Convergence Analysis of a Modified Forward-Backward Splitting Algorithm for Minimization and Application to Image Recovery

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr class="thead"><td class="align_left" rowspan="2">Algorithms</td><td class="align_center" colspan="5">Parameters</td></tr><tr class="thead"><td class="align_center"><span style="width: 6.63704ptpx;"><svg height="9.39034pt" id="M181" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 6.63704 9.39034" width="6.63704pt" 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="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 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388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"></path></g></svg></span></td><td class="align_center"><span style="width: 9.43046ptpx;"><svg height="10.9953pt" id="M185" style="vertical-align:-3.18148pt" version="1.1" viewbox="-0.0498162 -7.81382 9.43046 10.9953" width="9.43046pt" 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="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,4.368,3.132)"><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"></path></g></svg></span></td></tr><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td class="align_left">FISTA-BT</td><td class="align_center"><span style="width: 33.8176ptpx;"><svg height="11.6804pt" id="M186" style="vertical-align:-2.39207pt" version="1.1" viewbox="-0.0498162 -9.28833 33.8176 11.6804" width="33.8176pt" 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="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,-0.013,6.24,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.78,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.143,0)"><path d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.278,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z"></path></g></svg></span></td><td class="align_center">—</td><td class="align_center">—</td><td class="align_center">—</td><td class="align_center">1</td></tr><tr><td class="align_left">FISTA-CN</td><td class="align_center">—</td><td class="align_center">0.9</td><td class="align_center">0.4</td><td class="align_center">0.2</td><td class="align_center">1</td></tr><tr><td class="align_left">NAGA</td><td class="align_center"><span style="width: 33.8176ptpx;"><svg height="11.6804pt" id="M187" style="vertical-align:-2.39207pt" version="1.1" viewbox="-0.0498162 -9.28833 33.8176 11.6804" width="33.8176pt" 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="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,-0.013,6.24,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.78,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.143,0)"><path d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.278,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z"></path></g></svg></span></td><td class="align_center">—</td><td class="align_center">—</td><td class="align_center">—</td><td class="align_center">1</td></tr><tr><td class="align_left">NMFBS</td><td class="align_center">—</td><td class="align_center">0.9</td><td class="align_center">0.9</td><td class="align_center">0.2</td><td class="align_center">1</td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Chosen parameters of each algorithm.</div>

Computational and Mathematical Methods

tab1

Table 1

Table 1: Convergence Analysis of a Modified Forward-Backward Splitting Algorithm for Minimization and Application to Image Recovery 