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tv_filter.hxx |

1 /************************************************************************/
2 /* */
3 /* Copyright 2012 by Frank Lenzen & */
4 /* Ullrich Koethe */
5 /* */
6 /* This file is part of the VIGRA computer vision library. */
7 /* The VIGRA Website is */
8 /* http://hci.iwr.uni-heidelberg.de/vigra/ */
9 /* Please direct questions, bug reports, and contributions to */
10 /* ullrich.koethe@iwr.uni-heidelberg.de or */
11 /* vigra@informatik.uni-hamburg.de */
12 /* */
13 /* Permission is hereby granted, free of charge, to any person */
14 /* obtaining a copy of this software and associated documentation */
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20 /* conditions: */
21 /* */
22 /* The above copyright notice and this permission notice shall be */
23 /* included in all copies or substantial portions of the */
24 /* Software. */
25 /* */
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34 /* */
35 /************************************************************************/
36
37 #ifndef VIGRA_TV_FILTER_HXX
38 #define VIGRA_TV_FILTER_HXX
39
40 #include <iostream>
41 #include <cmath>
42 #include "config.hxx"
43 #include "impex.hxx"
44 #include "separableconvolution.hxx"
45 #include "multi_array.hxx"
46 #include "multi_math.hxx"
47 #include "eigensystem.hxx"
48 #include "convolution.hxx"
49 #include "fixedpoint.hxx"
50 #include "project2ellipse.hxx"
51
52 #ifndef VIGRA_MIXED_2ND_DERIVATIVES
53 #define VIGRA_MIXED_2ND_DERIVATIVES 1
54 #endif
55
56 #define setZeroX(A) A.subarray(Shape2(width-1,0),Shape2(width,height))*=0;
57 #define setZeroY(A) A.subarray(Shape2(0,height-1),Shape2(width,height))*=0;
58
59 namespace vigra{
60
61
62
63 /** \addtogroup NonLinearDiffusion
64 */
65
66 //@{
67
68 /********************************************************/
69 /* */
70 /* totalVariationFilter */
71 /* */
72 /********************************************************/
73
74 /** \brief Performs standard Total Variation Regularization
75
76 The algorithm minimizes
77
78 \f[
79 \min_u \int_\Omega \frac{1}{2} (u-f)^2\;dx + \alpha TV(u)\qquad\qquad (1)
80 \f]
81 where <em>\f$ f=f(x)\f$</em> are the two dimensional noisy data,
82 <em> \f$ u=u(x)\f$</em> are the smoothed data,<em>\f$ \alpha \ge 0 \f$</em>
83 is the filter parameter and <em>\f$ TV(u)\f$ </em> is the total variation semi-norm.
84
85 <b> Declarations:</b>
86
87 \code
88 namespace vigra {
89 template <class stride1,class stride2>
90 void totalVariationFilter(MultiArrayView<2,double,stride1> data,
91 MultiArrayView<2,double,stride2> out,
92 double alpha,
93 int steps,
94 double eps=0);
95 void totalVariationFilter(MultiArrayView<2,double,stride1> data,
96 MultiArrayView<2,double,stride2> weight,
97 MultiArrayView<2,double,stride3> out,
98 double alpha,
99 int steps,
100 double eps=0);
101 }
102 \endcode
103
104 \ref totalVariationFilter() implements a primal-dual algorithm to solve (1).
105
106 Input:
107 <table>
108 <tr><td><em>data</em>: </td><td> input data to be smoothed. </td></tr>
109 <tr><td><em>alpha</em>: </td><td> smoothing parameter.</td></tr>
110 <tr><td><em>steps</em>: </td><td> maximal number of iteration steps. </td></tr>
111 <tr><td><em>eps</em>: </td><td> The algorithm stops, if the primal-dual gap is below the threshold <em>eps</em>.
112 </table>
113
114 Output:
115
116 <em>out</em> contains the filtered data.
117
118 In addition, a point-wise weight (\f$ \ge 0 \f$) for the data term can be provided (overloaded function).
119
120 <b> Usage:</b>
121
122 <b>\#include</b> <vigra/tv_filter.hxx>
123
124 \code
125 MultiArray<2,double> data(Shape2(width,height)); // to be initialized
126 MultiArray<2,double> out(Shape2(width,height));
127 MultiArray<2,double> weight(Shape2(width,height))); //optional argument in overloaded function, to be initialized if used
128 int steps; // to be initialized
129 double alpha,eps; // to be initialized
130
131 totalVariationFilter(data,out,alpha,steps,eps);
132 \endcode
133 or
134 \code
135 totalVariationFilter(data,weight,out,alpha,steps,eps);
136 \endcode
137
138 */
139 doxygen_overloaded_function(template <...> void totalVariationFilter)
140
141 template <class stride1,class stride2>
142 void totalVariationFilter(MultiArrayView<2,double,stride1> data,MultiArrayView<2,double,stride2> out, double alpha, int steps, double eps=0){
143
144 using namespace multi_math;
145 int width=data.shape(0),height=data.shape(1);
146
147 MultiArray<2,double> temp1(data.shape()),temp2(data.shape()),vx(data.shape()),vy(data.shape()),u_bar(data.shape());
148 Kernel1D<double> Lx,LTx;
149 Lx.initExplicitly(-1,0)=1,-1; // = Right sided finite differences for d/dx and d/dy
150 Lx.setBorderTreatment(BORDER_TREATMENT_REFLECT); // with hom. Neumann boundary conditions
151 LTx.initExplicitly(0,1)=-1,1; // = Left sided finite differences for -d/dx and -d/dy
152 LTx.setBorderTreatment(BORDER_TREATMENT_ZEROPAD); // with hom. Dirichlet b. c.
153
154 out=data;
155 u_bar=data;
156
157 double tau=1.0 / std::max(alpha,1.) / std::sqrt(8.0) * 0.06;
158 double sigma=1.0 / std::sqrt(8.0) / 0.06;
159
160 for (int i=0;i<steps;i++){
161
162 separableConvolveX(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
163 vx+=(sigma*temp1);
164 separableConvolveY(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
165 vy+=(sigma*temp1);
166
167 //project to constraint set
168 for (int y=0;y<data.shape(1);y++){
169 for (int x=0;x<data.shape(0);x++){
170 double l=hypot(vx(x,y),vy(x,y));
171 if (l>1){
172 vx(x,y)/=l;
173 vy(x,y)/=l;
174 }
175 }
176 }
177
178 separableConvolveX(srcImageRange(vx),destImage(temp1),kernel1d(LTx));
179 separableConvolveY(srcImageRange(vy),destImage(temp2),kernel1d(LTx));
180 u_bar=out;
181 out-=tau*(out-data+alpha*(temp1+temp2));
182 u_bar=2*out-u_bar; //cf. Chambolle/Pock and Popov's algorithm
183
184
185 //stopping criterion
186 if (eps>0){
187 separableConvolveX(srcImageRange(out),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
188 separableConvolveY(srcImageRange(out),destImage(temp2),kernel1d(Lx));setZeroY(temp2);
189
190 double f_primal=0,f_dual=0;
191 for (int y=0;y<data.shape(1);y++){
192 for (int x=0;x<data.shape(0);x++){
193 f_primal+=.5*(out(x,y)-data(x,y))*(out(x,y)-data(x,y))+alpha*hypot(temp1(x,y),temp2(x,y));
194 }
195 }
196 separableConvolveX(srcImageRange(vx),destImage(temp1),kernel1d(LTx));
197 separableConvolveY(srcImageRange(vy),destImage(temp2),kernel1d(LTx));
198 for (int y=0;y<data.shape(1);y++){
199 for (int x=0;x<data.shape(0);x++){
200 double divv=temp1(x,y)+temp2(x,y);
201 f_dual+=-.5*alpha*alpha*(divv*divv)+alpha*data(x,y)*divv;
202 }
203 }
204 if (f_primal>0 && (f_primal-f_dual)/f_primal<eps){
205 break;
206 }
207 }
208 }
209 }
210
211 template <class stride1,class stride2, class stride3>
212 void totalVariationFilter(MultiArrayView<2,double,stride1> data,MultiArrayView<2,double,stride2> weight, MultiArrayView<2,double,stride3> out,double alpha, int steps, double eps=0){
213
214 using namespace multi_math;
215 int width=data.shape(0),height=data.shape(1);
216
217 MultiArray<2,double> temp1(data.shape()),temp2(data.shape()),vx(data.shape()),vy(data.shape()),u_bar(data.shape());
218 Kernel1D<double> Lx,LTx;
219 Lx.initExplicitly(-1,0)=1,-1; // = Right sided finite differences for d/dx and d/dy
220 Lx.setBorderTreatment(BORDER_TREATMENT_REFLECT); // with hom. Neumann boundary conditions
221 LTx.initExplicitly(0,1)=-1,1; // = Left sided finite differences for -d/dx and -d/dy
222 LTx.setBorderTreatment(BORDER_TREATMENT_ZEROPAD); // with hom. Dirichlet b. c.
223
224 out=data;
225 u_bar=data;
226
227 double tau=1.0 / std::max(alpha,1.) / std::sqrt(8.0) * 0.06;
228 double sigma=1.0 / std::sqrt(8.0) / 0.06;
229
230 for (int i=0;i<steps;i++){
231 separableConvolveX(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
232 vx+=(sigma*temp1);
233 separableConvolveY(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
234 vy+=(sigma*temp1);
235
236 //project to constraint set
237 for (int y=0;y<data.shape(1);y++){
238 for (int x=0;x<data.shape(0);x++){
239 double l=hypot(vx(x,y),vy(x,y));
240 if (l>1){
241 vx(x,y)/=l;
242 vy(x,y)/=l;
243 }
244 }
245 }
246
247 separableConvolveX(srcImageRange(vx),destImage(temp1),kernel1d(LTx));
248 separableConvolveY(srcImageRange(vy),destImage(temp2),kernel1d(LTx));
249 u_bar=out;
250 out-=tau*(weight*(out-data)+alpha*(temp1+temp2));
251 u_bar=2*out-u_bar;
252
253
254 //stopping criterion
255 if (eps>0){
256 separableConvolveX(srcImageRange(out),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
257 separableConvolveY(srcImageRange(out),destImage(temp2),kernel1d(Lx));setZeroY(temp2);
258
259 double f_primal=0,f_dual=0;
260 for (int y=0;y<data.shape(1);y++){
261 for (int x=0;x<data.shape(0);x++){
262 f_primal+=.5*weight(x,y)*(out(x,y)-data(x,y))*(out(x,y)-data(x,y))+alpha*hypot(temp1(x,y),temp2(x,y));
263 }
264 }
265 separableConvolveX(srcImageRange(vx),destImage(temp1),kernel1d(LTx));
266 separableConvolveY(srcImageRange(vy),destImage(temp2),kernel1d(LTx));
267 for (int y=0;y<data.shape(1);y++){
268 for (int x=0;x<data.shape(0);x++){
269 double divv=temp1(x,y)+temp2(x,y);
270 f_dual+=-.5*alpha*alpha*(weight(x,y)*divv*divv)+alpha*data(x,y)*divv;
271 }
272 }
273 if (f_primal>0 && (f_primal-f_dual)/f_primal<eps){
274 break;
275 }
276 }
277 }
278 }
279 //<!--\f$ \alpha(x)=\beta(x)=\beta_{par}\f$ in homogeneous regions without edges,
280 //and \f$ \alpha(x)=\alpha_{par}\f$ at edges.-->
281
282
283 /********************************************************/
284 /* */
285 /* getAnisotropy */
286 /* */
287 /********************************************************/
288
289 /** \brief Sets up directional data for anisotropic regularization
290
291 This routine provides a two-dimensional normalized vector field \f$ v \f$, which is normal to edges in the given data,
292 found as the eigenvector of the structure tensor belonging to the largest eigenvalue.
293 \f$ v \f$ is encoded by a scalar field \f$ \varphi \f$ of angles, i.e.
294 \f$ v(x)=(\cos(\varphi(x)),\sin(\varphi(x)))^\top \f$.
295
296 In addition, two scalar fields \f$ \alpha \f$ and \f$ \beta \f$ are generated from
297 scalar parameters \f$ \alpha_{par}\f$ and \f$ \beta_{par}\f$, such that
298 <center>
299 <table>
300 <tr> <td>\f$ \alpha(x)= \alpha_{par}\f$ at edges,</td></tr>
301 <tr> <td>\f$ \alpha(x)= \beta_{par}\f$ in homogeneous regions,</td></tr>
302 <tr> <td>\f$ \beta(x)=\beta_{par}\f$ .</td></tr>
303 </table>
304 </center>
305
306 <b> Declarations:</b>
307
308 \code
309 namespace vigra {
310 void getAnisotropy(MultiArrayView<2,double,stride1> data,
311 MultiArrayView<2,double,stride2> phi,
312 MultiArrayView<2,double,stride3> alpha,
313 MultiArrayView<2,double,stride4> beta,
314 double alpha_par,
315 double beta_par,
316 double sigma_par,
317 double rho_par,
318 double K_par);
319 }
320 \endcode
321
322 Output:
323 <table>
324 <tr><td>Three scalar fields <em>phi</em>, <em>alpha</em> and <em>beta</em>.</td></tr>
325 </table>
326
327 Input:
328 <table>
329 <tr><td><em>data</em>:</td><td>two-dimensional scalar field.</td></tr>
330 <tr><td><em>alpha_par,beta_par</em>:</td><td>two positive values for setting up the scalar fields alpha and beta</tr>
331 <tr><td><em>sigma_par</em>:</td><td> non-negative parameter for presmoothing the data.</td></tr>
332 <tr><td> <em>rho_par</em>:</td><td> non-negative parameter for presmoothing the structure tensor.</td></tr>
333 <tr><td><em>K_par</em>:</td><td> positive edge sensitivity parameter.</td></tr>
334 </table>
335
336 (see \ref anisotropicTotalVariationFilter() and \ref secondOrderTotalVariationFilter() for usage in an application).
337 */
338 doxygen_overloaded_function(template <...> void getAnisotropy)
339
340 template <class stride1,class stride2,class stride3,class stride4>
341 void getAnisotropy(MultiArrayView<2,double,stride1> data,MultiArrayView<2,double,stride2> phi,
342 MultiArrayView<2,double,stride3> alpha, MultiArrayView<2,double,stride4> beta,
343 double alpha_par, double beta_par, double sigma_par, double rho_par, double K_par){
344
345 using namespace multi_math;
346
347 MultiArray<2,double> smooth(data.shape()),tmp(data.shape());
348 vigra::Kernel1D<double> gauss;
349
350
351 gauss.initGaussian(sigma_par);
352 separableConvolveX(srcImageRange(data), destImage(tmp), kernel1d(gauss));
353 separableConvolveY(srcImageRange(tmp), destImage(smooth), kernel1d(gauss));
354
355 MultiArray<2,double> stxx(data.shape()),stxy(data.shape()),styy(data.shape());
356
357 // calculate Structure Tensor at inner scale = sigma and outer scale = rho
358 vigra::structureTensor(srcImageRange(smooth),destImage(stxx), destImage(stxy), destImage(styy),1.,1.);
359
360 gauss.initGaussian(rho_par);
361 separableConvolveX(srcImageRange(stxx), destImage(tmp), kernel1d(gauss));
362 separableConvolveY(srcImageRange(tmp), destImage(stxx), kernel1d(gauss));
363 separableConvolveX(srcImageRange(stxy), destImage(tmp), kernel1d(gauss));
364 separableConvolveY(srcImageRange(tmp), destImage(stxy), kernel1d(gauss));
365 separableConvolveX(srcImageRange(styy), destImage(tmp), kernel1d(gauss));
366 separableConvolveY(srcImageRange(tmp), destImage(styy), kernel1d(gauss));
367
368 MultiArray<2,double> matrix(Shape2(2,2)),ev(Shape2(2,2)),ew(Shape2(2,1));
369
370 for (int y=0;y<data.shape(1);y++){
371 for (int x=0;x<data.shape(0);x++){
372
373 matrix(0,0)=stxx(x,y);
374 matrix(1,1)=styy(x,y);
375 matrix(0,1)=stxy(x,y);
376 matrix(1,0)=stxy(x,y);
377 vigra::symmetricEigensystemNoniterative(matrix,ew,ev);
378
379 phi(x,y)=std::atan2(ev(1,0),ev(0,0));
380 double coherence=ew(0,0)-ew(1,0);
381 double c=std::min(K_par*coherence,1.);
382 alpha(x,y)=alpha_par*c+(1-c)*beta_par;
383 beta(x,y)=beta_par;
384 }
385 }
386 }
387
388 /********************************************************/
389 /* */
390 /* anisotropicTotalVariationFilter */
391 /* */
392 /********************************************************/
393
394 /** \brief Performs Anisotropic Total Variation Regularization
395
396 The algorithm minimizes
397 \f[
398 \min_u \int_\Omega \frac{1}{2} (u-f)^2 + \sqrt{\nabla u^\top A \nabla u}\;dx\qquad\qquad(2)
399 \f]
400
401 where <em> \f$ f=f(x)\f$ </em> are the noisy data, <em> \f$ u=u(x)\f$ </em> are the smoothed data,<em>\f$ \nabla u \f$ </em>
402 is the image gradient in the sense of Total Variation and <em>\f$ A \f$ </em> is a locally varying symmetric, positive definite 2x2 matrix.
403
404 Matrix <em>\f$ A \f$ </em> is described by providing for each data point a normalized eigenvector (via angle \f$ \phi \f$)
405 and two eigenvalues \f$ \alpha>0 \f$ and \f$ \beta>0 \f$.
406
407 \ref getAnisotropy() can be use to set up such data \f$ \phi,\alpha,\beta \f$ by providing a vector field normal to edges.
408
409 <b> Declarations:</b>
410
411 \code
412 namespace vigra {
413 template <class stride1,class stride2,class stride3,class stride4,class stride5,class stride6>
414 void anisotropicTotalVariationFilter(MultiArrayView<2,double,stride1> data,
415 MultiArrayView<2,double,stride2> weight,
416 MultiArrayView<2,double,stride3> phi,
417 MultiArrayView<2,double,stride4> alpha,
418 MultiArrayView<2,double,stride5> beta,
419 MultiArrayView<2,double,stride6> out,
420 int steps);
421 }
422 \endcode
423
424 \ref anisotropicTotalVariationFilter() implements a primal-dual algorithm to solve (2).
425
426 Input:
427 <table>
428 <tr><td><em>data</em>:</td><td>input data to be filtered. </td></tr>
429 <tr><td><em>steps</em>:</td><td>iteration steps.</td></tr>
430 <tr><td><em>weight</em> :</td><td>a point-wise weight (\f$ \ge 0 \f$ ) for the data term.</td></tr>
431 <tr><td><em>phi</em>,<em>alpha</em> and <em>beta</em> :</td><td> describe matrix \f$ A \f$, see above.</td></tr>
432 </table>
433
434 Output:
435 <table>
436 <tr><td><em>out</em> :</td><td>contains filtered data.</td></tr>
437 </table>
438
439 <b> Usage:</b>
440
441 E.g. with a solution-dependent adaptivity cf. [1], by updating the matrix \f$ A=A(u)\f$
442 in an outer loop:
443
444 <b>\#include</b> <vigra/tv_filter.hxx>
445
446 \code
447 MultiArray<2,double> data(Shape2(width,height)); //to be initialized
448 MultiArray<2,double> out (Shape2(width,height));
449 MultiArray<2,double> weight(Shape2(width,height)); //to be initialized
450 MultiArray<2,double> phi (Shape2(width,height));
451 MultiArray<2,double> alpha(Shape2(width,height));
452 MultiArray<2,double> beta (Shape2(width,height));
453 double alpha0,beta0,sigma,rho,K; //to be initialized
454 int outer_steps,inner_steps;//to be initialized
455
456 out=data; // data serves as initial value
457
458 for (int i=0;i<outer_steps;i++){
459 getAnisotropy(out,phi,alpha,beta,alpha0,beta0,sigma,rho,K); // sets phi, alpha, beta
460 anisotropicTotalVariationFilter(data,weight,phi,alpha,beta,out,inner_steps);
461 }
462 \endcode
463
464 [1] Frank Lenzen, Florian Becker, Jan Lellmann, Stefania Petra and Christoph Schn&ouml;rr, A Class of Quasi-Variational Inequalities for Adaptive Image Denoising and Decomposition, Computational Optimization and Applications, Springer, 2012.
465 */
466 doxygen_overloaded_function(template <...> void anisotropicTotalVariationFilter)
467
468 template <class stride1,class stride2,class stride3,class stride4,class stride5,class stride6>
469 void anisotropicTotalVariationFilter(MultiArrayView<2,double,stride1> data,MultiArrayView<2,double,stride2> weight,
470 MultiArrayView<2,double,stride3> phi,MultiArrayView<2,double,stride4> alpha,
471 MultiArrayView<2,double,stride5> beta,MultiArrayView<2,double,stride6> out,
472 int steps){
473
474 using namespace multi_math;
475 int width=data.shape(0),height=data.shape(1);
476
477 MultiArray<2,double> temp1(data.shape()),temp2(data.shape()),vx(data.shape()),vy(data.shape()),u_bar(data.shape());
478 MultiArray<2,double> rx(data.shape()),ry(data.shape());
479
480 Kernel1D<double> Lx,LTx;
481 Lx.initExplicitly(-1,0)=1,-1; // = Right sided finite differences for d/dx and d/dy
482 Lx.setBorderTreatment(BORDER_TREATMENT_REFLECT); // with hom. Neumann boundary conditions
483 LTx.initExplicitly(0,1)=-1,1; // = Left sided finite differences for -d/dx and -d/dy
484 LTx.setBorderTreatment(BORDER_TREATMENT_ZEROPAD); // with hom. Dirichlet b. c.
485
486 u_bar=out;
487
488 double m=0;
489 for (int y=0;y<data.shape(1);y++){
490 for (int x=0;x<data.shape(0);x++){
491 m=std::max(m,alpha(x,y));
492 m=std::max(m,beta (x,y));
493 }
494 }
495 m=std::max(m,1.);
496 double tau=.9/m/std::sqrt(8.)*0.06;
497 double sigma=.9/m/std::sqrt(8.)/0.06;
498
499
500 for (int i=0;i<steps;i++){
501 separableConvolveX(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
502 vx+=(sigma*temp1);
503 separableConvolveY(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
504 vy+=(sigma*temp1);
505
506 //project to constraint set
507 for (int y=0;y<data.shape(1);y++){
508 for (int x=0;x<data.shape(0);x++){
509 double e1,e2,skp1,skp2;
510
511 e1=std::cos(phi(x,y));
512 e2=std::sin(phi(x,y));
513 skp1=vx(x,y)*e1+vy(x,y)*e2;
514 skp2=vx(x,y)*(-e2)+vy(x,y)*e1;
515 vigra::detail::projectEllipse2D (skp1,skp2,alpha(x,y),beta(x,y),0.001,100);
516
517 vx(x,y)=skp1*e1-skp2*e2;
518 vy(x,y)=skp1*e2+skp2*e1;
519 }
520 }
521
522 separableConvolveX(srcImageRange(vx),destImage(temp1),kernel1d(LTx));
523 separableConvolveY(srcImageRange(vy),destImage(temp2),kernel1d(LTx));
524 u_bar=out;
525 out-=tau*(weight*(out-data)+(temp1+temp2));
526 u_bar=2*out-u_bar; //cf. Chambolle/Pock and Popov's algorithm
527 }
528 }
529
530 /********************************************************/
531 /* */
532 /* secondOrderTotalVariationFilter */
533 /* */
534 /********************************************************/
535
536 /** \brief Performs Anisotropic Total Variation Regularization
537
538 The algorithm minimizes
539
540 \f[
541 \min_u \int_\Omega \frac{1}{2} (u-f)^2 + \sqrt{\nabla u^\top A \nabla u} + \gamma |Hu|_F\;dx \qquad\qquad (3)
542 \f]
543 where <em> \f$ f=f(x)\f$ </em> are the noisy data, <em> \f$ u=u(x)\f$ </em> are the smoothed data,<em>\f$ \nabla u \f$ </em>
544 is the image gradient in the sense of Total Variation, <em>\f$ A \f$ </em> is a locally varying
545 symmetric, positive-definite 2x2 matrix and <em>\f$ |Hu|_F \f$</em> is the Frobenius norm of the Hessian of \f$ u \f$.
546
547 Matrix <em>\f$ A \f$ </em> is described by providing for each data point a normalized eigenvector (via angle \f$ \phi \f$)
548 and two eigenvalues \f$ \alpha>0 \f$ and \f$ \beta>0 \f$.
549 \ref getAnisotropy() can be use to set up such data \f$ \phi,\alpha, \beta \f$ by providing a vector field normal to edges.
550
551 \f$ \gamma>0 \f$ is the locally varying regularization parameter for second order.
552
553 <b> Declarations:</b>
554
555 \code
556 namespace vigra {
557 template <class stride1,class stride2,...,class stride9>
558 void secondOrderTotalVariationFilter(MultiArrayView<2,double,stride1> data,
559 MultiArrayView<2,double,stride2> weight,
560 MultiArrayView<2,double,stride3> phi,
561 MultiArrayView<2,double,stride4> alpha,
562 MultiArrayView<2,double,stride5> beta,
563 MultiArrayView<2,double,stride6> gamma,
564 MultiArrayView<2,double,stride7> xedges,
565 MultiArrayView<2,double,stride8> yedges,
566 MultiArrayView<2,double,stride9> out,
567 int steps);
568 }
569 \endcode
570
571 \ref secondOrderTotalVariationFilter() implements a primal-dual algorithm to solve (3).
572
573 Input:
574 <table>
575 <tr><td><em>data</em>: </td><td> the input data to be filtered. </td></tr>
576 <tr><td><em>steps</em> : </td><td> number of iteration steps.</td></tr>
577 <tr><td><em>out</em> : </td><td>contains the filtered data.</td></tr>
578 <tr><td><em>weight</em> : </td><td> point-wise weight (\f$ \ge 0\f$ ) for the data term.</td></tr>
579 <tr><td><em>phi</em>,<em>alpha</em>,<em>beta</em>: </td><td> describe matrix \f$ A\f$, see above.</td></tr>
580 <tr><td><em> xedges </em> and <em> yedges </em>: </td><td> binary arrays indicating the
581 presence of horizontal (between (x,y) and (x+1,y)) and vertical edges (between (x,y) and (x,y+1)).
582 These data are considered in the calculation of \f$ Hu\f$, such that
583 finite differences across edges are artificially set to zero to avoid second order smoothing over edges.</td></tr>
584 </table>
585
586 <b> Usage:</b>
587
588 E.g. with a solution-dependent adaptivity (cf.[1]), by updating the matrix \f$ A=A(u)\f$
589 in an outer loop:
590
591 <b>\#include</b> <vigra/tv_filter.hxx>
592
593 \code
594 MultiArray<2,double> data(Shape2(width,height)); //to be initialized
595 MultiArray<2,double> out(Shape2(width,height));
596 MultiArray<2,double> weight(Shape2(width,height))); //to be initialized
597 MultiArray<2,double> phi(Shape2(width,height);
598 MultiArray<2,double> alpha(Shape2(width,height);
599 MultiArray<2,double> beta(Shape2(width,height));
600 MultiArray<2,double> gamma(Shape2(width,height)); //to be initialized
601 MultiArray<2,double> xedges(Shape2(width,height)); //to be initialized
602 MultiArray<2,double> yedges(Shape2(width,height)); //to be initialized
603
604
605 double alpha0,beta0,sigma,rho,K; //to be initialized
606 int outer_steps,inner_steps;//to be initialized
607
608 out=data; // data serves as initial value
609
610 for (int i=0;i<outer_steps;i++){
611
612 getAnisotropy(out,phi,alpha,beta,alpha0,beta0,sigma,rho,K); // sets phi, alpha, beta
613 secondOrderTotalVariationFilter(data,weight,phi,alpha,beta,gamma,xedges,yedges,out,inner_steps);
614 }
615 \endcode
616
617
618 [1] Frank Lenzen, Florian Becker, Jan Lellmann, Stefania Petra and Christoph Schn&ouml;rr, A Class of Quasi-Variational Inequalities for Adaptive Image Denoising and Decomposition, Computational Optimization and Applications, Springer, 2012.
619 */
620 doxygen_overloaded_function(template <...> void secondOrderTotalVariationFilter)
621
622 template <class stride1,class stride2,class stride3,class stride4,class stride5,class stride6,class stride7,class stride8,class stride9>
623 void secondOrderTotalVariationFilter(MultiArrayView<2,double,stride1> data,
624 MultiArrayView<2,double,stride2> weight,MultiArrayView<2,double,stride3> phi,
625 MultiArrayView<2,double,stride4> alpha,MultiArrayView<2,double,stride5> beta,
626 MultiArrayView<2,double,stride6> gamma,
627 MultiArrayView<2,double,stride7> xedges,MultiArrayView<2,double,stride8> yedges,
628 MultiArrayView<2,double,stride9> out,
629 int steps){
630
631 using namespace multi_math;
632 int width=data.shape(0),height=data.shape(1);
633
634 MultiArray<2,double> temp1(data.shape()),temp2(data.shape()),vx(data.shape()),vy(data.shape()),u_bar(data.shape());
635 MultiArray<2,double> rx(data.shape()),ry(data.shape());
636 MultiArray<2,double> wx(data.shape()),wy(data.shape()),wz(data.shape());
637
638
639 Kernel1D<double> Lx,LTx;
640 Lx.initExplicitly(-1,0)=1,-1; // = Right sided finite differences for d/dx and d/dy
641 Lx.setBorderTreatment(BORDER_TREATMENT_REFLECT); // with hom. Neumann boundary conditions
642 LTx.initExplicitly(0,1)=-1,1; // = Left sided finite differences for -d/dx and -d/dy
643 LTx.setBorderTreatment(BORDER_TREATMENT_ZEROPAD); // with hom. Dirichlet b. c.
644
645 u_bar=out;
646
647 double m=0;
648 for (int y=0;y<data.shape(1);y++){
649 for (int x=0;x<data.shape(0);x++){
650 m=std::max(m,alpha(x,y));
651 m=std::max(m,beta (x,y));
652 m=std::max(m,gamma(x,y));
653 }
654 }
655 m=std::max(m,1.);
656 double tau=.1/m;//std::sqrt(8)*0.06;
657 double sigma=.1;//m;/std::sqrt(8)/0.06;
658
659 //std::cout<<"tau= "<<tau<<std::endl;
660
661 for (int i=0;i<steps;i++){
662
663 separableConvolveX(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
664 vx+=(sigma*temp1);
665 separableConvolveY(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
666 vy+=(sigma*temp1);
667
668
669 // update wx
670 separableConvolveX(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
671 temp1*=xedges;
672 separableConvolveX(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
673 wx-=sigma*temp2;//(-Lx'*(xedges.*(Lx*u)));
674
675 //update wy
676 separableConvolveY(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
677 temp1*=yedges;
678 separableConvolveY(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
679 wy-=sigma*temp2;//(-Ly'*(yedges.*(Ly*u)));
680
681
682 //update wz
683 #if (VIGRA_MIXED_2ND_DERIVATIVES)
684 separableConvolveY(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
685 temp1*=yedges;
686 separableConvolveX(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
687 wz-=sigma*temp2;//-Lx'*(yedges.*(Ly*u))
688
689 separableConvolveX(srcImageRange(u_bar),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
690 temp1*=xedges;
691 separableConvolveY(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
692 wz-=sigma*temp2;//-Ly'*(xedges.*(Lx*u)));
693
694 #endif
695
696
697 //project to constraint sets
698 for (int y=0;y<data.shape(1);y++){
699 for (int x=0;x<data.shape(0);x++){
700 double e1,e2,skp1,skp2;
701
702 //project v
703 e1=std::cos(phi(x,y));
704 e2=std::sin(phi(x,y));
705 skp1=vx(x,y)*e1+vy(x,y)*e2;
706 skp2=vx(x,y)*(-e2)+vy(x,y)*e1;
707 vigra::detail::projectEllipse2D (skp1,skp2,alpha(x,y),beta(x,y),0.001,100);
708 vx(x,y)=skp1*e1-skp2*e2;
709 vy(x,y)=skp1*e2+skp2*e1;
710
711 //project w
712 double l=sqrt(wx(x,y)*wx(x,y)+wy(x,y)*wy(x,y)+wz(x,y)*wz(x,y));
713 if (l>gamma(x,y)){
714 wx(x,y)=gamma(x,y)*wx(x,y)/l;
715 wy(x,y)=gamma(x,y)*wy(x,y)/l;
716 #if (VIGRA_MIXED_2ND_DERIVATIVES)
717 wz(x,y)=gamma(x,y)*wz(x,y)/l;
718 #endif
719 }
720 }
721 }
722
723 separableConvolveX(srcImageRange(vx),destImage(temp1),kernel1d(LTx));
724 separableConvolveY(srcImageRange(vy),destImage(temp2),kernel1d(LTx));
725
726 u_bar=out;
727 out-=tau*(weight*(out-data)+temp1+temp2);
728
729
730 // update wx
731 separableConvolveX(srcImageRange(wx),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
732 temp1*=xedges;
733 separableConvolveX(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
734 out+=tau*temp2; // (-1)^2
735
736
737 //update wy
738 separableConvolveY(srcImageRange(wy),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
739 temp1*=yedges;
740 separableConvolveY(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
741 out+=tau*temp2;
742
743 //update wz
744 #if (VIGRA_MIXED_2ND_DERIVATIVES)
745
746 separableConvolveY(srcImageRange(wz),destImage(temp1),kernel1d(Lx));setZeroY(temp1);
747 temp1*=yedges;
748 separableConvolveX(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
749 out+=tau*temp2;
750
751 separableConvolveX(srcImageRange(wz),destImage(temp1),kernel1d(Lx));setZeroX(temp1);
752 temp1*=xedges;
753 separableConvolveY(srcImageRange(temp1),destImage(temp2),kernel1d(LTx));
754 out+=tau*temp2;
755
756 #endif
757
758 u_bar=2*out-u_bar; //cf. Chambolle/Pock and Popov's algorithm
759
760 }
761 }
762
763 //@}
764 } // closing namespace vigra
765
766 #endif // VIGRA_TV_FILTER_HXX
vigra::atan2
FixedPoint16< 2, OverflowHandling > atan2(FixedPoint16< IntBits, OverflowHandling > y, FixedPoint16< IntBits, OverflowHandling > x)
Arctangent. Accuracy better than 1/3 degree (9 significant bits).
Definition: fixedpoint.hxx:1654
vigra::Kernel1D
Generic 1 dimensional convolution kernel.
Definition: separableconvolution.hxx:52
vigra::linalg::sin
linalg::TemporaryMatrix< T > sin(MultiArrayView< 2, T, C > const &v)
vigra::Kernel1D::initGaussian
void initGaussian(double std_dev, value_type norm, double windowRatio=0.0)
Definition: separableconvolution.hxx:2253
vigra::getAnisotropy
void getAnisotropy(...)
Sets up directional data for anisotropic regularization.
vigra::linalg::symmetricEigensystemNoniterative
bool symmetricEigensystemNoniterative(MultiArrayView< 2, T, C1 > const &a, MultiArrayView< 2, T, C2 > &ew, MultiArrayView< 2, T, C3 > &ev)
Definition: eigensystem.hxx:1152
vigra::separableConvolveX
void separableConvolveX(...)
Performs a 1 dimensional convolution in x direction.
vigra::hypot
FixedPoint16< IntBits, OverflowHandling > hypot(FixedPoint16< IntBits, OverflowHandling > v1, FixedPoint16< IntBits, OverflowHandling > v2)
Length of hypotenuse.
Definition: fixedpoint.hxx:1640
vigra::anisotropicTotalVariationFilter
void anisotropicTotalVariationFilter(...)
Performs Anisotropic Total Variation Regularization.
vigra::doxygen_overloaded_function
doxygen_overloaded_function(template<...> void separableConvolveBlockwise) template< unsigned int N
Separated convolution on ChunkedArrays.
vigra::gamma
double gamma(double x)
The gamma function.
Definition: mathutil.hxx:1587
vigra::structureTensor
void structureTensor(...)
Calculate the Structure Tensor for each pixel of and image, using Gaussian (derivative) filters...
vigra::totalVariationFilter
void totalVariationFilter(...)
Performs standard Total Variation Regularization.
impex.hxx
image import and export functions
vigra::linalg::cos
linalg::TemporaryMatrix< T > cos(MultiArrayView< 2, T, C > const &v)
vigra::separableConvolveY
void separableConvolveY(...)
Performs a 1 dimensional convolution in y direction.
vigra::sqrt
SquareRootTraits< FixedPoint< IntBits, FracBits > >::SquareRootResult sqrt(FixedPoint< IntBits, FracBits > v)
square root.
Definition: fixedpoint.hxx:616
vigra::secondOrderTotalVariationFilter
void secondOrderTotalVariationFilter(...)
Performs Anisotropic Total Variation Regularization.

© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de) Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany | html generated using doxygen and Python vigra 1.11.0 (Fri May 19 2017)