**Integral formulations of volumetric transmittance: supplemental material**

This supplemental is structured as follows. The first section contains links to pages
comparing estimators across various analytical extinction functions and conveys the
differences between each estimator's variance, efficiency, and cost across these simple
cases. The second section contains a link to various pages comparing our transmittance
estimators against the current state of the art across a variety of practical scenes, while
the third section then compares the improvement gains from using stratification in each
scene from section two. The last section contains a table describing, across all scenes,
the performance of each estimator as well as the relative benefits when using stratification.

Canonical experiments
===================================================

<div class="row">
    <div class="column">
        <a href="const/index.html"><img src="renders/generated_graphs/const/const_ext_3_5.svg" title="Constant Extinction"/></a>
    </div>
    <div class="column">
        <a href="cos/index.html"><img src="renders/generated_graphs/cos/cos_ext_3_5.svg" title="Cosine Extinction"/></a>
    </div>
    <div class="column">
        <a href="lin_dec/index.html"><img src="renders/generated_graphs/lin_dec/lin_dec_ext_0_0.svg" title="Linear Extinction"/></a>
    </div>
</div>
<div class="row">
    <div class="column">
        <a href="gauss/index.html"><img src="renders/generated_graphs/gauss/gauss_ext_3_5.svg" title="Gaussian Extinction"/></a>
    </div>
    <div class="column">
        <a href="cos_high_freq/index.html"><img src="renders/generated_graphs/cos_high_freq/cos_high_freq_ext_0_0.svg" title="High Frequency Cosine"/></a>
    </div>
    <div class="column">
        <a href="hole/index.html"><img src="renders/generated_graphs/hole/hole_ext_3_5.svg" title="Hole Extinction"/></a>
    </div>
</div>

Rendering experiments
===========================================

<div class="row">
    <div class="column">
        <a href="pawn/index.html"><img src="renders/gt_pawn.jpeg" title="Pawn"/></a>
    </div>
    <div class="column">
        <a href="vase_down/index.html"><img src="renders/gt_vase_down.jpeg" title="Stained Glass Vase Scene Two"/></a>
    </div>
    <div class="column">
        <a href="vase_up/index.html"><img src="renders/gt_vase_up.jpeg" title="Stained Glass Vase Scene One"/></a>
    </div>
    <div class="column">
        <a href="vdb/index.html"><img src="renders/gt_vdb.jpeg" title="Smoke"/></a>
    </div>
    <!-- <div class="column">
        <a href="cloud/index.html"><img src="renders/gt_cloud.jpeg" title="Cornell"/></a>
    </div> -->
    <!-- <div class="column">
        <a href="fox/index.html"><img src="renders/gt_fox.jpeg" title="Stained Glass Fox"/></a>
    </div> -->
    <div class="column">
        <a href="sphere/index.html"><img src="renders/gt_sphere.jpeg" title="Globe"/></a>
    </div>
    <div class="column">
        <a href="fox/index.html"><img src="renders/gt_fox.jpeg" title="Stained Glass Fox"/></a>
    </div>
</div>

Stratification experiments
==================================================

<div class="row">
    <div class="column">
        <a href="pawn_strat/index.html"><img src="renders/gt_pawn.jpeg" title="Pawn"/></a>
    </div>
    <div class="column">
        <a href="vase_down_strat/index.html"><img src="renders/gt_vase_down.jpeg" title="Stained Glass Vase Scene Two"/></a>
    </div>
    <div class="column">
        <a href="vase_up_strat/index.html"><img src="renders/gt_vase_up.jpeg" title="Stained Glass Vase Scene One"/></a>
    </div>
    <div class="column">
        <a href="vdb_strat/index.html"><img src="renders/gt_vdb.jpeg" title="Smoke"/></a>
    </div>
    <!-- <div class="column">
        <a href="cloud_strat/index.html"><img src="renders/gt_cloud.jpeg" title="Cornell Box Cloud"/></a>
    </div> -->
    <!-- <div class="column">
        <a href="fox_strat/index.html"><img src="renders/gt_fox.jpeg" title="Stained Glass Fox"/></a>
    </div> -->
    <div class="column">
        <a href="sphere_strat/index.html"><img src="renders/gt_sphere.jpeg" title="Globe"/></a>
    </div>
    <div class="column">
        <a href="fox_strat/index.html"><img src="renders/gt_fox.jpeg" title="Stained Glass Fox"/></a>
    </div>

</div>

Error analysis
================================================================

Random sampling (lower is better for all metrics):
* [Root mean squared error (RMSE)](err/RMSE.html)
* [Mean squared error (MSE)](err/MSE.html)
* [Mean absolute difference (MAD)](err/MAD.html)
* [Mean absolute relative difference (MARD)](err/MARD.html)
* [Cost $\times$ variance (1/efficiency)](err/EFF.html)

Stratified sampling; numbers relative to random sampling (lower is better for all metrics):
* [Root mean squared error (RMSE)](strat/RMSE.html)
* [Mean squared error (MSE)](strat/MSE.html)
* [Mean absolute difference (MAD)](strat/MAD.html)
* [Mean absolute relative difference (MARD)](strat/MARD.html)
* [Cost $\times$ variance (i.e. 1/efficiency)](strat/EFF.html)

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