A Unified Shape-From-Shading Approach for 3D Surface Reconstruction Using Fast Eikonal Solvers

<div>Reflection geometry of a local surface point. <svg height="6.089pt" id="M8" style="vertical-align:-0.04982042pt" version="1.1" viewbox="-0.0498162 -6.03918 7.48522 6.089" width="7.48522pt" 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="M549 0V32C497 40 491 42 491 105V294C491 394 441 454 360 454C331 454 312 445 282 426C260 412 240 396 220 383V456C161 439 95 425 39 418V390C87 380 92 377 92 317V109C92 42 88 41 26 32V0H267V32C225 38 220 42 220 104V346C246 366 268 377 294 377C330 377 363 355 363 283V113C363 42 358 38 316 32V0H549Z"></path></g></svg> is the unit normal of the surface point <span class="nowrap"><svg height="8.68572pt" id="M9" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 8.15071 8.68572" width="8.15071pt" 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="M600 480C600 590 528 650 384 650H143L137 622C222 614 225 607 210 531L130 127C113 41 106 36 23 28L17 0H294L300 28C204 36 195 42 212 127L243 284L314 263C327 263 339 263 352 264C465 271 600 337 600 480ZM508 481C508 351 402 304 329 304C289 304 265 311 250 317L295 559C302 594 310 606 323 611C335 616 350 619 367 619C455 619 508 573 508 481Z"></path></g></svg>;</span> <svg height="12.7178pt" id="M10" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 36.2987 12.7178" width="36.2987pt" 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="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,-0.013,4.498,0)"><path d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.647,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.045,0)"><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,-0.013,20.188,0)"><path d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,27.208,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.607,0)"><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> and <svg height="12.7178pt" id="M11" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 33.6133 12.7178" width="33.6133pt" 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="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,-0.013,4.498,0)"><path d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.647,3.132)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.703,0)"><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,-0.013,18.846,0)"><path d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,25.866,3.132)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.922,0)"><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> are the camera direction <svg height="8.85534pt" id="M12" style="vertical-align:-0.1150398pt" version="1.1" viewbox="-0.0498162 -8.7403 9.37731 8.85534" width="9.37731pt" 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="M702 650H468V614C550 604 550 590 527 521C500 434 457 318 402 172H399C363 266 284 469 255 549C240 592 240 603 313 614V650H15V614C77 608 93 594 124 517C165 413 259 172 327 -11H377C455 195 546 422 579 500C619 595 637 607 702 614V650Z"></path></g></svg> and incident light direction <span class="nowrap"><svg height="8.79011pt" id="M13" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.7403 7.2895 8.79011" width="7.2895pt" 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="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g></svg>,</span> respectively; <svg height="8.85534pt" id="M14" style="vertical-align:-0.1150398pt" version="1.1" viewbox="-0.0498162 -8.7403 8.71181 8.85534" width="8.71181pt" 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="M660 23C638 31 621 43 599 66C580 87 558 119 523 175C493 221 456 287 441 317C508 344 566 392 566 480C566 542 539 586 499 613C458 640 408 650 320 650H39V615C112 608 121 598 121 518V134C121 51 110 42 35 35V0H337V35C261 42 252 51 252 134V280H271C303 280 321 267 337 237C366 184 393 130 419 89C460 26 498 -2 624 -8C637 -8 648 -9 654 -9L660 23ZM252 320V563C252 584 254 594 261 600C267 606 281 610 306 610C362 610 424 572 424 465C424 414 410 374 376 346C354 327 320 320 278 320H252Z"></path></g></svg> is reflected light direction; <svg height="9.25986pt" id="M15" style="vertical-align:-0.04981041pt" version="1.1" viewbox="-0.0498162 -9.21005 7.27645 9.25986" width="7.27645pt" 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="M531 0V32C480 40 474 41 474 105V288C474 398 423 454 336 454C286 454 237 424 203 390V703C143 688 72 674 19 667V637C71 631 75 632 75 571V105C75 41 69 38 12 32V0H249V32C207 38 203 46 203 101V352C225 373 251 380 275 380C314 380 347 354 347 277V101C347 46 344 39 296 32V0H531Z"></path></g></svg> is the unit angular bisector of <svg height="8.85534pt" id="M16" style="vertical-align:-0.1150398pt" version="1.1" viewbox="-0.0498162 -8.7403 9.37731 8.85534" width="9.37731pt" 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="M702 650H468V614C550 604 550 590 527 521C500 434 457 318 402 172H399C363 266 284 469 255 549C240 592 240 603 313 614V650H15V614C77 608 93 594 124 517C165 413 259 172 327 -11H377C455 195 546 422 579 500C619 595 637 607 702 614V650Z"></path></g></svg> and <span class="nowrap"><svg height="8.79011pt" id="M17" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.7403 7.2895 8.79011" width="7.2895pt" 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="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g></svg>;</span> that is, <span class="nowrap"><svg height="11.5564pt" id="M18" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 107.208 11.5564" width="107.208pt" 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="M531 0V32C480 40 474 41 474 105V288C474 398 423 454 336 454C286 454 237 424 203 390V703C143 688 72 674 19 667V637C71 631 75 632 75 571V105C75 41 69 38 12 32V0H249V32C207 38 203 46 203 101V352C225 373 251 380 275 380C314 380 347 354 347 277V101C347 46 344 39 296 32V0H531Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.782,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.045,0)"><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,-0.013,26.543,0)"><path d="M702 650H468V614C550 604 550 590 527 521C500 434 457 318 402 172H399C363 266 284 469 255 549C240 592 240 603 313 614V650H15V614C77 608 93 594 124 517C165 413 259 172 327 -11H377C455 195 546 422 579 500C619 595 637 607 702 614V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.691,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,49.228,0)"><path d="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g><g transform="matrix(.013,0,0,-0.013,56.391,0)"><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><g transform="matrix(.013,0,0,-0.013,60.889,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,66.25,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z"></path></g><g transform="matrix(.013,0,0,-0.013,71.593,0)"><path d="M702 650H468V614C550 604 550 590 527 521C500 434 457 318 402 172H399C363 266 284 469 255 549C240 592 240 603 313 614V650H15V614C77 608 93 594 124 517C165 413 259 172 327 -11H377C455 195 546 422 579 500C619 595 637 607 702 614V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,83.741,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,94.278,0)"><path d="M512 180C494 129 475 97 457 77C433 50 397 43 354 43C323 43 297 45 279 52C259 61 255 78 255 125V516C255 602 264 608 340 615V650H35V615C111 608 121 602 121 516V133C121 48 111 42 24 35V0H511C519 27 540 126 548 169L512 180Z"></path></g><g transform="matrix(.013,0,0,-0.013,101.441,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z"></path></g></svg>;</span> <svg height="9.49473pt" id="M19" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.84582 9.49473" width="6.84582pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g></svg> is the angle between <svg height="6.089pt" id="M20" style="vertical-align:-0.04982042pt" version="1.1" viewbox="-0.0498162 -6.03918 7.48522 6.089" width="7.48522pt" 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="M549 0V32C497 40 491 42 491 105V294C491 394 441 454 360 454C331 454 312 445 282 426C260 412 240 396 220 383V456C161 439 95 425 39 418V390C87 380 92 377 92 317V109C92 42 88 41 26 32V0H267V32C225 38 220 42 220 104V346C246 366 268 377 294 377C330 377 363 355 363 283V113C363 42 358 38 316 32V0H549Z"></path></g></svg> and <span class="nowrap"><svg height="9.25986pt" id="M21" style="vertical-align:-0.04981041pt" version="1.1" viewbox="-0.0498162 -9.21005 7.27645 9.25986" width="7.27645pt" 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="M531 0V32C480 40 474 41 474 105V288C474 398 423 454 336 454C286 454 237 424 203 390V703C143 688 72 674 19 667V637C71 631 75 632 75 571V105C75 41 69 38 12 32V0H249V32C207 38 203 46 203 101V352C225 373 251 380 275 380C314 380 347 354 347 277V101C347 46 344 39 296 32V0H531Z"></path></g></svg>.</span></div>

International Journal of Optics

fig1

Figure 1

Figure 1: A Unified Shape-From-Shading Approach for 3D Surface Reconstruction Using Fast Eikonal Solvers 