DenseVector FitLeastSquaresBasis(double[] xdata, double[] ydata, FitFunction basis, int deg = 1)
{
var X = MakeBasis(xdata, basis, deg);
var Y = new DenseVector(ydata);
var A = (DenseMatrix)X.Transpose() * X;
var B = (DenseVector)(X.Transpose() * Y);
var W = (DenseVector)(A.Solve(B));
return(W);
}
DenseMatrix MakeBasis(double[] xdata, FitFunction basis, int deg)
{
int n = xdata.Length;
int columns = 0;
if (basis == FitFunction.polynomial)
{
columns = deg + 1;
}
else if (basis == FitFunction.noroozi)
{
columns = 3;
}
var X = new DenseMatrix(n, columns);
switch (basis)
{
case FitFunction.polynomial:
for (int j = 0; j <= deg; j++)
{
for (int i = 0; i < n; i++)
{
X[i, j] = Math.Pow(xdata[i], j);
}
}
break;
case FitFunction.noroozi:
int t;
t = 0;
for (int i = 0; i < n; i++)
{
X[i, 0] = Math.Pow(xdata[i], t);
}
t = 2;
for (int i = 0; i < n; i++)
{
X[i, 1] = Math.Pow(xdata[i], t);
}
t = 6;
for (int i = 0; i < n; i++)
{
X[i, 2] = Math.Pow(xdata[i], t);
}
break;
default:
break;
}
return(X);
}
double CalculatePerpendicularDistancePointToCurve(double[] xdata, double[] ydata, DenseVector W,
FitFunction basis, int deg, Graphics g)
{
const double stepSize = 0.01;
double minX = xdata.Min() - 20;
double maxX = xdata.Max() + 20;
int nCurve = (int)((maxX - minX) / stepSize);
int n = xdata.Length;
double[] plotX = new double[nCurve];
for (int i = 0; i < nCurve; i++)
{
plotX[i] = minX + i * stepSize;
}
var PlotX = MakeBasis(plotX, basis, deg);
var PlotY = PlotX * W;
double s = 0;
for (int i = 0; i < n; i++)
{
double minDist = double.MaxValue;
int minIndex = -1;
for (int j = 0; j < nCurve; j++)
{
var d = Math.Pow(xdata[i] - plotX[j], 2) + Math.Pow(ydata[i] - PlotY[j], 2);
if (d < minDist)
{
minDist = d;
minIndex = j;
}
}
s += minDist; //sum of squared error
//draw the perpendicular line
g.DrawLine(new Pen(new SolidBrush(Color.Red)), new PointF((float)xdata[i] * 3, (float)ydata[i] * 3),
new PointF((float)plotX[minIndex] * 3, (float)PlotY[minIndex] * 3));
}
double error = Math.Sqrt(s / n); //root mean squared error
return(error);
}
void DrawCurve(Graphics g, DenseVector W, double minX, double maxX, FitFunction basis, int deg = 0)
{
const double stepSize = 0.1;
int n = (int)((maxX - minX) / stepSize);
double[] xdata = new double[n];
for (int i = 0; i < n; i++)
{
xdata[i] = minX + stepSize * i;
}
var X = MakeBasis(xdata, basis, deg);
var Y = X * W;
for (int i = 0; i < n; i++)
{
DrawPoint2DOnControl(g, new PointF((float)xdata[i], (float)Y[i]), Color.Blue, 1);
}
}
/* central finite difference approximation to the jacobian of func */
static void FDIF_CENT_JAC_APPROX(
FitFunction func,
/* function to differentiate */
double[] p, /* I: current parameter estimate, mx1 */
double[] hxm, /* W/O: work array for evaluating func(p-delta), nx1 */
double[] hxp, /* W/O: work array for evaluating func(p+delta), nx1 */
double delta, /* increment for computing the jacobian */
double[] jac, /* O: array for storing approximated jacobian, nxm */
int m,
int n,
object adata)
{
int i, j;
double tmp;
double d;
for(j=0; j<m; ++j)
{
/* determine d=max(1E-04*|p[j]|, delta), see HZ */
d=(1E-04)*p[j]; // force evaluation
d=(d);
if(d<delta)
d=delta;
tmp=p[j];
p[j]-=d;
func(p, hxm, adata);
p[j]=tmp+d;
func(p, hxp, adata);
p[j]=tmp; /* restore */
d=(0.5)/d; /* invert so that divisions can be carried out faster as multiplications */
for(i=0; i<n; ++i)
{
jac[i*m+j]=(hxp[i]-hxm[i])*d;
}
}
}
static public int LEVMAR_DIF(
FitFunction func, /* functional relation describing measurements. A p \in R^m yields a \hat{x} \in R^n */
double [] p, /* I/O: initial parameter estimates. On output has the estimated solution */
double [] x, /* I: measurement vector */
int itmax, /* I: maximum number of iterations */
double[] opts, /* I: opts[0-4] = minim. options [\mu, \epsilon1, \epsilon2, \epsilon3, \delta]. Respectively the
* scale factor for initial \mu, stopping thresholds for ||J^T e||_inf, ||Dp||_2 and ||e||_2 and
* the step used in difference approximation to the jacobian. Set to NULL for defaults to be used.
* If \delta<0, the jacobian is approximated with central differences which are more accurate
* (but slower!) compared to the forward differences employed by default.
*/
double[] info,
/* O: information regarding the minimization. Set to NULL if don't care
* info[0]= ||e||_2 at initial p.
* info[1-4]=[ ||e||_2, ||J^T e||_inf, ||Dp||_2, mu/max[J^T J]_ii ], all computed at estimated p.
* info[5]= # iterations,
* info[6]=reason for terminating: 1 - stopped by small gradient J^T e
* 2 - stopped by small Dp
* 3 - stopped by itmax
* 4 - singular matrix. Restart from current p with increased mu
* 5 - no further error reduction is possible. Restart with increased mu
* 6 - stopped by small ||e||_2
* info[7]= # function evaluations
* info[8]= # jacobian evaluations
*/
ref object workingmemory, /* working memory, allocate if NULL */
double[] covar, /* O: Covariance matrix corresponding to LS solution; mxm. Set to NULL if not needed. */
object adata) /* pointer to possibly additional data, passed uninterpreted to func.
* Set to NULL if not needed
*/
{
int m = p.Length; /* I: parameter vector dimension (i.e. #unknowns) */
int n = x.Length; /* I: measurement vector dimension */
int i, j, k, l;
bool issolved;
/* temp work arrays */
double[] e; /* nx1 */
double[] hx; /* \hat{x}_i, nx1 */
double[] jacTe; /* J^T e_i mx1 */
double[] jac; /* nxm */
double[] jacTjac; /* mxm */
double[] Dp; /* mx1 */
double[] diag_jacTjac; /* diagonal of J^T J, mx1 */
double[] pDp; /* p + Dp, mx1 */
double[] wrk; /* nx1 */
bool using_ffdif=true;
double[] wrk2=null; /* nx1, used for differentiating with central differences only */
double mu, /* damping constant */
tmp; /* mainly used in matrix & vector multiplications */
double p_eL2, jacTe_inf, pDp_eL2; /* ||e(p)||_2, ||J^T e||_inf, ||e(p+Dp)||_2 */
double p_L2, Dp_L2=LM_REAL_MAX, dF, dL;
double tau, eps1, eps2, eps2_sq, eps3, delta;
double init_p_eL2;
int nu, nu2, stop, nfev, njap=0, K=(m>=10)? m: 10, updjac;
bool updp=true;
bool newjac;
int nm=n*m;
mu=jacTe_inf=p_L2=0.0; /* -Wall */
stop = updjac = 0; newjac = false; /* -Wall */
if(n<m)
{
throw new ArithmeticException(string.Format("Cannot solve a problem with fewer measurements {0} than unknowns {1}", n, m));
}
if(opts!=null)
{
tau=opts[0];
eps1=opts[1];
eps2=opts[2];
eps2_sq=opts[2]*opts[2];
eps3=opts[3];
delta=opts[4];
if(delta<0.0)
{
delta=-delta; /* make positive */
using_ffdif=false; /* use central differencing */
wrk2 = new double[n];
}
}
else
{ // use default values
tau=LM_INIT_MU;
eps1=LM_STOP_THRESH;
eps2=LM_STOP_THRESH;
eps2_sq=LM_STOP_THRESH*LM_STOP_THRESH;
eps3=LM_STOP_THRESH;
delta=LM_DIFF_DELTA;
}
WorkArrays work = workingmemory as WorkArrays;
if(null==work)
{
work = new WorkArrays(n, m);
workingmemory = work;
}
/* set up work arrays */
e=work.e;
hx = work.hx;
jacTe = work.jacTe;
jac = work.jac;
jacTjac = work.jacTjac;
Dp = work.Dp;
diag_jacTjac = work.diag_jacTjac;
pDp = work.pDp;
wrk = work.wrk;
/* compute e=x - f(p) and its L2 norm */
func(p, hx, adata); nfev=1;
for(i=0, p_eL2=0.0; i<n; ++i)
{
e[i]=tmp=x[i]-hx[i];
p_eL2+=tmp*tmp;
}
init_p_eL2=p_eL2;
nu=20; /* force computation of J */
for(k=0; k<itmax; ++k)
{
/* Note that p and e have been updated at a previous iteration */
if(p_eL2<=eps3)
{ /* error is small */
stop=6;
break;
}
/* Compute the jacobian J at p, J^T J, J^T e, ||J^T e||_inf and ||p||^2.
* The symmetry of J^T J is again exploited for speed
*/
if((updp && nu>16) || updjac==K)
{ /* compute difference approximation to J */
if(using_ffdif)
{ /* use forward differences */
FDIF_FORW_JAC_APPROX(func, p, hx, wrk, delta, jac, m, n, adata);
++njap; nfev+=m;
}
else
{ /* use central differences */
FDIF_CENT_JAC_APPROX(func, p, wrk, wrk2, delta, jac, m, n, adata);
++njap; nfev+=2*m;
}
nu=2; updjac=0; updp=false; newjac=true;
}
if(newjac)
{ /* jacobian has changed, recompute J^T J, J^t e, etc */
newjac=false;
/* J^T J, J^T e */
if(nm<=__BLOCKSZ__SQ)
{ // this is a small problem
/* This is the straightforward way to compute J^T J, J^T e. However, due to
* its noncontinuous memory access pattern, it incures many cache misses when
* applied to large minimization problems (i.e. problems involving a large
* number of free variables and measurements), in which J is too large to
* fit in the L1 cache. For such problems, a cache-efficient blocking scheme
* is preferable.
*
* Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
* performance problem.
*
* On the other hand, the straightforward algorithm is faster on small
* problems since in this case it avoids the overheads of blocking.
*/
for(i=0; i<m; ++i)
{
for(j=i; j<m; ++j)
{
int lm;
for(l=0, tmp=0.0; l<n; ++l)
{
lm=l*m;
tmp+=jac[lm+i]*jac[lm+j];
}
jacTjac[i*m+j]=jacTjac[j*m+i]=tmp;
}
/* J^T e */
for(l=0, tmp=0.0; l<n; ++l)
tmp+=jac[l*m+i]*e[l];
jacTe[i]=tmp;
}
}
else
{ // this is a large problem
/* Cache efficient computation of J^T J based on blocking
*/
TRANS_MAT_MAT_MULT(jac, jacTjac, n, m, __BLOCKSZ__,null);
/* cache efficient computation of J^T e */
for(i=0; i<m; ++i)
jacTe[i]=0.0;
for(i=0; i<n; ++i)
{
int jacrow;
for(l=0, jacrow=i*m, tmp=e[i]; l<m; ++l)
jacTe[l]+=jac[l+jacrow]*tmp;
}
}
/* Compute ||J^T e||_inf and ||p||^2 */
for(i=0, p_L2=jacTe_inf=0.0; i<m; ++i)
{
if(jacTe_inf < (tmp=Math.Abs(jacTe[i]))) jacTe_inf=tmp;
diag_jacTjac[i]=jacTjac[i*m+i]; /* save diagonal entries so that augmentation can be later canceled */
p_L2+=p[i]*p[i];
}
//p_L2=sqrt(p_L2);
}
#if false
if(!(k%100)){
printf("Current estimate: ");
for(i=0; i<m; ++i)
printf("%.9g ", p[i]);
printf("-- errors %.9g %0.9g\n", jacTe_inf, p_eL2);
}
#endif
/* check for convergence */
if((jacTe_inf <= eps1))
{
Dp_L2=0.0; /* no increment for p in this case */
stop=1;
break;
}
/* compute initial damping factor */
if(k==0)
{
for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
if(diag_jacTjac[i]>tmp) tmp=diag_jacTjac[i]; /* find max diagonal element */
mu=tau*tmp;
}
/* determine increment using adaptive damping */
/* augment normal equations */
for(i=0; i<m; ++i)
jacTjac[i*m+i]+=mu;
/* solve augmented equations */
#if HAVE_LAPACK
/* 5 alternatives are available: LU, Cholesky, 2 variants of QR decomposition and SVD.
* Cholesky is the fastest but might be inaccurate; QR is slower but more accurate;
* SVD is the slowest but most accurate; LU offers a tradeoff between accuracy and speed
*/
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m);
//issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m);
#else
/* use the LU included with levmar */
issolved=0!=AX_EQ_B_LU(jacTjac, jacTe, Dp, m);
#endif // HAVE_LAPACK
if(issolved)
{
/* compute p's new estimate and ||Dp||^2 */
for(i=0, Dp_L2=0.0; i<m; ++i)
{
pDp[i]=p[i] + (tmp=Dp[i]);
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2)
{ /* relative change in p is small, stop */
//if(Dp_L2<=eps2*(p_L2 + eps2)){ /* relative change in p is small, stop */
stop=2;
break;
}
if(Dp_L2>=(p_L2+eps2)/(EPSILON*EPSILON))
{ /* almost singular */
//if(Dp_L2>=(p_L2+eps2)/CNST(EPSILON)){ /* almost singular */
stop=4;
break;
}
func(pDp, wrk, adata); ++nfev; /* evaluate function at p + Dp */
for(i=0, pDp_eL2=0.0; i<n; ++i)
{ /* compute ||e(pDp)||_2 */
tmp=x[i]-wrk[i];
pDp_eL2+=tmp*tmp;
}
dF=p_eL2-pDp_eL2;
if(updp || dF>0)
{ /* update jac */
for(i=0; i<n; ++i)
{
for(l=0, tmp=0.0; l<m; ++l)
tmp+=jac[i*m+l]*Dp[l]; /* (J * Dp)[i] */
tmp=(wrk[i] - hx[i] - tmp)/Dp_L2; /* (f(p+dp)[i] - f(p)[i] - (J * Dp)[i])/(dp^T*dp) */
for(j=0; j<m; ++j)
jac[i*m+j]+=tmp*Dp[j];
}
++updjac;
newjac=true;
}
for(i=0, dL=0.0; i<m; ++i)
dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);
if(dL>0.0 && dF>0.0)
{ /* reduction in error, increment is accepted */
dF=(2.0*dF/dL-1.0);
tmp=dF*dF*dF;
tmp=1.0-tmp*tmp*dF;
mu=mu*( (tmp>=ONE_THIRD)? tmp : ONE_THIRD );
nu=2;
for(i=0 ; i<m; ++i) /* update p's estimate */
p[i]=pDp[i];
for(i=0; i<n; ++i)
{ /* update e, hx and ||e||_2 */
e[i]=x[i]-wrk[i];
hx[i]=wrk[i];
}
p_eL2=pDp_eL2;
updp=true;
continue;
}
}
/* if this point is reached, either the linear system could not be solved or
* the error did not reduce; in any case, the increment must be rejected
*/
mu*=nu;
nu2=nu<<1; // 2*nu;
if(nu2<=nu)
{ /* nu has wrapped around (overflown). Thanks to Frank Jordan for spotting this case */
stop=5;
break;
}
nu=nu2;
for(i=0; i<m; ++i) /* restore diagonal J^T J entries */
jacTjac[i*m+i]=diag_jacTjac[i];
}
if(k>=itmax) stop=3;
for(i=0; i<m; ++i) /* restore diagonal J^T J entries */
jacTjac[i*m+i]=diag_jacTjac[i];
if(info!=null)
{
info[0]=init_p_eL2;
info[1]=p_eL2;
info[2]=jacTe_inf;
info[3]=Dp_L2;
for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
if(tmp<jacTjac[i*m+i]) tmp=jacTjac[i*m+i];
info[4]=mu/tmp;
info[5]=(double)k;
info[6]=(double)stop;
info[7]=(double)nfev;
info[8]=(double)njap;
}
/* covariance matrix */
if(covar!=null)
{
LEVMAR_COVAR(jacTjac, covar, p_eL2, m, n);
}
return (stop!=4)? k : -1;
}
/*
* This function seeks the parameter vector p that best describes the measurements vector x.
* More precisely, given a vector function func : R^m --> R^n with n>=m,
* it finds p s.t. func(p) ~= x, i.e. the squared second order (i.e. L2) norm of
* e=x-func(p) is minimized.
*
* This function requires an analytic jacobian. In case the latter is unavailable,
* use LEVMAR_DIF() bellow
*
* Returns the number of iterations (>=0) if successfull, -1 if failed
*
* For more details, see H.B. Nielsen's (http://www.imm.dtu.dk/~hbn) IMM/DTU
* tutorial at http://www.imm.dtu.dk/courses/02611/nllsq.pdf
*/
public static int LEVMAR_DER(
FitFunction func, /* functional relation describing measurements. A p \in R^m yields a \hat{x} \in R^n */
JacobianFunction jacf, /* function to evaluate the jacobian \part x / \part p */
double []p, /* I/O: initial parameter estimates. On output has the estimated solution */
double []x, /* I: measurement vector */
double []weights, /* vector of the weights used to scale the fit differences, can be null */
int itmax, /* I: maximum number of iterations */
double[] opts, /* I: minim. options [\mu, \epsilon1, \epsilon2, \epsilon3]. Respectively the scale factor for initial \mu,
* stopping thresholds for ||J^T e||_inf, ||Dp||_2 and ||e||_2. Set to NULL for defaults to be used
*/
double[] info,
/* O: information regarding the minimization. Set to NULL if don't care
* info[0]= ||e||_2 at initial p.
* info[1-4]=[ ||e||_2, ||J^T e||_inf, ||Dp||_2, mu/max[J^T J]_ii ], all computed at estimated p.
* info[5]= # iterations,
* info[6]=reason for terminating: 1 - stopped by small gradient J^T e
* 2 - stopped by small Dp
* 3 - stopped by itmax
* 4 - singular matrix. Restart from current p with increased mu
* 5 - no further error reduction is possible. Restart with increased mu
* 6 - stopped by small ||e||_2
* info[7]= # function evaluations
* info[8]= # jacobian evaluations
*/
ref object workingmemory, /* working memory, allocate if NULL */
double[] covar, /* O: Covariance matrix corresponding to LS solution; mxm. Set to NULL if not needed. */
object adata) /* pointer to possibly additional data, passed uninterpreted to func & jacf.
* Set to NULL if not needed
*/
{
int m = p.Length; /* I: parameter vector dimension (i.e. #unknowns) */
int n = x.Length; /* I: measurement vector dimension */
int i, j, k, l;
int issolved;
/* temp work arrays */
double[] e, /* nx1 */
hx, /* \hat{x}_i, nx1 */
jacTe, /* J^T e_i mx1 */
jac, /* nxm */
jacTjac, /* mxm */
Dp, /* mx1 */
diag_jacTjac, /* diagonal of J^T J, mx1 */
pDp; /* p + Dp, mx1 */
double mu, /* damping constant */
tmp; /* mainly used in matrix & vector multiplications */
double p_eL2, jacTe_inf, pDp_eL2; /* ||e(p)||_2, ||J^T e||_inf, ||e(p+Dp)||_2 */
double p_L2, Dp_L2=LM_REAL_MAX, dF, dL;
double tau, eps1, eps2, eps2_sq, eps3;
double init_p_eL2;
int nu=2, nu2, stop, nfev, njev=0;
int nm=n*m;
mu=jacTe_inf=0.0; /* -Wall */
if(n<m)
{
throw new ArithmeticException(string.Format("Cannot solve a problem with fewer measurements {0} than unknowns {1}", n, m));
}
if(null==jacf)
{
throw new ArgumentException("No function specified for computing the jacobian. If no such function is available, use LEVMAR_DIF instead");
}
if(null!=opts)
{
tau=opts[0];
eps1=opts[1];
eps2=opts[2];
eps2_sq=opts[2]*opts[2];
eps3=opts[3];
}
else
{ // use default values
tau=(LM_INIT_MU);
eps1=(LM_STOP_THRESH);
eps2=(LM_STOP_THRESH);
eps2_sq=(LM_STOP_THRESH)*(LM_STOP_THRESH);
eps3=(LM_STOP_THRESH);
}
/* set up work arrays */
WorkArrays work = workingmemory as WorkArrays;
if(null==work)
{
work = new WorkArrays(n, m);
workingmemory = work;
}
/* set up work arrays */
e=work.e;
hx = work.hx;
jacTe = work.jacTe;
jac = work.jac;
jacTjac = work.jacTjac;
Dp = work.Dp;
diag_jacTjac = work.diag_jacTjac;
pDp = work.pDp;
/* compute e=x - f(p) and its L2 norm */
func(p, hx, adata); nfev=1;
if (weights == null)
{
for (i = 0, p_eL2 = 0.0; i < n; ++i)
{
e[i] = tmp = x[i] - hx[i];
p_eL2 += tmp * tmp;
}
}
else
{
for (i = 0, p_eL2 = 0.0; i < n; ++i)
{
e[i] = tmp = (x[i] - hx[i])*weights[i];
p_eL2 += tmp * tmp;
}
}
init_p_eL2=p_eL2;
for(k=stop=0; k<itmax && 0==stop; ++k)
{
/* Note that p and e have been updated at a previous iteration */
if(p_eL2<=eps3)
{ /* error is small */
stop=6;
break;
}
/* Compute the jacobian J at p, J^T J, J^T e, ||J^T e||_inf and ||p||^2.
* Since J^T J is symmetric, its computation can be speeded up by computing
* only its upper triangular part and copying it to the lower part
*/
jacf(p, jac, adata); ++njev;
/* J^T J, J^T e */
if(nm<__BLOCKSZ__SQ)
{ // this is a small problem
/* This is the straightforward way to compute J^T J, J^T e. However, due to
* its noncontinuous memory access pattern, it incures many cache misses when
* applied to large minimization problems (i.e. problems involving a large
* number of free variables and measurements), in which J is too large to
* fit in the L1 cache. For such problems, a cache-efficient blocking scheme
* is preferable.
*
* Thanks to John Nitao of Lawrence Livermore Lab for pointing out this
* performance problem.
*
* On the other hand, the straightforward algorithm is faster on small
* problems since in this case it avoids the overheads of blocking.
*/
for(i=0; i<m; ++i)
{
for(j=i; j<m; ++j)
{
int lm;
if (weights == null)
{
for (l = 0, tmp = 0.0; l < n; ++l)
{
lm = l * m;
tmp += jac[lm + i] * jac[lm + j];
}
}
else
{
for (l = 0, tmp = 0.0; l < n; ++l)
{
lm = l * m;
tmp += jac[lm + i] * jac[lm + j] * weights[i] * weights[i];
}
}
/* store tmp in the corresponding upper and lower part elements */
jacTjac[i*m+j]=jacTjac[j*m+i]=tmp;
}
/* J^T e */
for(l=0, tmp=0.0; l<n; ++l)
tmp+=jac[l*m+i]*e[l];
jacTe[i]=tmp;
}
}
else
{ // this is a large problem
/* Cache efficient computation of J^T J based on blocking
*/
TRANS_MAT_MAT_MULT(jac, jacTjac, n, m, __BLOCKSZ__,weights);
/* cache efficient computation of J^T e */
for(i=0; i<m; ++i)
jacTe[i]=0.0;
for(i=0; i<n; ++i)
{
int jacrow;
for(l=0, jacrow=i*m, tmp=e[i]; l<m; ++l)
jacTe[l]+=jac[jacrow+l]*tmp;
}
}
/* Compute ||J^T e||_inf and ||p||^2 */
for(i=0, p_L2=jacTe_inf=0.0; i<m; ++i)
{
if(jacTe_inf < (tmp=Math.Abs(jacTe[i]))) jacTe_inf=tmp;
diag_jacTjac[i]=jacTjac[i*m+i]; /* save diagonal entries so that augmentation can be later canceled */
p_L2+=p[i]*p[i];
}
//p_L2=sqrt(p_L2);
#if false
if(!(k%100)){
printf("Current estimate: ");
for(i=0; i<m; ++i)
printf("%.9g ", p[i]);
printf("-- errors %.9g %0.9g\n", jacTe_inf, p_eL2);
}
#endif
/* check for convergence */
if((jacTe_inf <= eps1))
{
Dp_L2=0.0; /* no increment for p in this case */
stop=1;
break;
}
/* compute initial damping factor */
if(k==0)
{
for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
if(diag_jacTjac[i]>tmp) tmp=diag_jacTjac[i]; /* find max diagonal element */
mu=tau*tmp;
}
/* determine increment using adaptive damping */
while(true)
{
/* augment normal equations */
for(i=0; i<m; ++i)
jacTjac[i*m+i]+=mu;
/* solve augmented equations */
#if HAVE_LAPACK
/* 5 alternatives are available: LU, Cholesky, 2 variants of QR decomposition and SVD.
* Cholesky is the fastest but might be inaccurate; QR is slower but more accurate;
* SVD is the slowest but most accurate; LU offers a tradeoff between accuracy and speed
*/
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_CHOL(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QR(jacTjac, jacTe, Dp, m);
//issolved=AX_EQ_B_QRLS(jacTjac, jacTe, Dp, m, m);
//issolved=AX_EQ_B_SVD(jacTjac, jacTe, Dp, m);
#else
/* use the LU included with levmar */
issolved=AX_EQ_B_LU(jacTjac, jacTe, Dp, m);
#endif // HAVE_LAPACK
if(0!=issolved)
{
/* compute p's new estimate and ||Dp||^2 */
for(i=0, Dp_L2=0.0; i<m; ++i)
{
pDp[i]=p[i] + (tmp=Dp[i]);
Dp_L2+=tmp*tmp;
}
//Dp_L2=sqrt(Dp_L2);
if(Dp_L2<=eps2_sq*p_L2)
{ /* relative change in p is small, stop */
//if(Dp_L2<=eps2*(p_L2 + eps2)){ /* relative change in p is small, stop */
stop=2;
break;
}
if(Dp_L2>=(p_L2+eps2)/((EPSILON)*(EPSILON)))
{ /* almost singular */
//if(Dp_L2>=(p_L2+eps2)/CNST(EPSILON)){ /* almost singular */
stop=4;
break;
}
func(pDp, hx, adata); ++nfev; /* evaluate function at p + Dp */
if (weights == null)
{
for (i = 0, pDp_eL2 = 0.0; i < n; ++i)
{ /* compute ||e(pDp)||_2 */
hx[i] = tmp = x[i] - hx[i];
pDp_eL2 += tmp * tmp;
}
}
else // use weights
{
for (i = 0, pDp_eL2 = 0.0; i < n; ++i)
{ /* compute ||e(pDp)||_2 */
hx[i] = tmp = (x[i] - hx[i])*weights[i];
pDp_eL2 += tmp * tmp;
}
}
for(i=0, dL=0.0; i<m; ++i)
dL+=Dp[i]*(mu*Dp[i]+jacTe[i]);
dF=p_eL2-pDp_eL2;
if(dL>0.0 && dF>0.0)
{ /* reduction in error, increment is accepted */
tmp=((2.0)*dF/dL-(1.0));
tmp=(1.0)-tmp*tmp*tmp;
mu=mu*( (tmp>=(ONE_THIRD))? tmp : (ONE_THIRD) );
nu=2;
for(i=0 ; i<m; ++i) /* update p's estimate */
p[i]=pDp[i];
for(i=0; i<n; ++i) /* update e and ||e||_2 */
e[i]=hx[i];
p_eL2=pDp_eL2;
break;
}
}
/* if this point is reached, either the linear system could not be solved or
* the error did not reduce; in any case, the increment must be rejected
*/
mu*=nu;
nu2=nu<<1; // 2*nu;
if(nu2<=nu)
{ /* nu has wrapped around (overflown). Thanks to Frank Jordan for spotting this case */
stop=5;
break;
}
nu=nu2;
for(i=0; i<m; ++i) /* restore diagonal J^T J entries */
jacTjac[i*m+i]=diag_jacTjac[i];
} /* inner loop */
}
if(k>=itmax) stop=3;
for(i=0; i<m; ++i) /* restore diagonal J^T J entries */
jacTjac[i*m+i]=diag_jacTjac[i];
if(null!=info)
{
info[0]=init_p_eL2;
info[1]=p_eL2;
info[2]=jacTe_inf;
info[3]=Dp_L2;
for(i=0, tmp=LM_REAL_MIN; i<m; ++i)
if(tmp<jacTjac[i*m+i]) tmp=jacTjac[i*m+i];
info[4]=mu/tmp;
info[5]=(double)k;
info[6]=(double)stop;
info[7]=(double)nfev;
info[8]=(double)njev;
}
/* covariance matrix */
if(null!=covar)
{
LEVMAR_COVAR(jacTjac, covar, p_eL2, m, n);
}
return (stop!=4)? k : -1;
}
GetIndependentVariables()
{
for (int i = 0; i < NumberOfIndependentVariables; ++i)
{
int k = i;
string nameOfVariable = null != FitFunction && i < FitFunction.NumberOfIndependentVariables ? FitFunction.IndependentVariableName(i) : string.Empty;
yield return(
nameOfVariable,
_independentVariables[k]?.Document,
void CalculateCurveFit(Common.Vbo handle)
{
int num = handle.selectedVertices.Count;
var pointsMat = new MathNet.Numerics.LinearAlgebra.Double.DenseMatrix(num, 3);
var points = new Vector3[num];
//get selected points
for (int i = 0; i < num; i++)
{
pointsMat[i, 0] = handle.verticesData.vertices[handle.selectedVertices[i]].X;
pointsMat[i, 1] = handle.verticesData.vertices[handle.selectedVertices[i]].Y;
pointsMat[i, 2] = handle.verticesData.vertices[handle.selectedVertices[i]].Z;
points[i] = handle.verticesData.vertices[handle.selectedVertices[i]];
}
//find the plane which holds the least distance to all ponits using SVD
Common.Plane curve_plane = NormalPlane(pointsMat);
curve_plane.valid = true;
Planes[GetSelectedVbOIndex() - 1][CURVEPLANE_INDEX] = curve_plane;
if (Planes[GetSelectedVbOIndex() - 1][OCCLUSALPLANE_INDEX] != null &&
Planes[GetSelectedVbOIndex() - 1][OCCLUSALPLANE_INDEX].valid)
{
double angle = Planes[GetSelectedVbOIndex() - 1][CURVEPLANE_INDEX].Angle2Plane(Planes[0][OCCLUSALPLANE_INDEX]);
lb_curve2occlusalPlane.Text = "Angle to Occlusal Plane: " + angle.ToString("F2");
}
//calculate sum of distances to this plane
double sumOfSquaredDistances = 0;
double[] z_distances = new double[num];
for (int i = 0; i < num; ++i)
{
double d = curve_plane.Distance2Point(points[i]);
z_distances[i] = d;
sumOfSquaredDistances += Math.Pow(d, 2);
}
double rmse_z = Math.Sqrt(sumOfSquaredDistances / num);
// find the x and y direction on the plane
var p0 = new Vector3(curve_plane.ProjectPointOnPlane(new Vector3(0, 0, 0)));
var px = new Vector3(curve_plane.ProjectPointOnPlane(points[num - 1] - points[0])); //new Vector3(1, 0, 0)));
var py = new Vector3(curve_plane.ProjectPointOnPlane(Vector3.Cross(curve_plane.GetNormal(), px))); //new Vector3(0, 1, 0)));
var pvx = (px - p0);
var pvy = (py - p0);
pvx.Normalize();
pvy.Normalize();
//project 3D points on the plane and create a 2D point set
var points2DX = new double[num];
var points2DY = new double[num];
for (int i = 0; i < num; ++i)
{
points2DX[i] = Vector3.Dot(pvx, points[i] - p0);
points2DY[i] = Vector3.Dot(pvy, points[i] - p0);
}
//check and fix inverse y mode
if (points2DY[0] < points2DY[points2DY.Length / 2])
{
points2DY = points2DY.Select(el => - el).ToArray();
}
//move points to fit on the screen
var xMin = points2DX.Min();
var yMin = points2DY.Min();
for (int i = 0; i < num; ++i)
{
points2DX[i] -= xMin;
points2DY[i] -= yMin;
points2DX[i] += 10;
points2DY[i] += 10;
}
//drawing projected points
var graphics = pl_curveFit.CreateGraphics();
graphics.Clear(Color.White);
for (int i = 0; i < num; ++i)
{
DrawPoint2DOnControl(graphics, new PointF((float)points2DX[i], (float)points2DY[i]), Color.Black, 4);
}
//DrawPoint2DOnControl(graphics, new PointF((float)10, (float)20), Color.Red, 10);
//change of parameter
var diffx = Diff(points2DX);
var diffy = Diff(points2DY);
diffx = diffx.Select(el => el * el).ToArray();
diffy = diffy.Select(el => el * el).ToArray();
double[] t = new double[num];
for (int i = 0; i < num; ++i)
{
if (i > 0)
{
t[i] = Math.Sqrt(diffx[i] + diffy[i]) + t[i - 1];
}
else
{
t[i] = Math.Sqrt(diffx[i] + diffy[i]);
}
}
//fit curve params
int deg = (int)nUpDown_order.Value;
FitFunction fitFunction = config.fitFunction;
//fit x and y to t
DenseVector Wx;
DenseVector Wy;
try
{
//f = Fit.Polynomial(xdata, ydata, order);
Wx = FitLeastSquaresBasis(t, points2DX, fitFunction, deg);
Wy = FitLeastSquaresBasis(t, points2DY, fitFunction, deg);
}
catch (Exception e)
{
Common.Logger.Log("MainForm", "curveFit.cs", "CalculateCurveFit",
e.Message);
MessageBox.Show("The order of polynomial is too high and " +
"results are unstable. Please use a lower " +
"degree polynomial.",
"Error in Curve Fitting");
return;
}
// create polynomial curve struct (for later to match to the wire brackets)
CurveFit[GetSelectedVbOIndex() - 1].CoeffsX = Wx;
CurveFit[GetSelectedVbOIndex() - 1].CoeffsY = Wy;
CurveFit[GetSelectedVbOIndex() - 1].degree = deg;
CurveFit[GetSelectedVbOIndex() - 1].minT = t.Min() - 5;
CurveFit[GetSelectedVbOIndex() - 1].maxT = t.Max() + 5;
CurveFit[GetSelectedVbOIndex() - 1].fitFunction = fitFunction;
//calculate xx and yy
//const double stepSize = 0.01;
//double mint = t.Min();
//double maxt = t.Max();
//int numCurve = (int)((maxt - mint) / stepSize);
//double[] tt = new double[numCurve];
//for (int i = 0; i < numCurve; i++)
//{
// tt[i] = mint + stepSize * i;
//}
//var TT = MakeBasis(tt, fitFunction, deg);
//var XX = TT * Wx;
//var YY = TT * Wy;
Tuple <DenseVector, DenseVector> tempTuple = ParametricPolynomial_to_2DPoints(CurveFit[GetSelectedVbOIndex() - 1]);
DenseVector XX = tempTuple.Item1;
DenseVector YY = tempTuple.Item2;
DrawCurve(graphics, XX, YY, Color.Blue);
//least squares
//DenseVector W;
//try
//{
// //f = Fit.Polynomial(xdata, ydata, order);
// W = FitLeastSquaresBasis(xdata, ydata, fitFunction, deg);
//}
//catch (Exception e)
//{
// MessageBox.Show("The order of polynomial is too high and results are unstable. Please use a lower degree polynomial.", "Error in Curve Fitting");
// return;
//}
//draw the blue curve
//DrawCurve(graphics, W, xdata.Min()-4, xdata.Max()+4, fitFunction, deg);
//calculate Error
//var rmse_xy = CalculatePerpendicularDistancePointToCurve(xdata, ydata, W, fitFunction, deg, graphics);
double[] xy_distances;
int[] closestPointsIndexOnCurve;
var rmse_xy = CalculatePerpendicularDistancePointToCurve(points2DX, points2DY, XX, YY, graphics,
out xy_distances, out closestPointsIndexOnCurve);
//fill the list view with point distances
lv_pointDistance.Clear();
lv_pointDistance.Columns.Add("Pt", 20, HorizontalAlignment.Center);
lv_pointDistance.Columns.Add("off", 40, HorizontalAlignment.Center);
lv_pointDistance.Columns.Add("intra", 40, HorizontalAlignment.Center);
for (int i = 0; i < num; ++i)
{
lv_pointDistance.Items.Add(new ListViewItem(new string[]
{
i.ToString(),
z_distances[i].ToString("F2"),
xy_distances[i].ToString("F2")
}));
}
if (lv_pointDistance.Items.Count > 0)
{
lv_pointDistance.Items[0].Selected = true;
}
lb_curvefit_rmse_xy.Text = "in-plane RMS Error: " + rmse_xy.ToString("F2");
lb_curvefit_rmse_z.Text = "off-plane RMS Error: " + rmse_z.ToString("F2");
// write results to disk
string resultsDir = "Results";
if (!System.IO.Directory.Exists(resultsDir))
{
System.IO.Directory.CreateDirectory(resultsDir);
}
string save_path = System.IO.Path.Combine(
resultsDir, "CurveFitResults_" + GetSelectedVbOIndex().ToString() + ".csv");
var writer = new StreamWriter(save_path);
writer.WriteLine("Point Number,X,Y,Z,Projected on WirePlane (X),Projected on WirePlane (Y)," +
"Nearest Point on 2D Wire (projected wire) (X),Nearest Point on 2D Wire(projected wire)(Y)," +
"InPlane Distance,OffPlane Distance");
for (int i = 0; i < num; ++i)
{
writer.Write(i.ToString() + ",");
writer.Write(points[i].X.ToString() + ",");
writer.Write(points[i].Y.ToString() + ",");
writer.Write(points[i].Z.ToString() + ",");
writer.Write(points2DX[i].ToString() + ",");
writer.Write(points2DY[i].ToString() + ",");
writer.Write(XX[closestPointsIndexOnCurve[i]].ToString() + ",");
writer.Write(YY[closestPointsIndexOnCurve[i]].ToString() + ",");
writer.Write(xy_distances[i].ToString() + ",");
writer.Write(z_distances[i].ToString());
writer.WriteLine("");
}
writer.Close();
status.Text = "Results of curve fitting are saved in: " + save_path;
}
