*Due Date*: Tues May 28th 11:59PM

# Sampling Efficiency

In this assignment you will evaluate the efficiency of different sampling strategies for Monte-Carlo integration. While there is very little coding in this assignment, you will probably need to script `pbrt`

to collect the right data.
The data collection itself may take some time, so plan ahead to make sure you have time to complete rendering. You can download the necessary scene files and reference images here.

## Step 1: Sampling and Variance

Variance of a random variable $X$ is defined as $E[ (X - E[X])^2 ]$. To compute the variance of an image rendered by `pbrt`

we can compare it, pixel by pixel, to a reference image that we consider to be ground truth for $E[X]$. We've included three EXR images that will serve as ground truth for this assignment (`occluder_ref.exr`

, `occluder_shadow_ref.exr`

, and `manykilleroos_ref.exr`

). These scenes were rendered with 16384 samples per pixel to ensure that they are very close to the correct image. To measure the variance of any image you render, you can use the `exrdiff`

program that is included with pbrt:

```
exrdiff image_ref.exr yourimage.exr
```

This will print something like:

```
Images differ: 173884 big (64.40%), 179244 small (66.39%)
avg 1 = 0.351801, avg2 = 0.361679 (-2.807667% delta)
MSE = 0.126821
```

Look at the code for `exrdiff`

and convince yourself that the MSE (mean-squared error) value it calculates is equivalent to the variance. You man need to modify `exrdiff`

to ensure it always prints the MSE value regardless of how close the two images are.

As we increase the number of samples taken, we expect the the variance to go down. We will measure this change empirically by modifying the `pixelsamples`

variable in the scene files. The file `occluder.pbrt`

contains a scene with a single area-light, a square occluder, and a large ground plane:

Render this scene varying the number of samples from 1 to 64, and compute the variance of each image. You only need to measure powers of 2 since many of the samplers in pbrt will round this number to a power of 2 anyway.

The scene file is setup to use a low-discrepancy sampler which tries to generate well-distributed non-uniform points such that no two points are close together. Let's compare this to a purely random sampler, by changing "lowdiscrepancy" to "random" in the scene file, and finding the variance for each image.

## Deliverables

First, submit a *graph* showing the variance for both the random and low-discrepancy samplers as you increase `pixelsamples`

from 1 to 64 for the `occluder.pbrt`

scene. You will probably want to use log scale for both axes since you are only considering powers of 2. Furthermore, you should answer the following questions:

When using Monte-Carlo sampling with uniformly sampled random variables, how does the variance change as a function of the number of samples taken, N? Does the data you collected reflect this behavior?

How does the low-discrepancy sampler perform in comparison to random sampling? Why does distributing the samples evenly through space change the variance in comparison to random sampling?

## Step 2: Evaluating Efficiency

In lecture, we discussed different strategies for sampling the value for a pixel. In one variation, we compute an intersection with the scene and a ray, and then evaluate the radiance along the ray using $N$ samples from each area light. In another variation, rather than taking multiple samples, we take only a single sample ($N = 1$), and compensate by increasing the number of samples taken per pixel.

Your task for this step is to compare the efficiency of these approaches on the three scenes we have included: `occluder.pbrt`

, `occluder_shadow.pbrt`

(a different view of the first scene), and `manykilleroos.pbrt`

:

You should consider strategies from $N = 1$ sample per light to $N = 64$ samples, again in powers of 2. One strategy is considered more efficient than another if, given a fixed amount of time, it can compute an image with less variance than the image produce by the other strategy (alternatively, given a target variance, it is more efficient if it can compute the image in less time).

Evaluate each strategy (1 to 64 samples per light) by rendering images for a range of pixelsamples (1 to 64 samples per pixel). For each image, you should measure the amount of time it takes to render, as well as the variance of the result. To limit the number of scenes you have to render, you can limit yourself to scenes where the [number of light samples] x [number of pixel samples] <= 64. You should only time the rendering of the scene, not the setup costs. This is particular important for the manykilleroo scene, which has complex geometry.

## Deliverables

For each scene, you should construct a scatter plot with time on the x-axis and variance on the y-axis. Each point in the plot should correspond to one of the images you generated (i.e. an image with a specific sampling strategy and number of samples per pixel). Furthermore, you should answer the following questions:

For each scene, which approach(s) (e.g 1 light sample per pixel sample, 2 light samples per pixel sample, ...) perform(s) most efficiently. Why do you think that is the case? Backup your answer using data from the graphs you have generated.

Is the same approach best across all the scenes? Why or why not? Are any approaches particularly bad? Why?

What characteristics in the scene make the path tracing ($N = 1$) method more efficient? What characteristics make strategies with a large number of light samples more efficient?

## Submission

To submit your work, create a `.zip`

or `.tar.gz`

file that contains a writeup with your plots and answers to the questions posed above. Use the assignment submission page to upload and submit your work.
You can submit multiple times if you wish to make changes after your initial submission.
A record of the submission including the date submitted and an md5 checksum of your bundle
will appear on the submission page when it is successful.

## Grading

This assignment will be graded on a 4 point scale:

- 1 point: A writeup with major errors.
- 2 points: A correct writeup but step 1 or step 2 is incomplete.
- 3 points: A complete writeup with but explanations are not clear, or contain minor errors.
- 4 points: A correct, clear, and complete writeup