Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/amp/volume-2018/2096058/figures/2096058.fig.006a.svgz" name="2096058.fig.006a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b><svg height="12.4698pt" id="M197" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 11.2181 12.4698" width="11.2181pt" 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="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" id="g113-230"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.149,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 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477 762 532 770 556C785 600 798 616 862 619L870 650Z" id="g50-79"></path></g></svg> satisfy (<a href="https://static.hindawi.com/articles/amp/volume-2018/2096058/figures/#EEq12">27</a>)</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/amp/volume-2018/2096058/figures/2096058.fig.006b.svgz" name="2096058.fig.006b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b><svg height="12.4698pt" id="M199" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 32.4048 12.4698" width="32.4048pt" 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="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" id="g113-230"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.149,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.013,0,0,-0.013,14.727,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,25.99,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g></svg> and <svg height="12.5794pt" id="M200" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 37.1872 12.5794" width="37.1872pt" 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="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 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<div>Map of <svg height="11.5564pt" id="M195" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 36.1272 11.5564" width="36.1272pt" 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="M567 0V44H236V319H535V363H236V618H567V662H70V0H567ZM192 44H114V618H192V44Z" id="g197-16"></path></g><g transform="matrix(.013,0,0,-0.013,8.086,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,12.584,0)"><path d="M410 152L393 124C355 58 321 44 251 44H187C264 179 330 297 407 428L398 442H153C114 441 100 447 92 463H70C60 419 44 357 30 308L60 295L77 328C106 384 130 399 197 399C214 399 232 399 249 399C173 265 106 147 24 16L37 0H392C404 31 420 82 439 138L410 152Z" id="g13-120"></path></g><g transform="matrix(.013,0,0,-0.013,18.369,0)"><path d="M114 412C81 412 58 388 58 355C58 321 81 297 113 297S169 321 169 355C169 388 145 412 114 412ZM95 130C70 130 46 114 46 88C46 72 54 65 59 64C93 56 121 33 121 -3C121 -41 93 -68 45 -88L56 -118C117 -99 186 -56 186 22C186 91 131 130 95 130Z" id="g113-60"></path></g><g transform="matrix(.013,0,0,-0.013,23.512,0)"><path d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" id="g113-71"></path></g><g transform="matrix(.013,0,0,-0.013,31.407,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> for when inhomogeneity is <span class="nowrap"><svg height="11.8174pt" id="M196" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 10.5526 11.8174" width="10.5526pt" 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="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z" id="g113-132"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.486,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg>.</span></div>

Advances in Mathematical Physics

fig6

Figure 6

Figure 6: Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions 