Newer
Older
/* optimise.cc
Mark Jenkinson, FMRIB Image Analysis Group
Copyright (C) 1999-2000 University of Oxford */
/* CCOPYRIGHT */
// Mathematical optimisation functions
namespace MISCMATHS {
// The following lines are ignored by the current SGI compiler
// (version egcs-2.91.57)
// A temporary fix of including the std:: in front of all abs() etc
// has been done for now
using std::abs;
using std::sqrt;
using std::exp;
using std::log;
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
bool estquadmin(float &xnew, float x1, float xmid, float x2,
float y1, float ymid, float y2)
{
// Finds the estimated quadratic minimum's position
float ad=0.0, bd=0.0, det=0.0;
ad = (xmid - x2)*(ymid - y1) - (xmid - x1)*(ymid - y2);
bd = -(xmid*xmid - x2*x2)*(ymid - y1) + (xmid*xmid - x1*x1)*(ymid - y2);
det = (xmid - x2)*(x2 -x1)*(x1 - xmid);
if ((fabs(det)>1e-15) && (ad/det < 0)) { // quadratic only has a maxima
xnew = 0.0;
return false;
}
if (fabs(ad)>1e-15) {
xnew = -bd/(2*ad);
return true;
} else { // near linear condition -> get closer to an end point
xnew = 0.0;
return false;
}
return false;
}
float extrapolatept(float x1, float xmid, float x2)
{
// xmid must be between x1 and x2
// use the golden ratio (scale similar result)
const float extensionratio = 0.3819660;
float xnew;
if (fabs(x2-xmid)>fabs(x1-xmid)) {
xnew = extensionratio * x2 + (1 - extensionratio) * xmid;
} else {
xnew = extensionratio * x1 + (1 - extensionratio) * xmid;
}
return xnew;
}
float nextpt(float x1, float xmid, float x2, float y1, float ymid, float y2)
{
// x1 and x2 are the bounds, xmid is between them
float xnew;
bool quadok=false;
quadok = estquadmin(xnew,x1,xmid,x2,y1,ymid,y2);
// check to see that the quadratic result is in the range
if ((!quadok) || (xnew < Min(x1,x2)) || (xnew > Max(x1,x2))) {
xnew = extrapolatept(x1,xmid,x2);
}
return xnew;
}
void findinitialbound(float &x1, float &xmid, float &x2,
float &y1, float &ymid, float &y2,
float (*func)(const ColumnVector &),
const ColumnVector &unitdir, const ColumnVector &pt)
{
const float extrapolationfactor = 1.6;
const float maxextrap = extrapolationfactor*2;
if (y1==0) y1 = (*func)(x1*unitdir + pt);
if (ymid==0) ymid = (*func)(xmid*unitdir + pt);
if (y1<ymid) { // swap a and b if this is the case
float tempx = x1, tempy = y1;
x1 = xmid; y1 = ymid;
xmid = tempx; ymid = tempy;
}
float newx2 = 0.0, newy2=0.0, maxx2=0.0;
float dir=1.0;
if (xmid<x1) dir=-1.0;
bool quadok;
x2 = xmid + extrapolationfactor*(xmid - x1);
y2 = (*func)(x2*unitdir + pt);
while (ymid > y2) { // note: must maintain y1 >= ymid
// cout << " <" << Min(x1,x2) << "," << xmid
// << "," << Max(x1,x2) << ">" << endl;
maxx2 = xmid + maxextrap*(x2 - xmid);
quadok = estquadmin(newx2,x1,xmid,x2,y1,ymid,y2);
if ((!quadok) || ((newx2 - x1)*dir<0) || ((newx2 - maxx2)*dir>0)) {
newx2 = xmid + extrapolationfactor*(x2-x1);
}
newy2 = (*func)(newx2*unitdir + pt);
if ((newx2 - xmid)*(newx2 - x1)<0) { // newx2 is between x1 and xmid
if (newy2 < ymid) { // found a bracket!
x2 = xmid; y2 = ymid;
xmid = newx2; ymid = newy2;
break;
} else { // can use newx2 as a new value for x1 (as newy2 >= ymid)
x1 = newx2; y1 = newy2;
}
} else { // newx2 is between xmid and maxx2
if (newy2 > ymid) { // found a bracket!
x2 = newx2; y2 = newy2;
break;
} else if ((newx2 - x2)*dir<0) { // newx2 closer to xmid than old x2
x1 = xmid; y1 = ymid;
xmid = newx2; ymid = newy2;
} else {
x1 = xmid; y1 = ymid;
xmid = x2; ymid = y2;
x2 = newx2; y2 = newy2;
}
}
}
if ( (y2<ymid) || (y1<ymid) ) {
cerr << "findinitialbound failed to bracket: current triplet is" << endl;
}
}
float optimise1d(ColumnVector &pt, const ColumnVector dir,
const ColumnVector tol, int &iterations_done,
float (*func)(const ColumnVector &), int max_iter,
{
// Golden Search Routine
// Must pass in the direction vector in N-space (dir), the initial
// N-dim point (pt), the acceptable tolerance (tol) and other
// stuff
// Note that the length of the direction vector is unimportant
float y1,y2,ymid;
float x1,x2,xmid;
// Calculate dot product of dir by tol
// st (x1-x2)*dir_tol = average number of tolerances between x1 and x2
float dir_tol = 0.0;
ColumnVector unitdir;
unitdir = dir/std::sqrt(dir.SumSquare());
for (int n=1; n<=tol.Nrows(); n++) {
if (fabs(tol(n))>1e-15) {
dir_tol += fabs(unitdir(n)/tol(n));
}
}
float unittol = fabs(1/dir_tol);
// set up initial points
xmid = 0.0;
x1 = boundguess * unittol; // initial guess (bound)
if (init_value==0.0) { init_value = (*func)(xmid*unitdir + pt); }
ymid = init_value;
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
y1 = (*func)(x1*unitdir + pt);
findinitialbound(x1,xmid,x2,y1,ymid,y2,func,unitdir,pt);
// cout << "(" << x1 << "," << y1 << ") ";
// cout << "(" << xmid << "," << ymid << ") ";
// cout << "(" << x2 << "," << y2 << ")" << endl;
float min_dist = 0.1 * unittol;
float xnew, ynew;
int it=0;
while ( ((++it)<=max_iter) && (fabs((x2-x1)/unittol)>1.0) )
{
// cout << " [" << Min(x1,x2) << "," << Max(x1,x2) << "]" << endl;
if (it>0) {
xnew = nextpt(x1,xmid,x2,y1,ymid,y2);
} else {
xnew = extrapolatept(x1,xmid,x2);
}
float dirn=1.0;
if (x2<x1) dirn=-1.0;
if (fabs(xnew - x1)<min_dist) {
xnew = x1 + dirn*min_dist;
}
if (fabs(xnew - x2)<min_dist) {
xnew = x2 - dirn*min_dist;
}
if (fabs(xnew - xmid)<min_dist) {
xnew = extrapolatept(x1,xmid,x2);
}
if (fabs(xmid - x1)<0.4*unittol) {
xnew = xmid + dirn*0.5*unittol;
}
if (fabs(xmid - x2)<0.4*unittol) {
xnew = xmid - dirn*0.5*unittol;
}
ynew = (*func)(xnew*unitdir + pt);
if ((xnew - xmid)*(x2 - xmid) > 0) { // is xnew between x2 and xmid ?
// swap x1 and x2 so that xnew is between x1 and xmid
float xtemp = x1; x1 = x2; x2 = xtemp;
float ytemp = y1; y1 = y2; y2 = ytemp;
}
if (ynew < ymid) {
// new interval is [xmid,x1] with xnew as best point in the middle
x2 = xmid; y2 = ymid;
xmid = xnew; ymid = ynew;
} else {
// new interval is [x2,xnew] with xmid as best point still
x1 = xnew; y1 = ynew;
}
}
iterations_done = it;
pt = xmid*unitdir + pt;
return ymid;
}
float optimise(ColumnVector &pt, int numopt, const ColumnVector &tol,
float (*func)(const ColumnVector &), int &iterations_done,
int max_iter, const ColumnVector& boundguess,
const string type)
// Note that numopt can be less than pt.Nrows() - e.g. 6 dof optimisation
// but with a 12 dimensional vector
// Calculate dot product of dir by tol
// st (x1-x2)*dir_tol = average number of tolerances between x1 and x2
ColumnVector inv_tol(tol.Nrows());
inv_tol = 0.0;
for (int n=1; n<=tol.Nrows(); n++) {
if (fabs(tol(n))>1e-15) {
inv_tol(n) = fabs(1.0/tol(n));
}
}
inv_tol /= (float) tol.Nrows();
Matrix dirs(pt.Nrows(),pt.Nrows());
dirs = IdentityMatrix(pt.Nrows());
ColumnVector dir(pt.Nrows()), initpt, deltaf(pt.Nrows());
deltaf=0.0f;
float fval=0.0, fval2=0.0, bndguess, finit=0.0, fend=0.0, fextrap=0.0;
while ((++it)<=max_iter)
{
initpt = pt;
bndguess = boundguess(Min(it,boundguess.Nrows())); // ceiling of nrows
for (int n=1; n<=numopt; n++) {
for (int m=1; m<=pt.Nrows(); m++) { dir(m) = dirs(m,n); }
fval2 = optimise1d(pt,dir,tol,lit,func,100,fval,bndguess);
deltaf(n)=fval2-fval;
if (n==1) { finit = fval; }
fval=fval2;
// check to see if the point has moved more than one average tolerance
float avtol = SP((initpt - pt),inv_tol).SumAbsoluteValue();
if (avtol < 1.0) break;
// if continuing then change the directions if using Powell's method
if (type=="powell")
{
// find direction of maximal change
int bestm=1;
for (int m=1; m<=numopt; m++) {
if (deltaf(m)<deltaf(bestm)) bestm=m;
}
fend=fval;
fextrap=(*func)(initpt + 2*(pt-initpt));
float df=fabs(deltaf(bestm));
if ( (2 * (finit-2*fend+fextrap) * (finit-fend-df)*(finit-fend-df)) < ( (finit-fextrap)*(finit-fextrap)*df ) ) {
if (fextrap<finit) {
cout << "Applying POWELL correction" << endl;
cout << "finit, fend, fextrap = " << finit << " , " << fend << " , " << fextrap << endl;
// do another minimisation
fval2 = optimise1d(pt,pt-initpt,tol,lit,func,100,fval,bndguess);
fval=fval2;
littot += lit;
// replace direction of maximum change with pt-initpt
for (int m=1; m<=pt.Nrows(); m++) {
dirs(m,bestm)=pt(m)-initpt(m);
}
}
}
}