1/*
2 * Copyright (C) 2011 The Android Open Source Project
3 *
4 * Licensed under the Apache License, Version 2.0 (the "License");
5 * you may not use this file except in compliance with the License.
6 * You may obtain a copy of the License at
7 *
8 *      http://www.apache.org/licenses/LICENSE-2.0
9 *
10 * Unless required by applicable law or agreed to in writing, software
11 * distributed under the License is distributed on an "AS IS" BASIS,
12 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13 * See the License for the specific language governing permissions and
14 * limitations under the License.
15 */
16
17/* $Id: db_metrics.h,v 1.3 2011/06/17 14:03:31 mbansal Exp $ */
18
19#ifndef DB_METRICS
20#define DB_METRICS
21
22
23
24/*****************************************************************
25*    Lean and mean begins here                                   *
26*****************************************************************/
27
28#include "db_utilities.h"
29/*!
30 * \defgroup LMMetrics (LM) Metrics
31 */
32/*\{*/
33
34
35
36
37/*!
38Compute function value fp and Jacobian J of robustifier given input value f*/
39inline void db_CauchyDerivative(double J[4],double fp[2],const double f[2],double one_over_scale2)
40{
41    double x2,y2,r,r2,r2s,one_over_r2,fu,r_fu,one_over_r_fu;
42    double one_plus_r2s,half_dfu_dx,half_dfu_dy,coeff,coeff2,coeff3;
43    int at_zero;
44
45    /*The robustifier takes the input (x,y) and makes a new
46    vector (xp,yp) where
47    xp=sqrt(log(1+(x^2+y^2)*one_over_scale2))*x/sqrt(x^2+y^2)
48    yp=sqrt(log(1+(x^2+y^2)*one_over_scale2))*y/sqrt(x^2+y^2)
49    The new vector has the property
50    xp^2+yp^2=log(1+(x^2+y^2)*one_over_scale2)
51    i.e. when it is square-summed it gives the robust
52    reprojection error
53    Define
54    r2=(x^2+y^2) and
55    r2s=r2*one_over_scale2
56    fu=log(1+r2s)/r2
57    then
58    xp=sqrt(fu)*x
59    yp=sqrt(fu)*y
60    and
61    d(r2)/dx=2x
62    d(r2)/dy=2y
63    and
64    dfu/dx=d(r2)/dx*(r2s/(1+r2s)-log(1+r2s))/(r2*r2)
65    dfu/dy=d(r2)/dy*(r2s/(1+r2s)-log(1+r2s))/(r2*r2)
66    and
67    d(xp)/dx=1/(2sqrt(fu))*(dfu/dx)*x+sqrt(fu)
68    d(xp)/dy=1/(2sqrt(fu))*(dfu/dy)*x
69    d(yp)/dx=1/(2sqrt(fu))*(dfu/dx)*y
70    d(yp)/dy=1/(2sqrt(fu))*(dfu/dy)*y+sqrt(fu)
71    */
72
73    x2=db_sqr(f[0]);
74    y2=db_sqr(f[1]);
75    r2=x2+y2;
76    r=sqrt(r2);
77
78    if(r2<=0.0) at_zero=1;
79    else
80    {
81        one_over_r2=1.0/r2;
82        r2s=r2*one_over_scale2;
83        one_plus_r2s=1.0+r2s;
84        fu=log(one_plus_r2s)*one_over_r2;
85        r_fu=sqrt(fu);
86        if(r_fu<=0.0) at_zero=1;
87        else
88        {
89            one_over_r_fu=1.0/r_fu;
90            fp[0]=r_fu*f[0];
91            fp[1]=r_fu*f[1];
92            /*r2s is always >= 0*/
93            coeff=(r2s/one_plus_r2s*one_over_r2-fu)*one_over_r2;
94            half_dfu_dx=f[0]*coeff;
95            half_dfu_dy=f[1]*coeff;
96            coeff2=one_over_r_fu*half_dfu_dx;
97            coeff3=one_over_r_fu*half_dfu_dy;
98
99            J[0]=coeff2*f[0]+r_fu;
100            J[1]=coeff3*f[0];
101            J[2]=coeff2*f[1];
102            J[3]=coeff3*f[1]+r_fu;
103            at_zero=0;
104        }
105    }
106    if(at_zero)
107    {
108        /*Close to zero the robustifying mapping
109        becomes identity*sqrt(one_over_scale2)*/
110        fp[0]=0.0;
111        fp[1]=0.0;
112        J[0]=sqrt(one_over_scale2);
113        J[1]=0.0;
114        J[2]=0.0;
115        J[3]=J[0];
116    }
117}
118
119inline double db_SquaredReprojectionErrorHomography(const double y[2],const double H[9],const double x[3])
120{
121    double x0,x1,x2,mult;
122    double sd;
123
124    x0=H[0]*x[0]+H[1]*x[1]+H[2]*x[2];
125    x1=H[3]*x[0]+H[4]*x[1]+H[5]*x[2];
126    x2=H[6]*x[0]+H[7]*x[1]+H[8]*x[2];
127    mult=1.0/((x2!=0.0)?x2:1.0);
128    sd=db_sqr((y[0]-x0*mult))+db_sqr((y[1]-x1*mult));
129
130    return(sd);
131}
132
133inline double db_SquaredInhomogenousHomographyError(const double y[2],const double H[9],const double x[2])
134{
135    double x0,x1,x2,mult;
136    double sd;
137
138    x0=H[0]*x[0]+H[1]*x[1]+H[2];
139    x1=H[3]*x[0]+H[4]*x[1]+H[5];
140    x2=H[6]*x[0]+H[7]*x[1]+H[8];
141    mult=1.0/((x2!=0.0)?x2:1.0);
142    sd=db_sqr((y[0]-x0*mult))+db_sqr((y[1]-x1*mult));
143
144    return(sd);
145}
146
147/*!
148Return a constant divided by likelihood of a Cauchy distributed
149reprojection error given the image point y, homography H, image point
150point x and the squared scale coefficient one_over_scale2=1.0/(scale*scale)
151where scale is the half width at half maximum (hWahM) of the
152Cauchy distribution*/
153inline double db_ExpCauchyInhomogenousHomographyError(const double y[2],const double H[9],const double x[2],
154                                                      double one_over_scale2)
155{
156    double sd;
157    sd=db_SquaredInhomogenousHomographyError(y,H,x);
158    return(1.0+sd*one_over_scale2);
159}
160
161/*!
162Compute residual vector f between image point y and homography Hx of
163image point x. Also compute Jacobian of f with respect
164to an update dx of H*/
165inline void db_DerivativeInhomHomographyError(double Jf_dx[18],double f[2],const double y[2],const double H[9],
166                                              const double x[2])
167{
168    double xh,yh,zh,mult,mult2,xh_mult2,yh_mult2;
169    /*The Jacobian of the inhomogenous coordinates with respect to
170    the homogenous is
171    [1/zh  0  -xh/(zh*zh)]
172    [ 0  1/zh -yh/(zh*zh)]
173    The Jacobian of the homogenous coordinates with respect to dH is
174    [x0 x1 1  0  0 0  0  0 0]
175    [ 0  0 0 x0 x1 1  0  0 0]
176    [ 0  0 0  0  0 0 x0 x1 1]
177    The output Jacobian is minus their product, i.e.
178    [-x0/zh -x1/zh -1/zh    0      0     0    x0*xh/(zh*zh) x1*xh/(zh*zh) xh/(zh*zh)]
179    [   0      0     0   -x0/zh -x1/zh -1/zh  x0*yh/(zh*zh) x1*yh/(zh*zh) yh/(zh*zh)]*/
180
181    /*Compute warped point, which is the same as
182    homogenous coordinates of reprojection*/
183    xh=H[0]*x[0]+H[1]*x[1]+H[2];
184    yh=H[3]*x[0]+H[4]*x[1]+H[5];
185    zh=H[6]*x[0]+H[7]*x[1]+H[8];
186    mult=1.0/((zh!=0.0)?zh:1.0);
187    /*Compute inhomogenous residual*/
188    f[0]=y[0]-xh*mult;
189    f[1]=y[1]-yh*mult;
190    /*Compute Jacobian*/
191    mult2=mult*mult;
192    xh_mult2=xh*mult2;
193    yh_mult2=yh*mult2;
194    Jf_dx[0]= -x[0]*mult;
195    Jf_dx[1]= -x[1]*mult;
196    Jf_dx[2]= -mult;
197    Jf_dx[3]=0;
198    Jf_dx[4]=0;
199    Jf_dx[5]=0;
200    Jf_dx[6]=x[0]*xh_mult2;
201    Jf_dx[7]=x[1]*xh_mult2;
202    Jf_dx[8]=xh_mult2;
203    Jf_dx[9]=0;
204    Jf_dx[10]=0;
205    Jf_dx[11]=0;
206    Jf_dx[12]=Jf_dx[0];
207    Jf_dx[13]=Jf_dx[1];
208    Jf_dx[14]=Jf_dx[2];
209    Jf_dx[15]=x[0]*yh_mult2;
210    Jf_dx[16]=x[1]*yh_mult2;
211    Jf_dx[17]=yh_mult2;
212}
213
214/*!
215Compute robust residual vector f between image point y and homography Hx of
216image point x. Also compute Jacobian of f with respect
217to an update dH of H*/
218inline void db_DerivativeCauchyInhomHomographyReprojection(double Jf_dx[18],double f[2],const double y[2],const double H[9],
219                                                           const double x[2],double one_over_scale2)
220{
221    double Jf_dx_loc[18],f_loc[2];
222    double J[4],J0,J1,J2,J3;
223
224    /*Compute reprojection Jacobian*/
225    db_DerivativeInhomHomographyError(Jf_dx_loc,f_loc,y,H,x);
226    /*Compute robustifier Jacobian*/
227    db_CauchyDerivative(J,f,f_loc,one_over_scale2);
228
229    /*Multiply the robustifier Jacobian with
230    the reprojection Jacobian*/
231    J0=J[0];J1=J[1];J2=J[2];J3=J[3];
232    Jf_dx[0]=J0*Jf_dx_loc[0];
233    Jf_dx[1]=J0*Jf_dx_loc[1];
234    Jf_dx[2]=J0*Jf_dx_loc[2];
235    Jf_dx[3]=                J1*Jf_dx_loc[12];
236    Jf_dx[4]=                J1*Jf_dx_loc[13];
237    Jf_dx[5]=                J1*Jf_dx_loc[14];
238    Jf_dx[6]=J0*Jf_dx_loc[6]+J1*Jf_dx_loc[15];
239    Jf_dx[7]=J0*Jf_dx_loc[7]+J1*Jf_dx_loc[16];
240    Jf_dx[8]=J0*Jf_dx_loc[8]+J1*Jf_dx_loc[17];
241    Jf_dx[9]= J2*Jf_dx_loc[0];
242    Jf_dx[10]=J2*Jf_dx_loc[1];
243    Jf_dx[11]=J2*Jf_dx_loc[2];
244    Jf_dx[12]=                J3*Jf_dx_loc[12];
245    Jf_dx[13]=                J3*Jf_dx_loc[13];
246    Jf_dx[14]=                J3*Jf_dx_loc[14];
247    Jf_dx[15]=J2*Jf_dx_loc[6]+J3*Jf_dx_loc[15];
248    Jf_dx[16]=J2*Jf_dx_loc[7]+J3*Jf_dx_loc[16];
249    Jf_dx[17]=J2*Jf_dx_loc[8]+J3*Jf_dx_loc[17];
250}
251/*!
252Compute residual vector f between image point y and rotation of
253image point x by R. Also compute Jacobian of f with respect
254to an update dx of R*/
255inline void db_DerivativeInhomRotationReprojection(double Jf_dx[6],double f[2],const double y[2],const double R[9],
256                                                   const double x[2])
257{
258    double xh,yh,zh,mult,mult2,xh_mult2,yh_mult2;
259    /*The Jacobian of the inhomogenous coordinates with respect to
260    the homogenous is
261    [1/zh  0  -xh/(zh*zh)]
262    [ 0  1/zh -yh/(zh*zh)]
263    The Jacobian at zero of the homogenous coordinates with respect to
264    [sin(phi) sin(ohm) sin(kap)] is
265    [-rx2   0   rx1 ]
266    [  0   rx2 -rx0 ]
267    [ rx0 -rx1   0  ]
268    The output Jacobian is minus their product, i.e.
269    [1+xh*xh/(zh*zh) -xh*yh/(zh*zh)   -yh/zh]
270    [xh*yh/(zh*zh)   -1-yh*yh/(zh*zh)  xh/zh]*/
271
272    /*Compute rotated point, which is the same as
273    homogenous coordinates of reprojection*/
274    xh=R[0]*x[0]+R[1]*x[1]+R[2];
275    yh=R[3]*x[0]+R[4]*x[1]+R[5];
276    zh=R[6]*x[0]+R[7]*x[1]+R[8];
277    mult=1.0/((zh!=0.0)?zh:1.0);
278    /*Compute inhomogenous residual*/
279    f[0]=y[0]-xh*mult;
280    f[1]=y[1]-yh*mult;
281    /*Compute Jacobian*/
282    mult2=mult*mult;
283    xh_mult2=xh*mult2;
284    yh_mult2=yh*mult2;
285    Jf_dx[0]= 1.0+xh*xh_mult2;
286    Jf_dx[1]= -yh*xh_mult2;
287    Jf_dx[2]= -yh*mult;
288    Jf_dx[3]= -Jf_dx[1];
289    Jf_dx[4]= -1-yh*yh_mult2;
290    Jf_dx[5]= xh*mult;
291}
292
293/*!
294Compute robust residual vector f between image point y and rotation of
295image point x by R. Also compute Jacobian of f with respect
296to an update dx of R*/
297inline void db_DerivativeCauchyInhomRotationReprojection(double Jf_dx[6],double f[2],const double y[2],const double R[9],
298                                                         const double x[2],double one_over_scale2)
299{
300    double Jf_dx_loc[6],f_loc[2];
301    double J[4],J0,J1,J2,J3;
302
303    /*Compute reprojection Jacobian*/
304    db_DerivativeInhomRotationReprojection(Jf_dx_loc,f_loc,y,R,x);
305    /*Compute robustifier Jacobian*/
306    db_CauchyDerivative(J,f,f_loc,one_over_scale2);
307
308    /*Multiply the robustifier Jacobian with
309    the reprojection Jacobian*/
310    J0=J[0];J1=J[1];J2=J[2];J3=J[3];
311    Jf_dx[0]=J0*Jf_dx_loc[0]+J1*Jf_dx_loc[3];
312    Jf_dx[1]=J0*Jf_dx_loc[1]+J1*Jf_dx_loc[4];
313    Jf_dx[2]=J0*Jf_dx_loc[2]+J1*Jf_dx_loc[5];
314    Jf_dx[3]=J2*Jf_dx_loc[0]+J3*Jf_dx_loc[3];
315    Jf_dx[4]=J2*Jf_dx_loc[1]+J3*Jf_dx_loc[4];
316    Jf_dx[5]=J2*Jf_dx_loc[2]+J3*Jf_dx_loc[5];
317}
318
319
320
321/*!
322// remove the outliers whose projection error is larger than pre-defined
323*/
324inline int db_RemoveOutliers_Homography(const double H[9], double *x_i,double *xp_i, double *wp,double *im, double *im_p, double *im_r, double *im_raw,double *im_raw_p,int point_count,double scale, double thresh=DB_OUTLIER_THRESHOLD)
325{
326    double temp_valueE, t2;
327    int c;
328    int k1=0;
329    int k2=0;
330    int k3=0;
331    int numinliers=0;
332    int ind1;
333    int ind2;
334    int ind3;
335    int isinlier;
336
337    // experimentally determined
338    t2=1.0/(thresh*thresh*thresh*thresh);
339
340    // count the inliers
341    for(c=0;c<point_count;c++)
342    {
343        ind1=c<<1;
344        ind2=c<<2;
345        ind3=3*c;
346
347        temp_valueE=db_SquaredInhomogenousHomographyError(im_p+ind3,H,im+ind3);
348
349        isinlier=((temp_valueE<=t2)?1:0);
350
351        // if it is inlier, then copy the 3d and 2d correspondences
352        if (isinlier)
353        {
354            numinliers++;
355
356            x_i[k1]=x_i[ind1];
357            x_i[k1+1]=x_i[ind1+1];
358
359            xp_i[k1]=xp_i[ind1];
360            xp_i[k1+1]=xp_i[ind1+1];
361
362            k1=k1+2;
363
364            // original normalized pixel coordinates
365            im[k3]=im[ind3];
366            im[k3+1]=im[ind3+1];
367            im[k3+2]=im[ind3+2];
368
369            im_r[k3]=im_r[ind3];
370            im_r[k3+1]=im_r[ind3+1];
371            im_r[k3+2]=im_r[ind3+2];
372
373            im_p[k3]=im_p[ind3];
374            im_p[k3+1]=im_p[ind3+1];
375            im_p[k3+2]=im_p[ind3+2];
376
377            // left and right raw pixel coordinates
378            im_raw[k3] = im_raw[ind3];
379            im_raw[k3+1] = im_raw[ind3+1];
380            im_raw[k3+2] = im_raw[ind3+2]; // the index
381
382            im_raw_p[k3] = im_raw_p[ind3];
383            im_raw_p[k3+1] = im_raw_p[ind3+1];
384            im_raw_p[k3+2] = im_raw_p[ind3+2]; // the index
385
386            k3=k3+3;
387
388            // 3D coordinates
389            wp[k2]=wp[ind2];
390            wp[k2+1]=wp[ind2+1];
391            wp[k2+2]=wp[ind2+2];
392            wp[k2+3]=wp[ind2+3];
393
394            k2=k2+4;
395
396        }
397    }
398
399    return numinliers;
400}
401
402
403
404
405
406/*\}*/
407
408#endif /* DB_METRICS */
409