├── GenerateUncertaintyData.m
├── LICENSE
├── README.md
├── RobustUncertaintyAwareMultiviewTriangulation_ReleaseCode.m
└── uncertainty.mat
/GenerateUncertaintyData.m:
--------------------------------------------------------------------------------
1 | clear all; close all; clc;
2 |
3 | load_precomputed_results = true;
4 |
5 | if (load_precomputed_results)
6 | load uncertainty.mat;
7 | load results_inliers.mat;
8 | else
9 |
10 | tic
11 | % Camera follows TUM rgb-d dataset parameters
12 | focal_length = 525;
13 | img_width = 640;
14 | img_height = 480;
15 | K = [focal_length 0 0; 0 focal_length 0; 0 0 1];
16 |
17 | noise_levels = [1:10, 12:2:30];
18 | dists = [1:9, 10:10:30];
19 |
20 | n_cameras = [2:10, 12:2:30, 35:5:50];
21 |
22 | n_simulations = [];
23 | for i = 1:length(n_cameras)
24 |
25 | if (n_cameras(i) <= 5)
26 | n_simulations = [n_simulations, 10];
27 | continue;
28 | end
29 |
30 | if (n_cameras(i) <= 20)
31 | n_simulations = [n_simulations, 5];
32 | continue;
33 | end
34 |
35 | n_simulations = [n_simulations, 3];
36 | end
37 |
38 | n_simulations = 1000*n_simulations;
39 |
40 | n_total_run = length(noise_levels)*length(dists)*sum(n_simulations);
41 |
42 | all_error_3D = nan(1,n_total_run);
43 | all_error_2D = nan(1,n_total_run);
44 | all_min_level_of_degeneracy = nan(1,n_total_run);
45 | all_n_camera = nan(1,n_total_run);
46 |
47 | idx = 0;
48 |
49 | for noise_level = noise_levels
50 |
51 | for dist = dists
52 | point_w = [0;0;dist];
53 |
54 | for n_camera = fliplr(n_cameras)
55 |
56 | disp(['noise_level = ', num2str(noise_level), ', dist = ', num2str(dist), ', n_camera = ', num2str(n_camera)])
57 |
58 |
59 | n_simulation = n_simulations(n_cameras == n_camera);
60 | for simulation = 1:n_simulation
61 |
62 | idx = idx + 1;
63 | cam_pos = cell(1,n_camera);
64 | cam_pos_mat = nan(3,n_camera);
65 | R = cell(1,n_camera);
66 | t = cell(1,n_camera);
67 | uv = cell(1,n_camera);
68 | uv_measured = cell(1,n_camera);
69 | ray = cell(1,n_camera);
70 | AA = cell(1,n_camera);
71 | bb = cell(1,n_camera);
72 |
73 | for i = 1:n_camera
74 |
75 | % Camera is located randomly inside a unit sphere.
76 | cam_pos{i} = rand(3,1)-0.5;
77 | cam_pos{i} = rand(1)*0.5*cam_pos{i}/norm(cam_pos{i});
78 |
79 | % The first two cameras are antipodal.
80 | if (i==1)
81 | cam_pos{1} = rand(3,1)-0.5;
82 | cam_pos{1} = 0.5*cam_pos{1}/norm(cam_pos{1});
83 | elseif (i==2)
84 | cam_pos{2} = -cam_pos{1};
85 | end
86 |
87 | while(true)
88 | R_temp = random_rotation(180);
89 | t_temp = -R_temp*cam_pos{i};
90 |
91 | point_c = R_temp*point_w+t_temp;
92 | uv_temp = K*[point_c(1)/point_c(3); point_c(2)/point_c(3);1];
93 | uv_temp = uv_temp(1:2);
94 | if (abs(uv_temp(1)) > img_width/2 || abs(uv_temp(2)) > img_height/2 || point_c(3) < 0)
95 | continue;
96 | end
97 |
98 | while(true)
99 | noise_dir = rand(2,1)-0.5;
100 | noise_dir = noise_dir/norm(noise_dir);
101 | uv_measured_temp = uv_temp+normrnd(0, noise_level)*noise_dir;
102 | if (abs(uv_measured_temp(1)) > img_width/2 || abs(uv_measured_temp(2)) > img_height/2)
103 | continue;
104 | end
105 | break;
106 | end
107 |
108 |
109 | break;
110 | end
111 |
112 | cam_pos_mat(:,i) = cam_pos{i};
113 | R{i} = R_temp;
114 | t{i} = t_temp;
115 | uv{i} = uv_temp;
116 | uv_measured{i} = uv_measured_temp;
117 | ray{i} = K\[uv_measured{i};1];
118 | ray{i} = ray{i}/norm(ray{i});
119 |
120 | % Compute AA and bb necesary for computing the Jacobian:
121 | r11 = R{i}(1,1); r12 = R{i}(1,2); r13 = R{i}(1,3);
122 | r21 = R{i}(2,1); r22 = R{i}(2,2); r23 = R{i}(2,3);
123 | r31 = R{i}(3,1); r32 = R{i}(3,2); r33 = R{i}(3,3);
124 | tx = t{i}(1); ty = t{i}(2); tz = t{i}(3);
125 | fx = K(1,1); skew = K(1,2); fy = K(2,2);
126 |
127 | a11 = [...
128 | 0, ...
129 | fx*(r11*r32-r31*r12)+skew*(r21*r32-r31*r22), ...
130 | fx*(r11*r33-r31*r13)+skew*(r21*r33-r31*r23), ...
131 | fx*(r11*tz-r31*tx)+skew*(r21*tz-r31*ty)];
132 | a21 = [...
133 | 0, ...
134 | fy*(r21*r32-r31*r22), ...
135 | fy*(r21*r33-r31*r23), ...
136 | +fy*(r21*tz-r31*ty)];
137 | a12 = [...
138 | fx*(r12*r31-r32*r11)+skew*(r22*r31-r32*r21),...
139 | 0,...
140 | fx*(r12*r33-r32*r13)+skew*(r22*r33-r32*r23),...
141 | fx*(r12*tz-r32*tx)+skew*(r22*tz-r32*ty)];
142 | a22 = [...
143 | fy*(r22*r31-r32*r21),...
144 | 0,...
145 | fy*(r22*r33-r32*r23),...
146 | fy*(r22*tz-r32*ty)];
147 | a13 = [...
148 | fx*(r13*r31-r33*r11)+skew*(r23*r31-r33*r21),...
149 | fx*(r13*r32-r33*r12)+skew*(r23*r32-r33*r22),...
150 | 0,...
151 | fx*(r13*tz-r33*tx)+skew*(r23*tz-r33*ty)];
152 | a23 = [...
153 | fy*(r23*r31-r33*r21),...
154 | fy*(r23*r32-r33*r22),...
155 | 0,...
156 | fy*(r23*tz-r33*ty)];
157 |
158 | AA{i} = [a11;a21;a12;a22;a13;a23];
159 | bb{i} = [r31, r32, r33, tz];
160 | end
161 |
162 |
163 | % figure;
164 | % axis equal
165 | % hold on
166 | % for i=1:n_camera
167 | % if (i<= n_inlier)
168 | % scatter3(cam_pos{i}(1),cam_pos{i}(2),cam_pos{i}(3), 'ko')
169 | % text(cam_pos{i}(1), cam_pos{i}(2), cam_pos{i}(3),num2str(i),'HorizontalAlignment','left','FontSize',20);
170 | %
171 | % endpoint = 15*R{i}'*ray{i}+cam_pos{i};
172 | % xx = [cam_pos{i}(1), endpoint(1)];
173 | % yy = [cam_pos{i}(2), endpoint(2)];
174 | % zz = [cam_pos{i}(3), endpoint(3)];
175 | % plot3(xx, yy, zz, 'Color', [0 0 0 0.5]);
176 | % else
177 | % scatter3(cam_pos{i}(1),cam_pos{i}(2),cam_pos{i}(3), 'ro')
178 | %
179 | % endpoint = R{i}'*ray{i}+cam_pos{i};
180 | % xx = [cam_pos{i}(1), endpoint(1)];
181 | % yy = [cam_pos{i}(2), endpoint(2)];
182 | % zz = [cam_pos{i}(3), endpoint(3)];
183 | % plot3(xx, yy, zz, 'Color', [1 0 0 0.5]);
184 | % end
185 | % end
186 | % scatter3(point_w(1), point_w(2), point_w(3), 200, 'kx')
187 | % view(360,0)
188 |
189 |
190 | %% DLT + GN for all inliers
191 |
192 | % Prepare the matrices.
193 | M0 = nan(1,n_camera);
194 | M1 = nan(1,n_camera);
195 | M2 = nan(4,n_camera);
196 | M3 = nan(4,n_camera);
197 | M4 = nan(4,n_camera);
198 | ray_w = cell(1,n_camera);
199 | for i = 1:n_camera
200 | ray_w{i} = K\[uv_measured{i};1];
201 | ray_w{i} = ray_w{i}/norm(ray_w{i});
202 | ray_w{i} = R{i}'*ray_w{i};
203 |
204 | M0(1,i) = uv_measured{i}(1)-K(1,3);
205 | M1(1,i) = uv_measured{i}(2)-K(2,3);
206 |
207 | P = [R{i}, t{i}];
208 | M2(:,i) = K(1,1)*P(1,:)'+K(1,2)*P(2,:)';
209 | M3(:,i) = K(2,1)*P(1,:)'+K(2,2)*P(2,:)';
210 | M4(:,i) = P(3,:)';
211 | end
212 |
213 | % Do DLT.
214 | A = zeros(2*n_camera, 4);
215 | for i = 1:n_camera
216 | Q = K*[R{i}, t{i}];
217 | u = uv_measured{i}(1);
218 | v = uv_measured{i}(2);
219 | A(2*i-1:2*i, :) = [u*Q(3,:)-Q(1,:); v*Q(3,:)-Q(2,:)];
220 | end
221 | [~,S_,V_] = svd(A);
222 | [~, min_idx] = min(diag(S_));
223 | x_dlt = V_(:, min_idx);
224 | x_dlt = x_dlt/x_dlt(end);
225 | x_dlt = x_dlt(1:3);
226 |
227 | % Do Gauss-Newton Optimization.
228 | mean_reproj_error = 0;
229 | x_gn = x_dlt;
230 | for it = 1:10
231 | N4 = [x_gn;1]'*M4;
232 | M5 = ([x_gn;1]'*M2)./N4 - M0;
233 | M6 = ([x_gn;1]'*M3)./N4 - M1;
234 | M5 = M5(1:n_camera);
235 | M6 = M6(1:n_camera);
236 | residuals = [M5;M6];
237 | squared_reproj_error_all = M5.*M5+M6.*M6;
238 |
239 | mean_reproj_error_prev = mean_reproj_error;
240 | mean_reproj_error = mean(sqrt(squared_reproj_error_all));
241 |
242 | if (it > 2 && abs(mean_reproj_error-mean_reproj_error_prev) < 0.1)
243 | break;
244 | end
245 |
246 | J_est = zeros(2*n_camera,3);
247 | for i = 1:n_camera
248 | J_temp = AA{i}*[x_gn;1]/(bb{i}*[x_gn;1])^2;
249 | J_est(2*i-1:2*i, :) = reshape(J_temp, [2,3]);
250 | end
251 | update = -pinv(J_est)*residuals(:);
252 | x_gn = x_gn+update;
253 | end
254 |
255 | % Evaluate the performance metrics.
256 | N4 = [x_gn;1]'*M4;
257 | M5 = ([x_gn;1]'*M2)./N4 - M0;
258 | M6 = ([x_gn;1]'*M3)./N4 - M1;
259 | squared_reproj_error_all = M5.*M5+M6.*M6;
260 | squared_reproj_error_inliers = squared_reproj_error_all(1:n_camera);
261 |
262 |
263 | all_error_3D(idx) = norm(point_w-x_gn);
264 | all_error_2D(idx) = mean(sqrt(squared_reproj_error_inliers));
265 | all_min_level_of_degeneracy(idx) = GetMinimumLevelOfDegeneracy(cam_pos, ray_w, 1:n_camera);
266 | all_n_camera(idx) = n_camera;
267 |
268 | end
269 |
270 | end
271 | end
272 | end
273 |
274 | angle_bin_edges = fliplr([0:19; 1:20]);
275 | noise_bin_edges = [0:19; 1:20];
276 | cell_colormap_results = cell(size(angle_bin_edges,2), size(noise_bin_edges,2), length(n_cameras));
277 |
278 | for i = 1:length(all_error_3D)
279 | angle_of_degeneracy = acosd(all_min_level_of_degeneracy(i));
280 | n_camera = all_n_camera(i);
281 | error_3D = all_error_3D(i);
282 | error_2D = all_error_2D(i);
283 | for j = 1:size(angle_bin_edges,2)
284 | if (angle_of_degeneracy < angle_bin_edges(1,j) || angle_of_degeneracy >= angle_bin_edges(2,j))
285 | continue;
286 | end
287 | for k = 1:size(noise_bin_edges,2)
288 | if (error_2D < noise_bin_edges(1,k) || error_2D >= noise_bin_edges(2,k))
289 | continue;
290 | end
291 | for l = 1:length(n_cameras)
292 | if (n_cameras(l) ~= n_camera)
293 | continue;
294 | end
295 | cell_colormap_results{j,k,l} = [cell_colormap_results{j,k,l}, error_3D];
296 | end
297 | end
298 | end
299 | end
300 |
301 | mat_colormap_results = zeros(size(cell_colormap_results));
302 | for j = 1:size(mat_colormap_results,1)
303 | for k = 1:size(mat_colormap_results,2)
304 | for l = 1:size(mat_colormap_results,3)
305 | mat_colormap_results(j,k,l) = min(1,sqrt(mean(cell_colormap_results{j,k,l}.^2)));
306 | end
307 | end
308 | end
309 |
310 | mat_colormap_results_smoothed = MonotoneSmooth(mat_colormap_results, 0.01, 1, 1, 0);
311 |
312 | uncertainty.angle_i = angle_bin_edges;
313 | uncertainty.reproj_j = noise_bin_edges;
314 | uncertainty.camera_k = n_cameras;
315 | uncertainty.result = mat_colormap_results_smoothed;
316 |
317 | save uncertainty.mat uncertainty
318 | save results_inliers.mat
319 | time_elsaped = toc;
320 | disp(['Took ', num2str(time_elsaped/3600), ' hours.'])
321 |
322 | end
323 |
324 |
325 |
326 | %out = TrilinearInterpolation(8.5, 3.5, 10, uncertainty)
327 |
328 |
329 |
330 |
331 |
332 | %% Plot
333 |
334 | figure;
335 | for i = 1:length(n_cameras)
336 | subplot(6,8,2*i-1)
337 | imagesc(mat_colormap_results(:,:,i))
338 | colormap(jet)
339 | caxis([0 1])
340 | set(gca,'xticklabel',{[]})
341 | set(gca,'yticklabel',{[]})
342 | title([num2str(n_cameras(i)), ' cams'])
343 |
344 | subplot(6,8,2*i)
345 | imagesc(mat_colormap_results_smoothed(:,:,i))
346 | colormap(jet)
347 | caxis([0 1])
348 | set(gca,'xticklabel',{[]})
349 | set(gca,'yticklabel',{[]})
350 | title([num2str(n_cameras(i)), ' cams (smoothed)'])
351 | end
352 |
353 |
354 | figure;
355 | plot_idx = 0;
356 | for i = 1:length(n_cameras)
357 | n_camera = n_cameras(i);
358 | if (ismember(n_cameras(i), [2, 3, 4, 5, 10, 20,30,50]))
359 | plot_idx = plot_idx + 1;
360 | subplot(2,4,plot_idx)
361 | imagesc(mat_colormap_results_smoothed(:,:,i))
362 | axis equal
363 | colormap(jet)
364 | colorbar
365 | caxis([0 1])
366 | xticks(0.5:5:20.5)
367 | xticklabels({'0', '5', '10', '15', '20'})
368 | xlabel('Mean 2D error (pix)')
369 | yticks(0.5:5:20.5)
370 | yticklabels({'20','15', '10', '5','0'})
371 | ylabel('Angle of Degeneracy (deg)')
372 | title([num2str(n_cameras(i)), ' cams'])
373 | end
374 | end
375 |
376 | % for i = 1:length(n_cameras)
377 | % disp(['n_camera = ', num2str(n_cameras(i))])
378 | % latex_table_raw = latex(vpa(sym(mat_colormap_results(:,:,i)),2))
379 | % latex_table_smoothed = latex(vpa(sym(mat_colormap_results_smoothed(:,:,i)),2))
380 | % end
381 |
382 |
383 | %%
384 | function out = random_rotation(max_angle_deg)
385 |
386 | axis = rand(3,1)-0.5;
387 | axis = axis/norm(axis);
388 | angle = rand*max_angle_deg/180*pi;
389 | rotvec = angle*axis;
390 | out = rotation_from_rotvec(rotvec);
391 |
392 | end
393 |
394 | function R = rotation_from_rotvec(in)
395 | angle = norm(in);
396 |
397 | if (angle==0)
398 | R = eye(3);
399 | else
400 | unit_axis = in/angle;
401 | so3 = SkewSymmetricMatrix(unit_axis);
402 | R = eye(3)+so3*sin(angle)+so3^2*(1-cos(angle));
403 | end
404 | end
405 |
406 |
407 | function out = SkewSymmetricMatrix(in)
408 | out=[0 -in(3) in(2) ; in(3) 0 -in(1) ; -in(2) in(1) 0 ];
409 | end
410 |
411 |
412 | function mLoD = GetMinimumLevelOfDegeneracy(cam_pos, ray_w, idx)
413 |
414 | idx = reshape(idx, [1, numel(idx)]);
415 |
416 | mLoD = inf;
417 | for i = idx
418 | for j = idx
419 | if (j<=i)
420 | continue;
421 | end
422 | t_ij = cam_pos{i}-cam_pos{j};
423 | t_ij = t_ij/norm(t_ij);
424 | f_i = ray_w{i};
425 | f_j = ray_w{j};
426 |
427 | LoD = abs(f_i'*f_j);
428 | % LoD = max(abs([t_ij'*f_i, t_ij'*f_j, f_i'*f_j]));
429 | if (LoD < mLoD)
430 | mLoD = LoD;
431 | end
432 | end
433 | end
434 |
435 | end
436 |
437 |
438 | function out = MonotoneSmooth(in, thr, x_increasing, y_increasing, z_increasing)
439 |
440 | out = in;
441 | for it = 1:100000
442 | out_prev = out;
443 |
444 | for i = 1:size(out_prev,1)
445 | for j = 1:size(out_prev,2)
446 | for k = 1:size(out_prev,3)
447 | candidates = [];
448 |
449 | if (x_increasing)
450 | if (i-1 >= 1)
451 | if (out_prev(i-1,j,k) <= out_prev(i,j,k))
452 | candidates = [candidates, out_prev(i,j,k)];
453 | else
454 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i-1,j,k))];
455 | end
456 | end
457 | if (i+1 <= size(out_prev,1))
458 | if (out_prev(i+1,j,k) >= out_prev(i,j,k))
459 | candidates = [candidates, out_prev(i,j,k)];
460 | else
461 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i+1,j,k))];
462 | end
463 | end
464 | else
465 | if (i-1 >= 1)
466 | if (out_prev(i-1,j,k) >= out_prev(i,j,k))
467 | candidates = [candidates, out_prev(i,j,k)];
468 | else
469 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i-1,j,k))];
470 | end
471 | end
472 | if (i+1 <= size(out_prev,1))
473 | if (out_prev(i+1,j,k) <= out_prev(i,j,k))
474 | candidates = [candidates, out_prev(i,j,k)];
475 | else
476 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i+1,j,k))];
477 | end
478 | end
479 | end
480 |
481 | if (y_increasing)
482 | if (j-1 >= 1)
483 | if (out_prev(i,j-1,k) <= out_prev(i,j,k))
484 | candidates = [candidates, out_prev(i,j,k)];
485 | else
486 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j-1,k))];
487 | end
488 | end
489 | if (j+1 <= size(out_prev,2))
490 | if (out_prev(i,j+1,k) >= out_prev(i,j,k))
491 | candidates = [candidates, out_prev(i,j,k)];
492 | else
493 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j+1,k))];
494 | end
495 | end
496 | else
497 | if (j-1 >= 1)
498 | if (out_prev(i,j-1,k) >= out_prev(i,j,k))
499 | candidates = [candidates, out_prev(i,j,k)];
500 | else
501 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j-1,k))];
502 | end
503 | end
504 | if (j+1 <= size(out_prev,2))
505 | if (out_prev(i,j+1,k) <= out_prev(i,j,k))
506 | candidates = [candidates, out_prev(i,j,k)];
507 | else
508 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j+1,k))];
509 | end
510 | end
511 | end
512 |
513 | if (z_increasing)
514 | if (k-1 >= 1)
515 | if (out_prev(i,j,k-1) <= out_prev(i,j,k))
516 | candidates = [candidates, out_prev(i,j,k)];
517 | else
518 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j,k-1))];
519 | end
520 | end
521 | if (k+1 <= size(out_prev,3))
522 | if (out_prev(i,j,k+1) >= out_prev(i,j,k))
523 | candidates = [candidates, out_prev(i,j,k)];
524 | else
525 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j,k+1))];
526 | end
527 | end
528 | else
529 | if (k-1 >= 1)
530 | if (out_prev(i,j,k-1) >= out_prev(i,j,k))
531 | candidates = [candidates, out_prev(i,j,k)];
532 | else
533 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j,k-1))];
534 | end
535 | end
536 | if (k+1 <= size(out_prev,3))
537 | if (out_prev(i,j,k+1) <= out_prev(i,j,k))
538 | candidates = [candidates, out_prev(i,j,k)];
539 | else
540 | candidates = [candidates, 0.5*(out_prev(i,j,k)+out_prev(i,j,k+1))];
541 | end
542 | end
543 | end
544 |
545 | out(i,j,k) = mean(candidates);
546 | end
547 | end
548 | end
549 |
550 |
551 | % Check if monotonicity is enforced for every value
552 |
553 | completed = true;
554 | for i = 1:size(out,1)
555 | if (~completed)
556 | break;
557 | end
558 | for j = 1:size(out,2)
559 | if (~completed)
560 | break;
561 | end
562 | for k = 1:size(out,3)
563 | if (~completed)
564 | break;
565 | end
566 |
567 | if (x_increasing)
568 | if (out(max(1,i-1),j,k) > out(i,j,k)+thr)
569 | completed = false;
570 | break;
571 | end
572 | if (out(min(size(out,1),i+1),j,k) < out(i,j,k)-thr)
573 | completed = false;
574 | break;
575 | end
576 | else
577 | if (out(max(1,i-1),j,k) < out(i,j,k)-thr)
578 | completed = false;
579 | break;
580 | end
581 | if (out(min(size(out,1),i+1),j,k) > out(i,j,k)+thr)
582 | completed = false;
583 | break;
584 | end
585 | end
586 |
587 | if (y_increasing)
588 | if (out(i,max(1,j-1),k) > out(i,j,k)+thr)
589 | completed = false;
590 | break;
591 | end
592 | if (out(i,min(size(out,2),j+1),k) < out(i,j,k)-thr)
593 | completed = false;
594 | break;
595 | end
596 | else
597 | if (out(i,max(1,j-1),k) < out(i,j,k)-thr)
598 | completed = false;
599 | break;
600 | end
601 | if (out(i,min(size(out,2),j+1),k) > out(i,j,k)+thr)
602 | completed = false;
603 | break;
604 | end
605 | end
606 |
607 | if (z_increasing)
608 | if (out(i,j,max(1,k-1)) > out(i,j,k)+thr)
609 | completed = false;
610 | break;
611 | end
612 | if (out(i,j,min(size(out,3),k+1)) < out(i,j,k)-thr)
613 | completed = false;
614 | break;
615 | end
616 | else
617 | if (out(i,j,max(1,k-1)) < out(i,j,k)-thr)
618 | completed = false;
619 | break;
620 | end
621 | if (out(i,j,min(size(out,3),k+1)) > out(i,j,k)+thr)
622 | completed = false;
623 | break;
624 | end
625 | end
626 |
627 | end
628 | end
629 | end
630 |
631 |
632 |
633 |
634 | if (completed)
635 | it
636 | break;
637 | end
638 |
639 |
640 |
641 |
642 | end
643 |
644 | end
645 |
646 |
647 | function [out] = TrilinearInterpolation(angle, reproj, n_cam, uncertainty)
648 |
649 | if (reproj > max(uncertainty.reproj_j(2,:)))
650 | out = 1;
651 | return;
652 | end
653 |
654 | angle_i = mean(uncertainty.angle_i);
655 | reproj_j = mean(uncertainty.reproj_j);
656 | camera_k = uncertainty.camera_k;
657 |
658 | [angle_lb, i_lb] = min(angle_i);
659 | [angle_ub, i_ub] = max(angle_i);
660 | for i = 1:length(angle_i)
661 | angle_candidate = angle_i(i);
662 | if (angle-angle_candidate >= 0 && angle_candidate > angle_lb)
663 | angle_lb = angle_candidate;
664 | i_lb = i;
665 | end
666 | if (angle-angle_candidate <= 0 && angle_candidate < angle_ub)
667 | angle_ub = angle_candidate;
668 | i_ub = i;
669 | end
670 | end
671 |
672 | [reproj_lb, j_lb] = min(reproj_j);
673 | [reproj_ub, j_ub] = max(reproj_j);
674 | for j = 1:length(reproj_j)
675 | reproj_candidate = reproj_j(j);
676 | if (reproj-reproj_candidate >= 0 && reproj_candidate > reproj_lb)
677 | reproj_lb = reproj_candidate;
678 | j_lb = j;
679 | end
680 | if (reproj-reproj_candidate <= 0 && reproj_candidate < reproj_ub)
681 | reproj_ub = reproj_candidate;
682 | j_ub = j;
683 | end
684 | end
685 |
686 | [n_cam_lb, k_lb] = min(camera_k);
687 | [n_cam_ub, k_ub] = max(camera_k);
688 | for k = 1:length(camera_k)
689 | n_cam_candidate = camera_k(k);
690 | if (n_cam-n_cam_candidate >= 0 && n_cam_candidate > n_cam_lb)
691 | n_cam_lb = n_cam_candidate;
692 | k_lb = k;
693 | end
694 | if (n_cam-n_cam_candidate <= 0 && n_cam_candidate < n_cam_ub)
695 | n_cam_ub = n_cam_candidate;
696 | k_ub = k;
697 | end
698 | end
699 |
700 |
701 | if (angle_ub == angle_lb)
702 | x_d = 0;
703 | else
704 | x_d = (angle-angle_lb)/(angle_ub-angle_lb);
705 | end
706 | if (reproj_ub == reproj_lb)
707 | y_d = 0;
708 | else
709 | y_d = (reproj-reproj_lb)/(reproj_ub-reproj_lb);
710 | end
711 | if (n_cam_ub == n_cam_lb)
712 | z_d = 0;
713 | else
714 | z_d = (n_cam-n_cam_lb)/(n_cam_ub-n_cam_lb);
715 | end
716 |
717 | c_000 = uncertainty.result(i_lb,j_lb,k_lb);
718 | c_001 = uncertainty.result(i_lb,j_lb,k_ub);
719 | c_010 = uncertainty.result(i_lb,j_ub,k_lb);
720 | c_011 = uncertainty.result(i_lb,j_ub,k_ub);
721 | c_100 = uncertainty.result(i_ub,j_lb,k_lb);
722 | c_101 = uncertainty.result(i_ub,j_lb,k_ub);
723 | c_110 = uncertainty.result(i_ub,j_ub,k_lb);
724 | c_111 = uncertainty.result(i_ub,j_ub,k_ub);
725 |
726 | c_00 = c_000*(1-x_d)+c_100*x_d;
727 | c_01 = c_001*(1-x_d)+c_101*x_d;
728 | c_10 = c_010*(1-x_d)+c_110*x_d;
729 | c_11 = c_011*(1-x_d)+c_111*x_d;
730 |
731 | c_0 = c_00*(1-y_d)+c_10*y_d;
732 | c_1 = c_01*(1-y_d)+c_11*y_d;
733 |
734 | out = c_0*(1-z_d)+c_1*z_d;
735 |
736 | % disp(['angle = ', num2str(angle), ', lb = ', num2str(angle_lb), ', ub = ', num2str(angle_ub), ' angle(i_lb) = ', num2str(angle_i(i_lb)), ', angle(i_ub) = ', num2str(angle_i(i_ub))])
737 | % disp(['reproj = ', num2str(reproj), ', lb = ', num2str(reproj_lb), ', ub = ', num2str(reproj_ub), ' reproj(j_lb) = ', num2str(reproj_j(j_lb)), ', reproj(j_ub) = ', num2str(reproj_j(j_ub))])
738 | % disp(['n_cam = ', num2str(n_cam), ', lb = ', num2str(n_cam_lb), ', ub = ', num2str(n_cam_ub), ' camera_k(k_lb) = ', num2str(camera_k(k_lb)), ', camera_k(k_ub) = ', num2str(camera_k(k_ub))])
739 |
740 |
741 | end
--------------------------------------------------------------------------------
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--------------------------------------------------------------------------------
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--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
1 | # RobustUncertaintyAwareMultiviewTriangulation
2 | MATLAB implementation of our multiview triangulation method proposed in "Robust Uncertainty-Aware Multiview Triangulation" ([arXiv](https://arxiv.org/abs/2008.01258)).
3 |
4 | #### Instruction to run the triangulation code:
5 |
6 | 1. Download the 3D uncertainty grid (learned from the simulations): [uncertainty.mat](https://github.com/sunghoon031/RobustUncertaintyAwareMultiviewTriangulation/blob/master/uncertainty.mat).
7 | 2. In the same folder, download the [script](https://github.com/sunghoon031/RobustUncertaintyAwareMultiviewTriangulation/blob/master/RobustUncertaintyAwareMultiviewTriangulation_ReleaseCode.m).
8 | 3. Run the script on Matlab.
9 |
10 | #### Instruction to run the uncertainty generation code:
11 |
12 | 1. Open GenerateUncertaintyData.m
13 | 2. Set `load_precomputed_results = false;`
14 | 3. Set `n_simulations` to the number of simulations you want to run for each number of cameras in `n_cameras`.
15 | 4. Run. If you use the default `n_simulations`, it will take a very long time to finish (possibly, more than a few days)...
16 | 5. This will create two files, `uncertainty.mat` and `results_inliers.mat`. **Caution: this will overwrite the original `uncertainty.mat`! Make sure you rename the filename if you want to avoid this!**
17 | 6. Set `load_precomputed_results = true;` and run. Then it will plot some of the results.
18 |
--------------------------------------------------------------------------------
/RobustUncertaintyAwareMultiviewTriangulation_ReleaseCode.m:
--------------------------------------------------------------------------------
1 | clc;
2 |
3 | %% Load the 3D uncertainty grid
4 | load uncertainty.mat;
5 |
6 | %% World point to be triangulated
7 | dist = 2+rand(1)*8; % between 2 and 10 unit
8 | point_w = [0;0;dist];
9 |
10 | %% Set up the cameras (50 cameras with inlying measurements + 50 cameras with outlying measurements)
11 | focal_length = 525;
12 | img_width = 640;
13 | img_height = 480;
14 |
15 | noise_level = 3; % pixel (Gaussian noise level)
16 | outlier_min_noise = 10; % pixel (Outliers contain noise greater than this)
17 |
18 | n_inliers = 50;
19 | n_outliers = 50;
20 | n_cameras = n_inliers + n_outliers;
21 |
22 | [R, t, K, uv_measured] = SimulateCameras(point_w, n_inliers, n_cameras, img_width, img_height, focal_length, noise_level, outlier_min_noise);
23 |
24 | %% Triangulate the point using our method.
25 |
26 | % Precomputation indepndent of the point
27 | tic;
28 | [cam_pos, P, A, M3, M4, M5] = Precompute(n_cameras, K, R, t);
29 | time_precompute = toc;
30 |
31 | % Gauss-Newton optimization:
32 | tic;
33 | [x_gn, uncertainty_3D_gn] = RunMultiviewTriangulation('GN', uncertainty, cam_pos, R,P,K,uv_measured,A, M3, M4, M5);
34 | time_gn = toc;
35 |
36 | % DLT optimization:
37 | tic;
38 | [x_dlt, uncertainty_3D_dlt] = RunMultiviewTriangulation('DLT', uncertainty, cam_pos, R,P,K,uv_measured,A, M3, M4, M5);
39 | time_dlt = toc;
40 |
41 |
42 | %% Evaluate the performance
43 |
44 | error_2D_gn = GetMean2DErrorOfTrueInliers(x_gn, n_cameras, n_inliers, K, P, uv_measured);
45 | error_2D_dlt = GetMean2DErrorOfTrueInliers(x_dlt, n_cameras, n_inliers, K, P, uv_measured);
46 |
47 | error_3D_gn = norm(x_gn-point_w);
48 | error_3D_dlt = norm(x_dlt-point_w);
49 |
50 | disp(['Point distance = ', num2str(dist), ' unit (1 unit = geometric span of the cameras).'])
51 | disp(['Number of cameras = ', num2str(n_cameras), ', Outlier ratio = ', num2str(n_outliers/n_cameras*100), '%.'])
52 | disp(['Precomputation took ', num2str(time_precompute*1000), ' ms.'])
53 | disp(['GN optimization: 2D error = ', num2str(error_2D_gn), ' pix, 3D error = ', num2str(error_3D_gn), ' unit, uncertainty = ', num2str(uncertainty_3D_gn), ', took ', num2str(time_gn*1000), ' ms.'])
54 | disp(['DLT optimization: 2D error = ', num2str(error_2D_dlt), ' pix, 3D error = ', num2str(error_3D_dlt), ' unit, uncertainty = ', num2str(uncertainty_3D_dlt), ', took ', num2str(time_dlt*1000), ' ms.'])
55 |
56 |
57 |
58 |
59 |
60 |
61 |
62 | %% Function definitions
63 |
64 | function R = RandomRotation(max_angle_rad)
65 |
66 | unit_axis = rand(3,1)-0.5;
67 | unit_axis = unit_axis/norm(unit_axis);
68 | angle = rand*max_angle_rad;
69 | R = RotationFromUnitAxisAngle(unit_axis, angle);
70 |
71 | end
72 |
73 | function R = RotationFromUnitAxisAngle(unit_axis, angle)
74 |
75 | if (angle==0)
76 | R = eye(3);
77 | else
78 | so3 = SkewSymmetricMatrix(unit_axis);
79 | R = eye(3)+so3*sin(angle)+so3^2*(1-cos(angle));
80 | end
81 | end
82 |
83 |
84 | function out = SkewSymmetricMatrix(in)
85 | out=[0 -in(3) in(2) ; in(3) 0 -in(1) ; -in(2) in(1) 0 ];
86 | end
87 |
88 | function [R, t, K, uv_measured] = SimulateCameras(point_w, n_inlier, n_cameras, img_width, img_height, focal_length, noise_level, outlier_min_noise)
89 |
90 | K_cam = [focal_length 0 0; 0 focal_length 0; 0 0 1];
91 |
92 |
93 | R = cell(1,n_cameras);
94 | t = cell(1,n_cameras);
95 | K = cell(1,n_cameras);
96 | uv_measured = cell(1,n_cameras);
97 |
98 | for i = 1:n_cameras
99 |
100 | K{i} = K_cam;
101 |
102 | % Camera is located randomly inside a unit sphere.
103 | cam_pos = rand(3,1)-0.5;
104 | cam_pos = rand(1)*0.5*cam_pos/norm(cam_pos);
105 |
106 | % The first two cameras are antipodal.
107 | if (i==1)
108 | cam_pos = rand(3,1)-0.5;
109 | cam_pos = 0.5*cam_pos/norm(cam_pos);
110 | elseif (i==2)
111 | cam_pos = -cam_pos;
112 | end
113 |
114 | while(true)
115 | R_temp = RandomRotation(pi);
116 | t_temp = -R_temp*cam_pos;
117 |
118 | point_c = R_temp*point_w+t_temp;
119 | uv = K{i}*[point_c(1)/point_c(3); point_c(2)/point_c(3);1];
120 | uv = uv(1:2);
121 | if (abs(uv(1)) > img_width/2 || abs(uv(2)) > img_height/2 || point_c(3) < 0)
122 | continue;
123 | end
124 |
125 | if (i <= n_inlier)
126 | while(true)
127 | noise_dir = rand(2,1)-0.5;
128 | noise_dir = noise_dir/norm(noise_dir);
129 | uv_with_noise = uv+normrnd(0, noise_level)*noise_dir;
130 | if (abs(uv_with_noise(1)) > img_width/2 || abs(uv_with_noise(2)) > img_height/2)
131 | continue;
132 | end
133 | break;
134 | end
135 | else
136 | while(true)
137 | uv_with_noise = [img_width*(rand(1)-0.5); img_height*(rand(1)-0.5)];
138 | noise_magnitude = norm(uv-uv_with_noise);
139 | if (noise_magnitude < outlier_min_noise)
140 | continue;
141 | end
142 | break;
143 | end
144 | end
145 |
146 | break;
147 | end
148 |
149 | R{i} = R_temp;
150 | t{i} = t_temp;
151 | uv_measured{i} = uv_with_noise;
152 | end
153 | end
154 |
155 |
156 | function [cam_pos, P, A, M3, M4, M5] = Precompute(n_cameras, K, R, t)
157 |
158 |
159 | cam_pos = cell(1,n_cameras);
160 | P = cell(1,n_cameras);
161 | A = cell(1,n_cameras); % necessary for computing the Jacobian
162 | M3 = nan(4,n_cameras);
163 | M4 = nan(4,n_cameras);
164 | M5 = nan(4,n_cameras);
165 |
166 | for i = 1:n_cameras
167 | r11 = R{i}(1,1); r12 = R{i}(1,2); r13 = R{i}(1,3);
168 | r21 = R{i}(2,1); r22 = R{i}(2,2); r23 = R{i}(2,3);
169 | r31 = R{i}(3,1); r32 = R{i}(3,2); r33 = R{i}(3,3);
170 | tx = t{i}(1); ty = t{i}(2); tz = t{i}(3);
171 | k11 = K{i}(1,1); k12 = K{i}(1,2); k21 = K{i}(2,1); k22 = K{i}(2,2);
172 |
173 | b1 = r11*[0;r32;r33;tz]-r31*[0;r12;r13;tx];
174 | b2 = r21*[0;r32;r33;tz]-r31*[0;r22;r23;ty];
175 | b3 = r12*[r31;0;r33;tz]-r32*[r11;0;r13;tx];
176 | b4 = r22*[r31;0;r33;tz]-r32*[r21;0;r23;ty];
177 | b5 = r13*[r31;r32;0;tz]-r33*[r11;r12;0;tx];
178 | b6 = r23*[r31;r32;0;tz]-r33*[r21;r22;0;ty];
179 | a1 = k11*b1+k12*b2;
180 | a2 = k21*b1+k22*b2;
181 | a3 = k11*b3+k12*b4;
182 | a4 = k21*b3+k22*b4;
183 | a5 = k11*b5+k12*b6;
184 | a6 = k21*b5+k22*b6;
185 | A{i} = [a1,a2,a3,a4,a5,a6];
186 |
187 | cam_pos{i} = -R{i}'*t{i};
188 | P{i} = [R{i}, t{i}];
189 |
190 | M3(:,i) = K{i}(1,1)*P{i}(1,:)'+K{i}(1,2)*P{i}(2,:)';
191 | M4(:,i) = K{i}(2,1)*P{i}(1,:)'+K{i}(2,2)*P{i}(2,:)';
192 | M5(:,i) = P{i}(3,:)';
193 | end
194 |
195 | end
196 |
197 |
198 |
199 | function [output_pairs] = GetSamplePairs(input_vec, max_n_samples, bool_random_forced)
200 | n_sample = length(input_vec);
201 | possible_n_pairs = n_sample*(n_sample-1)/2;
202 | if(~bool_random_forced && possible_n_pairs <= max_n_samples)
203 | output_pairs = zeros(2,possible_n_pairs);
204 | c = 0;
205 | for i = input_vec
206 | for j = input_vec
207 | if (i<=j)
208 | continue;
209 | end
210 | c = c + 1;
211 | output_pairs(:,c) = [i,j];
212 | end
213 | end
214 | else
215 | max_n_samples = min(max_n_samples, possible_n_pairs);
216 | output_pairs = zeros(2,max_n_samples);
217 | idx_shuffled = input_vec(randperm(n_sample));
218 | c = 0;
219 | gap = 1;
220 | sample1_idx = 0;
221 | while (c n_sample || sample2_idx > n_sample)
225 | sample1_idx = 0;
226 | gap = gap + 1;
227 | continue;
228 | end
229 | sample1 = idx_shuffled(sample1_idx);
230 | sample2 = idx_shuffled(sample2_idx);
231 | c = c + 1;
232 | output_pairs(:,c) = [sample1,sample2];
233 | end
234 | end
235 | end
236 |
237 | function error_2D = GetMean2DErrorOfTrueInliers(x, n_cameras, n_inlier, K, P, uv_measured)
238 |
239 | M1 = nan(1,n_cameras);
240 | M2 = nan(1,n_cameras);
241 | M3 = nan(4,n_cameras);
242 | M4 = nan(4,n_cameras);
243 | M5 = nan(4,n_cameras);
244 | for i = 1:n_cameras
245 | M1(1,i) = K{i}(1,3)-uv_measured{i}(1);
246 | M2(1,i) = K{i}(2,3)-uv_measured{i}(2);
247 | M3(:,i) = K{i}(1,1)*P{i}(1,:)'+K{i}(1,2)*P{i}(2,:)';
248 | M4(:,i) = K{i}(2,1)*P{i}(1,:)'+K{i}(2,2)*P{i}(2,:)';
249 | M5(:,i) = P{i}(3,:)';
250 | end
251 |
252 | M6 = [x;1]'*M5;
253 | M7 = M1+([x;1]'*M3)./M6;
254 | M8 = M2+([x;1]'*M4)./M6;
255 | squared_reproj_error_all = M7.*M7+M8.*M8;
256 |
257 | error_2D = 0;
258 | c = 0;
259 | for i = 1:n_inlier
260 | if (M6(i) <= 0)
261 | continue;
262 | end
263 | error_2D = error_2D + sqrt(squared_reproj_error_all(i));
264 | c = c + 1;
265 | end
266 | error_2D = error_2D/c;
267 |
268 | if (error_2D == 0)
269 | error_2D = nan;
270 | end
271 | end
272 |
273 | function error_2D = GetMean2DErrorOfEstimatedInliers(x, M1, M2, M3, M4, M5, idx_inlier)
274 | M6 = [x;1]'*M5;
275 | M7 = M1+([x;1]'*M3)./M6;
276 | M8 = M2+([x;1]'*M4)./M6;
277 | squared_reproj_error_all = M7.*M7+M8.*M8;
278 |
279 | error_2D = 0;
280 | c = 0;
281 | for i = idx_inlier
282 | if (M6(i) <= 0)
283 | continue;
284 | end
285 | error_2D = error_2D + sqrt(squared_reproj_error_all(i));
286 | c = c + 1;
287 | end
288 | error_2D = error_2D/c;
289 |
290 | if (error_2D == 0)
291 | error_2D = nan;
292 | end
293 | end
294 |
295 |
296 | function [min_level_of_degeneracy] = GetMinLevelOfDegeneracy(cam_pos, ray_w, K, uv_measured, R, idx, n_sample_mLoD)
297 |
298 | min_level_of_degeneracy = inf;
299 | idx = reshape(idx, [1, numel(idx)]);
300 |
301 | sample_pairs = GetSamplePairs(idx, n_sample_mLoD, false);
302 |
303 | for k = 1:size(sample_pairs,2)
304 | i = sample_pairs(1,k);
305 | j = sample_pairs(2,k);
306 |
307 | ray_i = ray_w(:,i);
308 | if (ray_i(1)==0 && ray_i(2)==0 && ray_i(3)==0)
309 | ray_i = K{i}\[uv_measured{i};1];
310 | ray_i = ray_i/norm(ray_i);
311 | ray_i = R{i}'*ray_i;
312 | ray_w(:,i) = ray_i;
313 | end
314 |
315 | ray_j = ray_w(:,j);
316 | if (ray_j(1)==0 && ray_j(2)==0 && ray_j(3)==0)
317 | ray_j = K{j}\[uv_measured{j};1];
318 | ray_j = ray_j/norm(ray_j);
319 | ray_j = R{j}'*ray_j;
320 | ray_w(:,j) = ray_j;
321 | end
322 |
323 | t_ij = cam_pos{i}-cam_pos{j};
324 | t_ij = t_ij/norm(t_ij);
325 | f_i = ray_i;
326 | f_j = ray_j;
327 |
328 | level_of_degeneracy = max(abs([t_ij'*f_i, t_ij'*f_j, f_i'*f_j]));
329 |
330 | if (level_of_degeneracy < min_level_of_degeneracy)
331 | min_level_of_degeneracy = level_of_degeneracy;
332 | end
333 | end
334 |
335 | end
336 |
337 |
338 |
339 | function [x_est, idx_consensus, max_n_consensus, min_cost, k_max, ray_w] = TwoViewRansac(j, k, x_est, idx_consensus, max_n_consensus, min_cost, k_max, confidence, n_cameras, epipolar_thr, cos_angle_thr, reproj_thr, M1, M2, M3, M4, M5, K, R, P, cam_pos, uv_measured,ray_w)
340 |
341 | ray_j = ray_w(:,j);
342 | if (ray_j(1)==0 && ray_j(2)==0 && ray_j(3)==0)
343 | ray_j = K{j}\[uv_measured{j};1];
344 | ray_j = ray_j/norm(ray_j);
345 | ray_j = R{j}'*ray_j;
346 |
347 | ray_w(:,j) = ray_j;
348 | end
349 |
350 | ray_k = ray_w(:,k);
351 | if (ray_k(1)==0 && ray_k(2)==0 && ray_k(3)==0)
352 | ray_k = K{k}\[uv_measured{k};1];
353 | ray_k = ray_k/norm(ray_k);
354 | ray_k = R{k}'*ray_k;
355 |
356 | ray_w(:,k) = ray_k;
357 | end
358 |
359 |
360 |
361 |
362 | % Epipolar error should be below threshold
363 | t_jk = cam_pos{j}-cam_pos{k};
364 | t_norm = norm(t_jk);
365 | t_jk = t_jk/t_norm;
366 | cross_fj_fk = [...
367 | ray_j(2)*ray_k(3)-ray_j(3)*ray_k(2);...
368 | ray_j(3)*ray_k(1)-ray_j(1)*ray_k(3);...
369 | ray_j(1)*ray_k(2)-ray_j(2)*ray_k(1)];
370 | e_epipolar = abs(t_jk'*cross_fj_fk);
371 |
372 | if (e_epipolar > epipolar_thr)
373 | return;
374 | end
375 |
376 | % Parallax must be below 90 deg, above threshold.
377 | p_jk = ray_j'*ray_k;
378 | if (p_jk < 0 || p_jk > cos_angle_thr)
379 | %disp(['Skip I:' num2str(i), ', ', num2str(j)])
380 | return;
381 | end
382 |
383 | % Point must be far from the epipoles
384 | q_jk = t_jk'*ray_j;
385 | r_jk = t_jk'*ray_k;
386 | if (abs(q_jk) > cos_angle_thr || abs(r_jk) > cos_angle_thr)
387 | %disp(['Skip II:' num2str(i), ', ', num2str(j)])
388 | return;
389 | end
390 |
391 | % Midpoint should satisfy the cheirality.
392 | tau_i = p_jk*r_jk-q_jk;
393 | tau_j = -p_jk*q_jk+r_jk;
394 | if (tau_i < 0 || tau_j < 0)
395 | %disp(['Skip III:' num2str(i), ', ', num2str(j)])
396 | return;
397 | end
398 |
399 | % Obtain the midpoint.
400 | s_jk = t_norm/(1-p_jk'*p_jk);
401 | lambda_j = s_jk*tau_i;
402 | lambda_k = s_jk*tau_j;
403 | x_anchor_j = cam_pos{j}+lambda_j*ray_j;
404 | x_anchor_k= cam_pos{k}+lambda_k*ray_k;
405 | x_mid = 0.5*(x_anchor_j+x_anchor_k);
406 |
407 |
408 | % Cheirality
409 |
410 | x_j = P{j}*[x_mid;1];
411 | x_k = P{k}*[x_mid;1];
412 |
413 | if (x_j(3) < 0 || x_k(3) < 0)
414 | %disp(['Skip III:' num2str(i), ', ', num2str(j)])
415 | return;
416 | end
417 |
418 | % Small reprojection error
419 | u_j = uv_measured{j}(1); v_j = uv_measured{j}(2);
420 | u_k = uv_measured{k}(1); v_k = uv_measured{k}(2);
421 |
422 | uv_reproj_j = K{j}*x_j/x_j(3);
423 | uv_error_j = [u_j;v_j;1]-uv_reproj_j;
424 | reproj_error_j = uv_error_j'*uv_error_j;
425 | if (reproj_error_j > reproj_thr)
426 | return;
427 | end
428 |
429 | uv_reproj_k = K{k}*x_k/x_k(3);
430 | uv_error_k = [u_k;v_k;1]-uv_reproj_k;
431 | reproj_error_k = uv_error_k'*uv_error_k;
432 | if (reproj_error_k > reproj_thr)
433 | return;
434 | end
435 |
436 | % Count the consensus.
437 | M6 = [x_mid;1]'*M5;
438 | M7 = M1 + ([x_mid;1]'*M3)./M6;
439 | M8 = M2 + ([x_mid;1]'*M4)./M6;
440 | squared_reproj_error_all = M7.*M7+M8.*M8;
441 | squared_reproj_error_all(M6<=0) = inf;
442 |
443 | cost = 0;
444 | for ii = 1:length(squared_reproj_error_all)
445 | cost = cost + min(squared_reproj_error_all(ii), reproj_thr);
446 | end
447 |
448 | bool_consensus = squared_reproj_error_all 0;
539 |
540 | if (bool_consensus_changed)
541 | c = 0;
542 | else
543 | c = c +1;
544 | end
545 |
546 | if (sum(bool_consensus) < 2)
547 | break;
548 | end
549 |
550 | if (c == 2 && abs(mean_reproj_error-mean_reproj_error_prev) < 0.1)
551 | break;
552 | end
553 |
554 | residuals_new = [];
555 | J_est = [];
556 |
557 | for i = 1:n_cameras
558 | if (bool_consensus(i))
559 | residuals_new = [residuals_new; residuals(:,i)];
560 |
561 | J_temp = A{i}'*[x_gn;1];
562 | J_temp = reshape(J_temp, [2,3])/((P{i}(3,:)*[x_gn;1])^2);
563 | J_est = [J_est; J_temp];
564 | end
565 | end
566 |
567 | update = -pinv(J_est)*residuals_new;
568 |
569 | x_gn = x_gn+update;
570 |
571 | end
572 |
573 | x_est = x_gn;
574 | idx_consensus = find(bool_consensus);
575 | max_n_consensus = sum(bool_consensus);
576 | end
577 |
578 | function [out] = TrilinearInterpolation(angle, reproj, n_cam, uncertainty)
579 |
580 | if (reproj > max(uncertainty.reproj_j(2,:)))
581 | out = 1;
582 | return;
583 | end
584 |
585 | angle_i = mean(uncertainty.angle_i);
586 | reproj_j = mean(uncertainty.reproj_j);
587 | camera_k = uncertainty.camera_k;
588 |
589 | [angle_lb, i_lb] = min(angle_i);
590 | [angle_ub, i_ub] = max(angle_i);
591 | for i = 1:length(angle_i)
592 | angle_candidate = angle_i(i);
593 | if (angle-angle_candidate >= 0 && angle_candidate > angle_lb)
594 | angle_lb = angle_candidate;
595 | i_lb = i;
596 | end
597 | if (angle-angle_candidate <= 0 && angle_candidate < angle_ub)
598 | angle_ub = angle_candidate;
599 | i_ub = i;
600 | end
601 | end
602 |
603 | [reproj_lb, j_lb] = min(reproj_j);
604 | [reproj_ub, j_ub] = max(reproj_j);
605 | for j = 1:length(reproj_j)
606 | reproj_candidate = reproj_j(j);
607 | if (reproj-reproj_candidate >= 0 && reproj_candidate > reproj_lb)
608 | reproj_lb = reproj_candidate;
609 | j_lb = j;
610 | end
611 | if (reproj-reproj_candidate <= 0 && reproj_candidate < reproj_ub)
612 | reproj_ub = reproj_candidate;
613 | j_ub = j;
614 | end
615 | end
616 |
617 | [n_cam_lb, k_lb] = min(camera_k);
618 | [n_cam_ub, k_ub] = max(camera_k);
619 | for k = 1:length(camera_k)
620 | n_cam_candidate = camera_k(k);
621 | if (n_cam-n_cam_candidate >= 0 && n_cam_candidate > n_cam_lb)
622 | n_cam_lb = n_cam_candidate;
623 | k_lb = k;
624 | end
625 | if (n_cam-n_cam_candidate <= 0 && n_cam_candidate < n_cam_ub)
626 | n_cam_ub = n_cam_candidate;
627 | k_ub = k;
628 | end
629 | end
630 |
631 |
632 | if (angle_ub == angle_lb)
633 | x_d = 0;
634 | else
635 | x_d = (angle-angle_lb)/(angle_ub-angle_lb);
636 | end
637 | if (reproj_ub == reproj_lb)
638 | y_d = 0;
639 | else
640 | y_d = (reproj-reproj_lb)/(reproj_ub-reproj_lb);
641 | end
642 | if (n_cam_ub == n_cam_lb)
643 | z_d = 0;
644 | else
645 | z_d = (n_cam-n_cam_lb)/(n_cam_ub-n_cam_lb);
646 | end
647 |
648 | c_000 = uncertainty.result(i_lb,j_lb,k_lb);
649 | c_001 = uncertainty.result(i_lb,j_lb,k_ub);
650 | c_010 = uncertainty.result(i_lb,j_ub,k_lb);
651 | c_011 = uncertainty.result(i_lb,j_ub,k_ub);
652 | c_100 = uncertainty.result(i_ub,j_lb,k_lb);
653 | c_101 = uncertainty.result(i_ub,j_lb,k_ub);
654 | c_110 = uncertainty.result(i_ub,j_ub,k_lb);
655 | c_111 = uncertainty.result(i_ub,j_ub,k_ub);
656 |
657 | c_00 = c_000*(1-x_d)+c_100*x_d;
658 | c_01 = c_001*(1-x_d)+c_101*x_d;
659 | c_10 = c_010*(1-x_d)+c_110*x_d;
660 | c_11 = c_011*(1-x_d)+c_111*x_d;
661 |
662 | c_0 = c_00*(1-y_d)+c_10*y_d;
663 | c_1 = c_01*(1-y_d)+c_11*y_d;
664 |
665 | out = c_0*(1-z_d)+c_1*z_d;
666 |
667 | % disp(['angle = ', num2str(angle), ', lb = ', num2str(angle_lb), ', ub = ', num2str(angle_ub), ' angle(i_lb) = ', num2str(angle_i(i_lb)), ', angle(i_ub) = ', num2str(angle_i(i_ub))])
668 | % disp(['reproj = ', num2str(reproj), ', lb = ', num2str(reproj_lb), ', ub = ', num2str(reproj_ub), ' reproj(j_lb) = ', num2str(reproj_j(j_lb)), ', reproj(j_ub) = ', num2str(reproj_j(j_ub))])
669 | % disp(['n_cam = ', num2str(n_cam), ', lb = ', num2str(n_cam_lb), ', ub = ', num2str(n_cam_ub), ' camera_k(k_lb) = ', num2str(camera_k(k_lb)), ', camera_k(k_ub) = ', num2str(camera_k(k_ub))])
670 |
671 |
672 | end
673 |
674 |
675 |
676 |
677 |
678 | function [x_est, uncertainty_3D] = RunMultiviewTriangulation(method, uncertainty, cam_pos, R,P,K,uv_measured,A, M3, M4, M5)
679 |
680 | %% 1. Initialize
681 |
682 | uncertainty_3D = nan;
683 | x_est = nan;
684 | idx_consensus = [];
685 |
686 | n_cameras = numel(cam_pos);
687 |
688 | % Construct matrix M1 and M2
689 | M1 = nan(1,n_cameras);
690 | M2 = nan(1,n_cameras);
691 | for i = 1:n_cameras
692 | M1(1,i) = K{i}(1,3)-uv_measured{i}(1);
693 | M2(1,i) = K{i}(2,3)-uv_measured{i}(2);
694 | end
695 |
696 |
697 | ray_w = zeros(3,n_cameras);
698 |
699 |
700 |
701 | %% 2. Run two-view RANSAC
702 |
703 | confidence = 0.99;
704 | cos_angle_thr = cosd(4);
705 | reproj_thr = 10^2;
706 | epipolar_thr = 0.01;
707 |
708 | n_sample = min(10000, n_cameras*(n_cameras-1)/2);
709 | n_sample_mLoD = 100;
710 | n_consensus_thr = 2;
711 |
712 | min_cost = inf;
713 | inlier_ratio_est = 3/n_cameras;
714 | k_max = log(1-confidence)/log(1-inlier_ratio_est^2);
715 | max_n_consensus = 0;
716 |
717 | sample_pairs = GetSamplePairs(1:n_cameras, inf, true);
718 |
719 | for k = 1:size(sample_pairs,2)
720 | if (k == n_sample)
721 | break;
722 | end
723 | if (k > k_max && max_n_consensus >= n_consensus_thr)
724 | break;
725 | end
726 | i = sample_pairs(1,k);
727 | j = sample_pairs(2,k);
728 | [x_est, idx_consensus, max_n_consensus, min_cost, k_max, ray_w] = TwoViewRansac(i, j, x_est, idx_consensus, max_n_consensus, min_cost, k_max, confidence, n_cameras, epipolar_thr, cos_angle_thr, reproj_thr, M1, M2, M3, M4, M5, K, R, P, cam_pos, uv_measured,ray_w);
729 |
730 | end
731 |
732 | if (max_n_consensus < n_consensus_thr)
733 | disp('Not enough inliers!')
734 | return;
735 | end
736 |
737 |
738 | %% 3. Refine the initial solution and the inlier set:
739 | switch method
740 | case 'GN'
741 | [x_est, idx_consensus, max_n_consensus] = GaussNewton(x_est, idx_consensus, A, P, n_cameras, reproj_thr, M1, M2, M3, M4, M5);
742 | case 'DLT'
743 | [x_est, idx_consensus, max_n_consensus] = DLT(x_est, idx_consensus, n_cameras, reproj_thr, M1, M2, M3, M4, M5, K, P, uv_measured);
744 | end
745 |
746 | %% 4. Estimate the 3D uncertainty:
747 | error_2D_estimated_inliers = GetMean2DErrorOfEstimatedInliers(x_est, M1, M2, M3, M4, M5, idx_consensus);
748 | mLoD_approx = GetMinLevelOfDegeneracy(cam_pos, ray_w, K, uv_measured, R, idx_consensus, n_sample_mLoD);
749 | uncertainty_3D = TrilinearInterpolation(acosd(mLoD_approx), error_2D_estimated_inliers, max_n_consensus, uncertainty);
750 |
751 |
752 | end
753 |
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