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Method

The raytracer used in the assignment is not my own, but Pharr&Humphrey's physically based raytracer pbrt. I opted to using theirs as opposed to my own, because that would provide me with a solid framework to try out some of the more advanced techniques for the next assignments.

Unfortunately, pbrt already has most of the well known sampling and reconstruction techniques implemented. Instead of rewriting my own versions of what was already implemented, I chose to implement some other techniques that may have inferior quality as more advanced ones, and try to find out in which cases they would break. In terms of other reconstruction techniques, I couldn't really find others in the literature, therefore I emphasize on implementing my own alternative sampling techniques, and discuss the influence of already-implemented reconstruction techniques on these.

Comparisons will be made on four basic scenes:

• A basic checkerboard
• Glass and mirror spheres above a textured plane
• Kangaroo in textured box with area light
• Buddha with area light

Sampling techniques

Uniform, jittered, best-candidate and low-discrepancy sampling

n-dimensional Hammersley sampling

I implemented a new Sampler plugin for pbrt, called HammersleyLDSampler. This is a low-discrepancy sampler, which differs from pbrt's in that it doesn't use a scrambling technique to decorrelate the different 1- and 2-dimensional samples. Instead, for each pixel, the HammersleyLDSampler generates a sequence of pixelSamples n-dimensional samples from an n-dimensional Hammersley sequence x(i), for all n dimensions---image plane, time, lens and integration samples (area lights, etc..). Therefore, you guarantee that all n dimensions of the sample are decorrelated (pixelSamples is the number of necessary samples in a pixel) .

In the first attempt, I generated a new sequence of samples for each pixel, by increasing i by pixelSamples, hoping that this would yield decorrelated samples between the different pixels. I also experimented with regular and folded radical inverse functions, yielding so called Hammersley-Zaremba sequences. It turns out that the latter have slightly less discrepancy, as can be seen in the blowups below. The following renderings show that my Hammersley performs equally well as pbrt's built-in scrambled LDSampler for 2 and 4 times supersampling, and is already significantly superior to jittered supersampling at 2 samples per pixel.

Once I tried out rendering some scenes involving area lights, I observed strange artifacts in the shadows, as can be observed in the following renderings:

These artifacts encouraged me to do some googling, which led me to a paper by Alexander Keller: Strictly Deterministic Sampling Methods in Computer Graphics. The figure shows the interesting observation that in fact, the individual components in the n-dimensional Hammersley sequences are not decorrelated!

First I didn't realize that this was the cause of the strange artifacts I had with the area lights and soft shadows, as the render of the same buddha scene with a point light doesn't show the same artefacts. Then, I realized that the samples assigned to the area light integrator in this particular scene correspond to the 8th and 9th component of the n-dimensional sample. Indeed, as the paper mentions, the correlation is only really apparent starting from the 7th component. Indeed, even if the Hammersley sequence guarantees that the components have low discrepancy in between components in the same sample, components between different samples can be highly correlated. Therefore, I had to come up with a way to decorrelate the components between samples.

Hammersley sampling with Cranley-Patterson rotations

I opted to reuse the initial sequence of pixelSamples samples (n-dimensional) for all pixels, transforming the original sequence randomly for each pixel. I do this by so called Cranley-Patterson rotations. These rotations compute X'(i) = (X(i) + r(i)) mod 1 in each dimension, where X(i) is the sample value and r(i) is a random number between zero and one.

I thus added an option to switch between both methods in the HammersleyLDSampler plugin, and re-rendered the previous scenes with Cranley-Patterson rotations turned on. As can be observed from the next renderings, the artifacts have gone away. Another advantage of this approach is computation efficiency. The table below summarizes a few rendering times for the checkerboard and scene on my Centrino 1.7Ghz laptop, parameterized by function technique and number of pixel samples. Because I no longer compute pixelSamples Hammersley sequences for each pixel, but only once, the computation time is significantly reduced. The Cranley-Patterson rotations can be computed very efficiently as a 'wrapping' operation to confine the result of the sum between 0 and 1. Also note that non-folded radical inverse is computed slightly more efficiently.

Here, I compare zooms of the shadows cast by the buddha figure, for the different sampling techniques so far, each with 16 light samples per pixel. It seems that the rotations are increasing the discrepancy of the samples in the pixels. Compared to the PBRT's samplers, mine seems to have a little more clumping artefacts here.

The last comparison shows that the HammersleyLDSampler ensures good distribution of samples over the pixel area over all dimensions. To see this, note from the renderings that equally good results are obtained weither we use a combination of 1 image sample per pixel and 16 light samples per pixel or 1 image sample per pixel and 16 light samples per pixel. This would not be the case for a jittered sampler, as is shown in figure 7.23 of Pharr and Humphrey's book.

Larcher based scrambled sampling

While pbrt's LDSampler uses an efficient technique to generate a scrambled Sobol sample for 2D samplers (using bit-ops only), this paper also mentions an equally efficient technique by Larcher and Pillichshammer. I added an option to the LDSampler plugin to support that technique as well as the Sobol technique. My timings and the following renders point out that both are equally good and equally efficient (in fact, the only difference is that the AND operation is replaced by an OR operation).

Reconstruction methods

As mentioned in Pharr and Humphrey's book, the choice of reconstruction filter for low-discrepancy filters can have surprising effects. The low-discrepancy sampler LDSampler, unlike the best-candidate sampler, doesn't guarantee that samples in adjacent pixels are placed well distributed in regard to their neighbors. When used with a box filter, this sampling method works well, but when a filter that spans multiple pixels and isn't a constant value is used, it becomes less effective. The following renderings test how well my own HammersleyLDSampler with Cranley-Patterson rotations distributes samples in regard to neighboring pixels. All renderings use 1 image sample and 64 light samples per pixel. I expected that it would do a better job, because of the rotations, but I don't really believe it does. The rendering filtered with a 2-pixel wide Mitchell filter seems to be 'blurrier'.

References and code

1. Physically based rendering, Pharr and Humphrey's Morgan Kaufmann
2. Strictly Deterministic Sampling Methods in Computer Graphics, Alexander Keller
3. Efficient Multidimensional Sampling, Alexander Keller and Thomas Kollig

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