begin, begin + size, d, temp);
        for (unsigned int i = 0; i < paramDim; ++i)
        {
            paramNew[i] = param[i];
        }
        outerIter = 0;
        if (rho < 0)
        {
            rho = 1;
        }
        do
        {
            ++outerIter;
            ++totalIter;
            if (rho > 0)
            {
                nu = 2;
                chi = newChi;
                for (size_t i = 0; i < paramDim; ++i)
                {
                    param[i] = paramNew[i];
                }
#ifndef PRECISIONLEVMAR
                if (std::sqrt(chi / size) < ScalarType(1e-5)) // chi very small? -> will be hard to improve
                {
                    //std::cout << "LevMar converged because of small chi" << std::endl;
                    break;
                }
#endif
                paramNorm = 0;
                for (size_t i = 0; i < paramDim; ++i)
                {
                    paramNorm += param[i] * param[i];
                }
                paramNorm = std::sqrt(paramNorm);
                // construct the needed matrices
                // F0 is the matrix constructed from param
                // F0 has gradient_i(param) as its ith row
                func.Derivatives(param, begin, begin + size, d, temp, F0);
                // U = F0_t * F0
                // v = F0_t * d(param) (d(param) = [d_i(param)])
                #pragma omp parallel for
                for (int i = 0; i < paramDim; ++i)
                {
                    for (size_t j = i; j < paramDim; ++j) // j = i since only upper triangle is needed
                    {
                        U[i * paramDim + j] = 0;
                        for (size_t k = 0; k < size; ++k)
                        {
                            U[i * paramDim + j] += F0[k * paramDim + i] *
                                                   F0[k * paramDim + j];
                        }
                    }
                }
                ScalarType vmag = 0; // magnitude of v
                #pragma omp parallel for
                for (int i = 0; i < paramDim; ++i)
                {
                    v[i] = 0;
                    for (size_t k = 0; k < size; ++k)
                    {
                        v[i] += F0[k * paramDim + i] * d[k];
                    }
                    v[i] *= -1;
#ifndef DOPARALLEL
                    vmag = std::max(abs(v[i]), vmag);
#endif
                }
#ifdef DOPARALLEL
                for (unsigned int i = 0; i < paramDim; ++i)
                {
                    vmag = std::max(abs(v[i]), vmag);
                }
#endif
                // and check for convergence with magnitude of v
#ifndef PRECISIONLEVMAR
                if (vmag < ScalarType(1e-6))
#else
                if (vmag < ScalarType(1e-8))
#endif
                {
                    //std::cout << "LevMar converged with small gradient" << std::endl;
                    //retVal = chi < initialChi;
                    //goto cleanup;
                    break;
                }
                if (outerIter == 1)
                {
                    // compute magnitue of F0
                    ScalarType fmag = abs(F0[0]);
                    for (size_t i = 1; i < paramDim * size; ++i)
                        if (fmag < abs(F0[i]))
                        {
                            fmag = abs(F0[i]);
                        }
                    lambda = 1e-3f * fmag;
                }
                else
                {
                    lambda *= std::max(ScalarType(.3), ScalarType(1 - std::pow(2 * rho - 1, 3)));
                }
            }
 
            memcpy(H, U, sizeof(ScalarType) * paramDim * paramDim);
            for (size_t i = 0; i < paramDim; ++i)
            {
                H[i * paramDim + i] += lambda;    // * (ScalarType(1) + H[i * paramDim + i]);
            }
            // now H is positive definite and symmetric
            // solve Hx = -v with Cholesky
            ScalarType xNorm = 0, L = 0;
            if (!Cholesky< ScalarType, paramDim >(H, p))
            {
                goto increment;
            }
            CholeskySolve< ScalarType, paramDim >(H, p, v, x);
 
            // magnitude of x small? If yes we are done
            for (size_t i = 0; i < paramDim; ++i)
            {
                xNorm += x[i] * x[i];
            }
            xNorm = std::sqrt(xNorm);
#ifndef PRECISIONLEVMAR
            if (xNorm <= ScalarType(1e-6) * (paramNorm + ScalarType(1e-6)))
#else
            if (xNorm <= ScalarType(1e-8) * (paramNorm + ScalarType(1e-8)))
#endif
            {
                //std::cout << "LevMar converged with small step" << std::endl;
                //goto cleanup;
                break;
            }
 
            for (size_t i = 0; i < paramDim; ++i)
            {
                paramNew[i] = param[i] + x[i];
            }
            func.Normalize(paramNew);
 
            // get new error
            newChi = func.Chi(paramNew, begin, begin + size, d, temp);
 
            // the following test is taken from
            // "Methods for non-linear least squares problems"
            // by Madsen, Nielsen, Tingleff
            L = 0;
            for (size_t i = 0; i < paramDim; ++i)
            {
                L += .5f * x[i] * (lambda * x[i] + v[i]);
            }
            rho = (chi - newChi) / L;
            if (rho > 0)
            {
                ++usefulIter;
#ifndef PRECISIONLEVMAR
                if ((chi - newChi) < 1e-4 * chi)
                {
                    //std::cout << "LevMar converged with small chi difference" << std::endl;
                    chi = newChi;
                    for (size_t i = 0; i < paramDim; ++i)
                    {
                        param[i] = paramNew[i];
                    }
                    break;
                }
#endif
                continue;
            }
 
increment:
            rho = -1;
            // increment lambda
            lambda = nu * lambda;
            size_t nu2 = nu << 1;
            if (nu2 < nu)
            {
                nu2 = 2;
            }
            nu = nu2;
        }
        while (outerIter < maxOuterIter);
        ++curSubset;
    }
    while (curSubset < subsetSizes.size());
    retVal = usefulIter > 0;
    delete[] F0;
    delete[] U;
    delete[] H ;
    delete[] v;
    delete[] d;
    delete[] temp;
    delete[] x;
    delete[] p;
    delete[] paramNew;
    return retVal;
}
 
template< class ScalarT >
ScalarT LevMar(unsigned int paramDim, unsigned int imgDim,
               const LevMarFunc< ScalarT >** funcs, ScalarT* param)
{
    ScalarT retVal = -1;
    size_t size = imgDim;
    float lambda = ScalarT(0.0001);
    ScalarT* F0 = new ScalarT[size * paramDim];
    ScalarT* U = new ScalarT[paramDim * paramDim];
    ScalarT* H = new ScalarT[paramDim * paramDim];
    ScalarT* v = new ScalarT[paramDim];
    ScalarT* d = new ScalarT[size];
    ScalarT* dNew = new ScalarT[size];
    ScalarT* x = new ScalarT[paramDim];
    ScalarT* p = new ScalarT[paramDim];
    ScalarT* paramNew = new ScalarT[paramDim];
    size_t outerIter = 0, maxOuterIter = 10;
    // get current error
    ScalarT chi = 0, newChi;
    for (size_t i = 0; i < size; ++i)
    {
        d[i] = (*(funcs[i]))(param);
        chi += d[i] * d[i];
    }
    do
    {
        ++outerIter;
        lambda *= ScalarT(0.04);
        // construct the needed matrices
        // F0 is the matrix constructed from param
        // F0 has gradient_i(param) as its ith row
        for (size_t i = 0; i < size; ++i)
        {
            (*(funcs[i]))(param, &F0[i * paramDim]);
        }
        // U = F0_t * F0
        for (size_t i = 0; i < paramDim; ++i)
            for (size_t j = i; j < paramDim; ++j) // j = i since only upper triangle is needed
            {
                U[i * paramDim + j] = 0;
                for (size_t k = 0; k < size; ++k)
                    U[i * paramDim + j] += F0[k * paramDim + i] *
                                           F0[k * paramDim + j];
            }
        // v = F_t * d(param) (d(param) = [d_i(param)])
        for (size_t i = 0; i < paramDim; ++i)
        {
            v[i] = 0;
            for (size_t k = 0; k < size; ++k)
            {
                v[i] += F0[k * paramDim + i] * d[k];
            }
            v[i] *= -1;
        }
        size_t iter = 0, maxIter = 10;
        do
        {
            ++iter;
            // increment lambda
            lambda = 10 * lambda;
            memcpy(H, U, sizeof(ScalarT) * paramDim * paramDim);
            for (size_t i = 0; i < paramDim; ++i)
            {
                H[i * paramDim + i] += lambda * (ScalarT(1) + H[i * paramDim + i]);
            }
            // now H is positive definite and symmetric
            // solve Hx = -v with Cholesky
            if (!Cholesky(H, paramDim, p))
            {
                goto cleanup;
            }
            CholeskySolve(H, paramDim, p, v, x);
            for (size_t i = 0; i < paramDim; ++i)
            {
                paramNew[i] = param[i] + x[i];
            }
            // get new error
            newChi = 0;
            for (size_t i = 0; i < size; ++i)
            {
                dNew[i] = (*(funcs[i]))(paramNew);
                newChi += dNew[i] * dNew[i];
            }
            // check for convergence
            /*float cvgTest = 0;
            for(size_t i = 0; i < paramDim; ++i)
            {
                //float c = param[i] - paramNew[i];
                cvgTest += x[i] * x[i];
            }
            if(std::sqrt(cvgTest) < 1e-6)
            {
                for(size_t i = 0; i < paramDim; ++i)
                    param[i] = paramNew[i];
                goto cleanup;
            }*/
            if (/*newChi <= chi &&*/ abs(chi - newChi)
                                     / chi < ScalarT(1e-4))
            {
                for (size_t i = 0; i < paramDim; ++i)
                {
                    param[i] = paramNew[i];
                }
                retVal = newChi;
                goto cleanup;
            }
        }
        while (newChi > chi && iter < maxIter);
        if (newChi < chi)
        {
            chi = newChi;
            for (size_t i = 0; i < paramDim; ++i)
            {
                param[i] = paramNew[i];
            }
            std::swap(d, dNew);
        }
    }
    while (outerIter < maxOuterIter);
cleanup:
    delete[] F0;
    delete[] U;
    delete[] H ;
    delete[] v;
    delete[] d;
    delete[] dNew;
    delete[] x;
    delete[] p;
    delete[] paramNew;
    return retVal;
}
 
#endif