Calibration of Double-Plane-Mirror Catadioptric Camera Based on Coaxial Parallel Circles

<div>Solving the orthogonal vanishing points <span class="nowrap"><svg height="11.3463pt" id="M356" style="vertical-align:-5.52893pt" version="1.1" viewbox="-0.0498162 -5.81737 10.5931 11.3463" width="10.5931pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C358 401 362 388 351 355C329 286 296 195 278 149C262 190 226 288 200 358C188 393 193 400 239 410V442H-8V410C40 400 53 388 71 344C120 229 165 113 215 -11H254C315 141 364 261 400 342C421 389 433 400 483 410V442Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.045,3.132)"><path d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z"></path></g></svg>,</span><svg height="14.8173pt" id="M357" style="vertical-align:-5.52897pt" version="1.1" viewbox="-0.0498162 -9.28833 62.4182 14.8173" width="62.4182pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C358 401 362 388 351 355C329 286 296 195 278 149C262 190 226 288 200 358C188 393 193 400 239 410V442H-8V410C40 400 53 388 71 344C120 229 165 113 215 -11H254C315 141 364 261 400 342C421 389 433 400 483 410V442Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.28,0)"><path d="M295 534L291 558C277 573 202 603 180 597L75 72L120 51L295 534Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.7,3.132)"><path d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.126,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.624,0)"><path d="M394 607C394 634 378 655 350 655C315 655 294 620 294 594C294 564 314 546 339 546C372 546 394 574 394 607ZM360 349C373 414 374 448 351 448C321 448 249 404 183 338L197 313C223 336 265 367 276 367C285 367 283 355 275 313C241 129 220 23 189 -102C175 -158 153 -197 124 -197C106 -197 85 -187 70 -177C64 -173 51 -177 44 -183C34 -193 23 -209 23 -228C23 -244 46 -261 67 -261C86 -261 127 -245 181 -197C237 -147 279 -52 303 66L360 349Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.756,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,40.019,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,46.259,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,51.402,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,57.642,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg>, ICPs <svg height="9.10325pt" id="M358" style="vertical-align:-3.181499pt" version="1.1" viewbox="-0.0498162 -5.92175 15.5621 9.10325" width="15.5621pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M823 0V32C771 40 767 43 767 100V293C767 395 719 454 632 454C603 454 576 441 554 426C530 409 507 394 482 376C464 421 424 454 368 454C339 454 312 442 281 422C259 409 240 395 218 379V456C157 439 89 425 42 418V389C89 382 93 373 93 315V105C93 42 88 39 30 32V0H265V32C227 38 220 43 220 100V344C248 366 273 376 297 376C334 376 366 354 366 282V100C366 41 362 38 315 32V0H546V32C501 38 494 43 494 100V302C494 317 493 331 492 342C524 369 549 376 569 376C601 376 639 359 639 277V105C639 42 632 38 588 32V0H823Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11,3.132)"><path d="M418 650H141L133 618C225 611 232 604 218 531L141 120C126 45 119 37 33 32L24 0H301L308 32C220 37 214 44 227 120L305 531C319 605 325 611 411 618L418 650Z"></path></g></svg> and <span class="nowrap"><svg height="10.4825pt" id="M359" style="vertical-align:-4.56075pt" version="1.1" viewbox="-0.0498162 -5.92175 15.7539 10.4825" width="15.7539pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M823 0V32C771 40 767 43 767 100V293C767 395 719 454 632 454C603 454 576 441 554 426C530 409 507 394 482 376C464 421 424 454 368 454C339 454 312 442 281 422C259 409 240 395 218 379V456C157 439 89 425 42 418V389C89 382 93 373 93 315V105C93 42 88 39 30 32V0H265V32C227 38 220 43 220 100V344C248 366 273 376 297 376C334 376 366 354 366 282V100C366 41 362 38 315 32V0H546V32C501 38 494 43 494 100V302C494 317 493 331 492 342C524 369 549 376 569 376C601 376 639 359 639 277V105C639 42 632 38 588 32V0H823Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11,3.132)"><path d="M434 650H168L162 618C248 609 258 608 243 528L174 155C163 93 153 39 138 1C110 -68 67 -98 24 -111L37 -147C55 -144 75 -136 94 -128C185 -87 232 -12 259 139L328 528C342 605 349 609 427 618L434 650Z"></path></g></svg>,</span> and the pole and polar with respect to the IAC, <svg height="11.1049pt" id="M360" style="vertical-align:-3.29108pt" version="1.1" viewbox="-0.0498162 -7.81382 15.7852 11.1049" width="15.7852pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C358 401 362 388 351 355C329 286 296 195 278 149C262 190 226 288 200 358C188 393 193 400 239 410V442H-8V410C40 400 53 388 71 344C120 229 165 113 215 -11H254C315 141 364 261 400 342C421 389 433 400 483 410V442Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.28,0)"><path d="M295 534L291 558C277 573 202 603 180 597L75 72L120 51L295 534Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.7,3.132)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g></svg> and <span class="nowrap"><svg height="12.509pt" id="M361" style="vertical-align:-3.27287pt" version="1.1" viewbox="-0.0498162 -9.23613 7.57753 12.509" width="7.57753pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M270 0V32C208 38 203 43 203 110V703C146 688 72 674 15 667V637C70 630 75 628 75 567V110C75 45 70 41 10 32V0H270Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,3.628,3.132)"><path d="M319 325C317 349 306 409 297 431C277 440 250 449 209 449C117 449 57 389 57 319C57 243 122 209 182 182C232 159 261 135 261 91C261 48 227 21 190 21C130 21 85 79 68 145L41 140C41 104 51 36 58 22C75 7 121 -12 172 -12C252 -12 337 35 337 126C337 195 286 231 210 262C166 281 126 304 126 348C126 388 152 417 191 417C240 417 274 378 294 318L319 325Z"></path></g></svg>.</span></div>

Journal of Sensors

fig10

Figure 10

Figure 10: Calibration of Double-Plane-Mirror Catadioptric Camera Based on Coaxial Parallel Circles 