三维重建方法概述
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2024-04-30 22:41:35
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</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6944em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span></span></span></span></span>来截断，类似下图：<br> <img src="https://img-blog.****img.cn/5ed828820afc435782df8bfd5f7ffc7b.png" alt="在这里插入图片描述"></li><li>用公式定义如下：<br> <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-1-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mn>1</mn><mo>,</mo></mtd><mtd><mrow><mtext>if&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>&gt;</mo><mi>&#x03B4;</mi></mrow></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mi>S</mi><mi>D</mi><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo>|</mo><mi>&#x03B4;</mi><mo>|</mo></mrow></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>&#xFF0C;</mo></mrow></mtd><mtd><mrow><mtext>if&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B4;</mi><mo>&#x2264;</mo><mi>x</mi><mo>&#x2264;</mo><mi>&#x03B4;</mi></mrow></mrow></mtd></mtr><mtr><mtd><mo>&#x2212;</mo><mn>1</mn><mo>,</mo></mtd><mtd><mrow><mtext>if&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>&lt;</mo><mi>&#x03B4;</mi></mrow></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 100%; display: inline-block; min-width: 11.494em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.635em, 1011.24em, 7.09em, -999.997em); top: -5.11em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-2"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.635em, 1011.24em, 7.09em, -999.997em); top: -5.11em; left: 50%; margin-left: -5.617em;"><span class="mrow" id="MathJax-Span-3"><span class="mo" id="MathJax-Span-4" style="vertical-align: 2.331em;"><span style="display: inline-block; position: relative; width: 0.914em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -3.085em; left: 0em;">⎧<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.757em; left: 0em;">⎩<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -1.668em; left: 0em;">⎨<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -2.832em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.757em; left: 0em;">⎪<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mtable" id="MathJax-Span-5" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 10.076em; height: 0px;"><span style="position: absolute; clip: rect(2.635em, 1003.39em, 6.938em, -999.997em); top: -4.958em; left: 0em;"><span style="display: inline-block; position: relative; width: 3.394em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.71em, 4.356em, -999.997em); top: -5.515em; left: 0em;"><span class="mtd" id="MathJax-Span-6"><span class="mrow" id="MathJax-Span-7"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.71em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.402em;"><span class="mn" id="MathJax-Span-8" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-9" style="font-family: MathJax_Main;">,</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.736em, 1003.39em, 4.761em, -999.997em); top: -3.997em; left: 0em;"><span class="mtd" id="MathJax-Span-19"><span class="mrow" id="MathJax-Span-20"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1003.39em, 4.761em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.668em;"><span class="mfrac" id="MathJax-Span-21"><span style="display: inline-block; position: relative; width: 2.635em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1002.48em, 4.305em, -999.997em); top: -4.554em; left: 50%; margin-left: -1.263em;"><span class="mrow" id="MathJax-Span-22"><span style="display: inline-block; position: relative; width: 2.534em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1002.48em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-23" style="font-size: 70.7%; font-family: MathJax_Math-italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-24" style="font-size: 70.7%; font-family: MathJax_Math-italic;">D</span><span class="mi" id="MathJax-Span-25" style="font-size: 70.7%; font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span class="mo" id="MathJax-Span-26" style="font-size: 70.7%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-27" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-28" style="font-size: 70.7%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1000.61em, 4.305em, -999.997em); top: -3.541em; left: 50%; margin-left: -0.352em;"><span class="mrow" id="MathJax-Span-29"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.61em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="mo" id="MathJax-Span-30" style="font-size: 70.7%; font-family: MathJax_Main;">|</span><span class="mi" id="MathJax-Span-31" style="font-size: 70.7%; font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-32" style="font-size: 70.7%; font-family: MathJax_Main;">|</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1002.63em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.635em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span class="texatom" id="MathJax-Span-33"><span class="mrow" id="MathJax-Span-34"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.04em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mo" id="MathJax-Span-35"><span style="font-family: STIXGeneral, "Arial Unicode MS", serif; font-size: 98%; font-style: normal; font-weight: normal;">，</span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1001.47em, 4.356em, -999.997em); top: -2.377em; left: 0em;"><span class="mtd" id="MathJax-Span-48"><span class="mrow" id="MathJax-Span-49"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1001.47em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.757em;"><span class="mo" id="MathJax-Span-50" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-51" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-52" style="font-family: MathJax_Main;">,</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.964em;"></span></span><span style="position: absolute; clip: rect(2.483em, 1005.67em, 6.685em, -999.997em); top: -4.857em; left: 4.407em;"><span style="display: inline-block; position: relative; width: 5.672em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.141em, 1003.14em, 4.204em, -999.997em); top: -5.515em; left: 0em;"><span class="mtd" id="MathJax-Span-10"><span class="mrow" id="MathJax-Span-11"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1003.14em, 4.204em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.567em;"><span class="mrow" id="MathJax-Span-12"><span class="mtext" id="MathJax-Span-13" style="font-family: MathJax_Main;">if </span><span class="texatom" id="MathJax-Span-14"><span class="mrow" id="MathJax-Span-15"><span style="display: inline-block; position: relative; width: 2.331em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1002.33em, 4.204em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-16" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-17" style="font-family: MathJax_Main; padding-left: 0.256em;">></span><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.141em, 1005.67em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="mtd" id="MathJax-Span-36"><span class="mrow" id="MathJax-Span-37"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1005.67em, 4.305em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.832em;"><span class="mrow" id="MathJax-Span-38"><span class="mtext" id="MathJax-Span-39" style="font-family: MathJax_Main;">if </span><span class="texatom" id="MathJax-Span-40"><span class="mrow" id="MathJax-Span-41"><span style="display: inline-block; position: relative; width: 4.812em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1004.81em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="mo" id="MathJax-Span-42" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-43" style="font-family: MathJax_Math-italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-44" style="font-family: MathJax_Main; padding-left: 0.256em;">≤</span><span class="mi" id="MathJax-Span-45" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">x</span><span class="mo" id="MathJax-Span-46" style="font-family: MathJax_Main; padding-left: 0.256em;">≤</span><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.141em, 1003.14em, 4.204em, -999.997em); top: -2.377em; left: 0em;"><span class="mtd" id="MathJax-Span-53"><span class="mrow" id="MathJax-Span-54"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1003.14em, 4.204em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.567em;"><span class="mrow" id="MathJax-Span-55"><span class="mtext" id="MathJax-Span-56" style="font-family: MathJax_Main;">if </span><span class="texatom" id="MathJax-Span-57"><span class="mrow" id="MathJax-Span-58"><span style="display: inline-block; position: relative; width: 2.331em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1002.33em, 4.204em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-60" style="font-family: MathJax_Main; padding-left: 0.256em;"><</span><span class="mi" id="MathJax-Span-61" style="font-family: MathJax_Math-italic; padding-left: 0.256em;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.862em;"></span></span></span></span><span class="mo" id="MathJax-Span-62"></span></span><span style="display: inline-block; width: 0px; height: 5.115em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.115em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.908em; border-left: 0px solid; width: 0px; height: 4.288em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mn>1</mn><mo>,</mo></mtd><mtd><mrow><mtext>if </mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>></mo><mi>δ</mi></mrow></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mi>S</mi><mi>D</mi><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo>|</mo><mi>δ</mi><mo>|</mo></mrow></mfrac><mrow class="MJX-TeXAtom-ORD"><mo>，</mo></mrow></mtd><mtd><mrow><mtext>if </mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>δ</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>δ</mi></mrow></mrow></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn><mo>,</mo></mtd><mtd><mrow><mtext>if </mtext><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo><</mo><mi>δ</mi></mrow></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">\begin{cases} 1, & \text{if $x > \delta$} \\ \frac{SDF(x)}{\left| \delta \right|}，& \text{if $ -\delta \le x \le \delta$}\\ -1, & \text{if $x < \delta$} \end{cases}</script>
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0576em;">TS</span><span class="mord mathnormal" style="margin-right: 0.0278em;">D</span><span class="mord mathnormal" style="margin-right: 0.1389em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 4.41em; vertical-align: -1.955em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.35em;"><span class="" style="top: -2.2em;"><span class="pstrut" style="height: 3.15em;"></span><span class="delimsizinginner delim-size4"><span class="">⎩</span></span></span><span class="" style="top: -2.192em;"><span class="pstrut" style="height: 3.15em;"></span><span class="" style="height: 0.316em; width: 0.8889em;">
<svg width="0.8889em" height="0.316em" style="width:0.8889em" viewBox="0 0 888.89 316" preserveAspectRatio="xMinYMin">
<path d="M384 0 H504 V316 H384z M384 0 H504 V316 H384z"></path>
</svg></span></span><span class="" style="top: -3.15em;"><span class="pstrut" style="height: 3.15em;"></span><span class="delimsizinginner delim-size4"><span class="">⎨</span></span></span><span class="" style="top: -4.292em;"><span class="pstrut" style="height: 3.15em;"></span><span class="" style="height: 0.316em; width: 0.8889em;">
<svg width="0.8889em" height="0.316em" style="width:0.8889em" viewBox="0 0 888.89 316" preserveAspectRatio="xMinYMin">
<path d="M384 0 H504 V316 H384z M384 0 H504 V316 H384z"></path>
</svg></span></span><span class="" style="top: -4.6em;"><span class="pstrut" style="height: 3.15em;"></span><span class="delimsizinginner delim-size4"><span class="">⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.85em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.455em;"><span class="" style="top: -4.457em;"><span class="pstrut" style="height: 3.01em;"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span><span class="" style="top: -3.015em;"><span class="pstrut" style="height: 3.01em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.01em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="minner mtight"><span class="mopen mtight delimcenter" style="top: 0em;"><span class="mtight">∣</span></span><span class="mord mathnormal mtight" style="margin-right: 0.0379em;">δ</span><span class="mclose mtight delimcenter" style="top: 0em;"><span class="mtight">∣</span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.485em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0576em;">S</span><span class="mord mathnormal mtight" style="margin-right: 0.0278em;">D</span><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">F</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.52em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord cjk_fallback">，</span></span></span><span class="" style="top: -1.487em;"><span class="pstrut" style="height: 3.01em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.955em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.455em;"><span class="" style="top: -4.457em;"><span class="pstrut" style="height: 3.01em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span></span></span></span><span class="" style="top: -3.015em;"><span class="pstrut" style="height: 3.01em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span></span></span></span><span class="" style="top: -1.487em;"><span class="pstrut" style="height: 3.01em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.955em;"><span class=""></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></li><li>在 2D 平面上，一个简单的 TSDF 场类似下面的样子：<br> <img src="https://img-blog.****img.cn/717d0686307c4da69d7e826924c30087.png" alt="在这里插入图片描述"></li></ul>
3.2 融合
3.2.1 权重
- 有了上述 TSDF 的定义，想要对于多个不同帧的同一个体素观测进行更新，首先要定义体素的概率，考虑到一个显然的事实，对于一个深度传感器来说，测量数据的方差与角度有关：
- 一种基本的权重定义如下：
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.363em; vertical-align: -0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal">an</span><span class="mord mathnormal">ce</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0359em;">v</span><span class="mclose">)</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0278em;">θ</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></li><li><span class="katex--inline"><span class="katex"><span class="katex-mathml"> </span><span class="katex-html"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal">an</span><span class="mord mathnormal">ce</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0359em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.8444em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span></li></ul>
3.2.2 更新
-
对于同一个体素多次不同观测，我们采用如下方式更新：
-
初始化
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>T</mi><mi>S</mi><mi>D</mi><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>W</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-63" style="width: 100%; display: inline-block; min-width: 7.596em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.382em, 1007.44em, 5.115em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-64"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.382em, 1007.44em, 5.115em, -999.997em); top: -3.997em; left: 50%; margin-left: -3.744em;"><span class="mrow" id="MathJax-Span-65"><span class="mo" id="MathJax-Span-66" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">{</span></span><span class="mtable" id="MathJax-Span-67" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 6.381em; height: 0px;"><span style="position: absolute; clip: rect(2.483em, 1006.33em, 5.065em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.381em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.091em, 1006.33em, 4.407em, -999.997em); top: -4.604em; left: 0em;"><span class="mtd" id="MathJax-Span-68"><span class="mrow" id="MathJax-Span-69"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.091em, 1006.33em, 4.407em, -999.997em); top: -3.997em; left: 50%; margin-left: -3.187em;"><span class="mi" id="MathJax-Span-70" style="font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span class="mi" id="MathJax-Span-71" style="font-family: MathJax_Math-italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span class="mi" id="MathJax-Span-72" style="font-family: MathJax_Math-italic;">D</span><span class="msubsup" id="MathJax-Span-73"><span style="display: inline-block; position: relative; width: 1.066em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-74" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.661em;"><span class="mn" id="MathJax-Span-75" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-76" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-77" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-78" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-79" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mn" id="MathJax-Span-80" style="font-family: MathJax_Main; padding-left: 0.256em;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.091em, 1004.46em, 4.407em, -999.997em); top: -3.339em; left: 0em;"><span class="mtd" id="MathJax-Span-81"><span class="mrow" id="MathJax-Span-82"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.091em, 1004.46em, 4.407em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.275em;"><span class="msubsup" id="MathJax-Span-83"><span style="display: inline-block; position: relative; width: 1.369em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1001.07em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-84" style="font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.964em;"><span class="mn" id="MathJax-Span-85" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-86" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-87" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-88" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-89" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mn" id="MathJax-Span-90" style="font-family: MathJax_Main; padding-left: 0.256em;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-91"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.03em; border-left: 0px solid; width: 0px; height: 2.533em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>T</mi><mi>S</mi><mi>D</mi><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>W</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{cases} TSDF_0(x) = 0 \\ W_0(x) = 0 \end{cases}</script> </span><span class="katex-html"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top: 0em;"><span class="delimsizing size4">{<!-- --></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.69em;"><span class="" style="top: -3.69em;"><span class="pstrut" style="height: 3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0576em;">TS</span><span class="mord mathnormal" style="margin-right: 0.0278em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.1389em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.1389em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord">0</span></span></span><span class="" style="top: -2.25em;"><span class="pstrut" style="height: 3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.1389em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.1389em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.19em;"><span class=""></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p> </li><li> <p>更新TSDF和W值<br> <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>T</mi><mi>S</mi><mi>D</mi><msub><mi>F</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>W</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x22C5;</mo><mi>T</mi><mi>S</mi><mi>D</mi><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>w</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x22C5;</mo><mi>t</mi><mi>s</mi><mi>d</mi><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>W</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>w</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>W</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>w</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-92" style="width: 100%; display: inline-block; min-width: 19.087em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.078em, 1018.68em, 5.419em, -999.997em); top: -3.997em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-93"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.078em, 1018.68em, 5.419em, -999.997em); top: -3.997em; left: 50%; margin-left: -9.312em;"><span class="mrow" id="MathJax-Span-94"><span class="mo" id="MathJax-Span-95" style="vertical-align: 0em;"><span style="font-family: MathJax_Size4;">{</span></span><span class="mtable" id="MathJax-Span-96" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 17.569em; height: 0px;"><span style="position: absolute; clip: rect(2.129em, 1017.57em, 5.369em, -999.997em); top: -3.997em; left: 0em;"><span style="display: inline-block; position: relative; width: 17.569em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.736em, 1017.57em, 4.761em, -999.997em); top: -4.604em; left: 0em;"><span class="mtd" id="MathJax-Span-97"><span class="mrow" id="MathJax-Span-98"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1017.57em, 4.761em, -999.997em); top: -3.997em; left: 50%; margin-left: -8.755em;"><span class="mi" id="MathJax-Span-99" style="font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span class="mi" id="MathJax-Span-100" style="font-family: MathJax_Math-italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span class="mi" id="MathJax-Span-101" style="font-family: MathJax_Math-italic;">D</span><span class="msubsup" id="MathJax-Span-102"><span style="display: inline-block; position: relative; width: 0.964em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-103" style="font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.661em;"><span class="mi" id="MathJax-Span-104" style="font-size: 70.7%; font-family: MathJax_Math-italic;">i</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-105" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-106" style="font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-107" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-108" style="font-family: MathJax_Main; padding-left: 0.256em;">=</span><span class="mfrac" id="MathJax-Span-109" style="padding-left: 0.256em;"><span style="display: inline-block; position: relative; width: 11.544em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1011.34em, 4.305em, -999.997em); top: -4.554em; left: 50%; margin-left: -5.718em;"><span class="mrow" id="MathJax-Span-110"><span style="display: inline-block; position: relative; width: 11.393em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1011.34em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="msubsup" id="MathJax-Span-111"><span style="display: inline-block; position: relative; width: 1.521em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-112" style="font-size: 70.7%; font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.661em;"><span class="texatom" id="MathJax-Span-113"><span class="mrow" id="MathJax-Span-114"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.495em, 1000.76em, 4.204em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-115" style="font-size: 50%; font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-116" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-117" style="font-size: 50%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-118" style="font-size: 70.7%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-119" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-120" style="font-size: 70.7%; font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-121" style="font-size: 70.7%; font-family: MathJax_Main;">⋅</span><span class="mi" id="MathJax-Span-122" style="font-size: 70.7%; font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.104em;"></span></span><span class="mi" id="MathJax-Span-123" style="font-size: 70.7%; font-family: MathJax_Math-italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-124" style="font-size: 70.7%; font-family: MathJax_Math-italic;">D</span><span class="msubsup" id="MathJax-Span-125"><span style="display: inline-block; position: relative; width: 1.319em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-126" style="font-size: 70.7%; font-family: MathJax_Math-italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.458em;"><span class="texatom" id="MathJax-Span-127"><span class="mrow" id="MathJax-Span-128"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.495em, 1000.76em, 4.204em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-129" style="font-size: 50%; font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-130" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-131" style="font-size: 50%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-132" style="font-size: 70.7%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-133" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-134" style="font-size: 70.7%; font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-135" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="msubsup" id="MathJax-Span-136"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(3.546em, 1000.51em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-137" style="font-size: 70.7%; font-family: MathJax_Math-italic;">w</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-138" style="font-size: 50%; font-family: MathJax_Math-italic;">i</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-139" style="font-size: 70.7%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-140" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-141" style="font-size: 70.7%; font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-142" style="font-size: 70.7%; font-family: MathJax_Main;">⋅</span><span class="mi" id="MathJax-Span-143" style="font-size: 70.7%; font-family: MathJax_Math-italic;">t</span><span class="mi" id="MathJax-Span-144" style="font-size: 70.7%; font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-145" style="font-size: 70.7%; font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-146"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.41em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-147" style="font-size: 70.7%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.357em;"><span class="mi" id="MathJax-Span-148" style="font-size: 50%; font-family: MathJax_Math-italic;">i</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-149" style="font-size: 70.7%; font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-150" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span class="mo" id="MathJax-Span-151" style="font-size: 70.7%; font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1004.66em, 4.305em, -999.997em); top: -3.541em; left: 50%; margin-left: -2.377em;"><span class="mrow" id="MathJax-Span-152"><span style="display: inline-block; position: relative; width: 4.71em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1004.66em, 4.305em, -999.997em); top: -3.997em; left: 0em;"><span class="msubsup" id="MathJax-Span-153"><span style="display: inline-block; position: relative; width: 1.521em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-154" style="font-size: 70.7%; font-family: MathJax_Math-italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.661em;"><span class="texatom" id="MathJax-Span-155"><span class="mrow" id="MathJax-Span-156"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.495em, 1000.76em, 4.204em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-157" style="font-size: 50%; font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-158" style="font-size: 50%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-159" style="font-size: 50%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span>
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