1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
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
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
|
/*
* Kabsch's algorithm: compute least-square-best-fit transformation (2D version)
* Copyright (C) 2003, Arno Formella (formella@ei.uvigo.es)
*
* 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.
*
* This is a gsl-based implementation of Kabsch's algorithm presented in:
*
* W. Kabsch, A solution for the best rotation to relate two sets of vectors,
* Acta Cryst. (1976), A32, 922-923
*
* W. Kabsch, A discussion of the solution for the best rotation to relate
* two sets of vectors, Acta Cryst. (1978), A34, 827-828
*
* The code is C and C++ compilable.
*
* More information about GSL:
*
* The project homepage is http://www.gnu.org/software/gsl/
* The development site is http://sources.redhat.com/gsl/
*/
#include <stdio.h>
#include "kabsch2d.h"
#include <gsl/gsl_eigen.h>
#include <gsl/gsl_blas.h>
#define NORM_EPS 0.00000001
int kabsch2d(
unsigned int size, /* the number of points */
gsl_matrix *X, /* the points to be moved */
gsl_matrix *Y, /* the points to move to */
gsl_matrix *U, /* the rotation matrix */
gsl_vector *t, /* the translation vector */
double *s /* the optimal scaling, if != 0 */
) {
unsigned int i,j,k;
int U_ok=1;
double n=1.0/size;
gsl_vector *cx=gsl_vector_alloc(2); /* centroid of X */
gsl_vector *cy=gsl_vector_alloc(2); /* centroid of Y */
gsl_matrix *R=gsl_matrix_alloc(2,2); /* Kabsch's R */
gsl_matrix *RTR=gsl_matrix_alloc(2,2); /* R_trans * R (and Kabsch's bk) */
gsl_eigen_symmv_workspace *espace=gsl_eigen_symmv_alloc(2);
gsl_matrix *evec=gsl_matrix_alloc(2,2); /* eigenvectors (and Kabsch's ak) */
gsl_vector *eval=gsl_vector_alloc(2); /* vector of eigenvalues */
/* compute centroid of X */
gsl_vector_set_zero(cx);
for(i=size;i>0;) {
gsl_vector_const_view row=gsl_matrix_const_row(X,--i);
gsl_vector_add(cx,&row.vector);
}
gsl_vector_scale(cx,n);
/* move X to origin */
for(i=size;i>0;) {
gsl_vector_view row=gsl_matrix_row(X,--i);
gsl_vector_sub(&row.vector,cx);
}
/* compute centroid of Y */
gsl_vector_set_zero(cy);
for(i=size;i>0;) {
gsl_vector_const_view row=gsl_matrix_const_row(Y,--i);
gsl_vector_add(cy,&row.vector);
}
gsl_vector_scale(cy,n);
/* move Y to origin */
for(i=size;i>0;) {
gsl_vector_view row=gsl_matrix_row(Y,--i);
gsl_vector_sub(&row.vector,cy);
}
if(size==1) {
/* just one point, so U is trival */
gsl_matrix_set_identity(U);
}
else {
/* compute R */
gsl_matrix_set_zero(R);
for(k=size;k>0;) {
--k;
for(i=2;i>0;) {
--i;
for(j=2;j>0;) {
--j;
gsl_matrix_set(R,i,j,
gsl_matrix_get(R,i,j)+
gsl_matrix_get(Y,k,i)*gsl_matrix_get(X,k,j)
);
}
}
}
/* compute RTR = R_trans * R */
gsl_matrix_set_zero(RTR);
gsl_blas_dgemm(CblasTrans,CblasNoTrans,1.0,R,R,0.0,RTR);
/* compute orthonormal eigenvectors */
gsl_eigen_symmv(RTR,eval,evec,espace); /* RTR will be modified! */
gsl_eigen_symmv_sort(eval,evec,GSL_EIGEN_SORT_VAL_DESC);
if(gsl_vector_get(eval,1)>NORM_EPS) {
/* compute ak's (as columns of evec) and bk's (as columns of RTR) */
double norm_b0,norm_b1;
gsl_vector_const_view a0=gsl_matrix_const_column(evec,0);
gsl_vector_const_view a1=gsl_matrix_const_column(evec,1);
gsl_vector_view b0=gsl_matrix_column(RTR,0);
gsl_vector_view b1=gsl_matrix_column(RTR,1);
gsl_blas_dgemv(CblasNoTrans,1.0,R,&a0.vector,0.0,&b0.vector);
norm_b0=gsl_blas_dnrm2(&b0.vector);
gsl_blas_dgemv(CblasNoTrans,1.0,R,&a1.vector,0.0,&b1.vector);
norm_b1=gsl_blas_dnrm2(&b1.vector);
if(norm_b0>NORM_EPS&&norm_b1>NORM_EPS) {
gsl_vector_scale(&b0.vector,1.0/norm_b0); /* b0 = ||R * a0|| */
gsl_vector_scale(&b1.vector,1.0/norm_b1); /* b1 = ||R * a1|| */
/* we reach this point only if all bk different from 0 */
/* compute U = B * A_trans (use RTR as B and evec as A) */
gsl_matrix_set_zero(U); /* to avoid nan */
gsl_blas_dgemm(CblasNoTrans,CblasTrans,1.0,RTR,evec,0.0,U);
}
else {
U_ok=0;
gsl_matrix_set_identity(U);
}
}
else {
U_ok=0;
gsl_matrix_set_identity(U);
}
}
if(s) {
/* let us compute the optimal scaling as well */
/* s = <Y,UX> / <UX,UX> */
*s=1.0;
if(U_ok&&size>1) {
double dom=0.0;
double nom=0.0;
double dom_i,nom_i;
gsl_vector *Uxi=gsl_vector_alloc(2);
for(i=size;i>0;) {
gsl_vector_const_view row_x=gsl_matrix_const_row(X,--i);
gsl_vector_const_view row_y=gsl_matrix_const_row(Y,i);
gsl_vector_set_zero(Uxi);
gsl_blas_dgemv(CblasNoTrans,1.0,U,&row_x.vector,1.0,Uxi);
gsl_blas_ddot(&row_y.vector,Uxi,&nom_i);
nom+=nom_i;
gsl_blas_ddot(Uxi,Uxi,&dom_i);
dom+=dom_i;
}
*s=nom/dom;
gsl_vector_free(Uxi);
}
gsl_vector_scale(cx,*s);
}
/* compute t = cy - s * U * cx */
gsl_vector_memcpy(t,cy);
gsl_blas_dgemv(CblasNoTrans,-1.0,U,cx,1.0,t);
gsl_vector_free(eval);
gsl_matrix_free(evec);
gsl_eigen_symmv_free(espace);
gsl_matrix_free(RTR);
gsl_matrix_free(R);
gsl_vector_free(cy);
gsl_vector_free(cx);
return U_ok;
}
|
