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This image was generated using a renderer developed during my PhD using the bidirectional path tracing algorithm.

In ray tracing based rendering algorithms the main performance bottleneck comes from the time it takes to perform intersection tests between rays and the scene geometry. As the number of elements used to represent surfaces within the scene increases so to does the computation complexity of performing these intersection tests.

Rather than testing a given ray against each surface element present in the scene, which would scale poorly with O(n) complexity, acceleration structures such as the KD-Tree and BHV-Tree can be used to reduce the number of intersection tests to roughly O(log n) on average. This is accomplished by traversing a binary-tree like hierarchy performing a comparatively cheap intersection test between a ray and a cutting plane or axis-aligned bounding-box. When the leaf nodes of the tree have been reached only the mesh elements contained within the leaf node need to be tested against the given ray. After an initial intersected element has been found all nodes of the tree which can only contain elements that would have to be further away than the current closest intersection can be ignored as the tree traversal continues, yielding significant performance gains.

The Stanford Dragon model shown in this image contains 871,414 triangular mesh elements. In the above image, 99 copies of the dragon are present. By a naive scene construction method this would total 86,269,986 triangles. This has the effect of greatly increasing the memory requirements and size of the resulting BVH-Tree. In an ideal world we would only store the dragon mesh and a BVH-Tree over its elements once, and have a system for replaying this data efficiently in arbitrary positions, rotations, and scales.

To this end I have implemented geometric instancing as a feature within the rendering software developed for my PhD research. This is done through a coordinate-space transformation and cross-linking injected into the acceleration structure traversal. The dragon mesh is loaded into the renderer, and a BVH-Tree is constructed for the single dragon model. The root node of the BVH-Tree's bounding box is then duplicated through independent affine-transformations into the size, rotation, and position of each of the desired dragons. Another BVH-Tree is then constructed over the these bounding boxes, along with the triangles making up the floor, walls, and light source of the scene. During traversal of the BVH-Tree, when one of the transformed bounding boxes is queried, the given ray is transformed by the inverse of the affine-transformation matrix for the current bounding box. This inverse transformed ray is then intersected with the BVH-Tree containing only the dragon mesh. The resulting intersection location can be found by scaling the original ray by the resulting distance to the closest intersection found, regardless of whether the intersection was within one of the nested bounding boxes.

With some additional work, each instanced mesh can have different materials applied to it through a stack based shading hierarchy which is also tracked during BVH-Tree traversal.

This image was generated using a renderer developed during my PhD using the bidirectional path tracing algorithm in submission of the 2015, SURF Research as Art Competition, hosted for reasearcher at Swansea University.

The image is rendered at 4k resolution and was rendered in approximately 8 hours to a high degree of convergence by utilizing 50 of the machines in the Swansea Computer Science Department's linux teaching lab as a computational cluster using the Message Passing Interface (MPI) framework to implement multi-node parallism into my rendering software.

This image was generated using a renderer developed during my PhD using the bidirectional path tracing algorithm.

The pig model is shaded with my interpretation of a glazed ceramic material. This combines a diffuse ceramic substrate with a glossy overlayer which are Fresnel blended during shading computation. The glaze overlay has a normal map applied to it for adding detailed surface variation.

This image was generated using a renderer developed during my PhD using the bidirectional path tracing algorithm.

Just a quick render after implementing an Anisotropic Metal material in my renderer. The test scene was inspired by one I saw on Kevin Beason's worklog blog.

Yesterday I stumbled upon a lesser known and far superior method for mapping points from a square to a disk. The common approach which is presented to you after a quick google for Draw Random Samples on a Disk is the clean and simple mapping from Cartesian to Polar coordinates; i.e.

Given a disk centered origin (0,0) with radius r

// Draw two uniform random numbers in the range [0,1)
R1 = RAND(0,1);
R2 = RAND(0,1);

// Map these values to polar space (phi,radius)
phi = R1 * 2PI;
radius = R2 * r;

// Map (phi,radius) in polar space to (x,y) in Cartesian space
x = cos(phi) * radius;
y = sin(phi) * radius;

The result of this sampling on a regular grid of samples is shown in the image below. The left plot shows the input points as simple ordered pairs in the range [0,1)^2, while the right plot shows these same points (colour for colour) mapped onto a unit disk using Polar mapping as described above.

As you can see the mapping is not ideal with many points being over-sampled at the poles (I wonder why they call is Polar coordinates), and with areas towards the radius left under-sampled. What we would actually like is a disk sampling strategy that keeps the uniformity seen in the square distribution while mapping the points onto the disk.

Enter, Concentric Disk Sampling. A Low Distortion Map Between Disk and Square, Shirley & Chiu 1997 presents the idea for warping the unit square into that of a unit circle. Their method is nice but it contains a lot of nested branching for determining which quadrant the current point lays within. Shirley mentions an improved variant of this mapping on his blog, accredited to Dave Cline. Cline's method only uses one if-else branch and is simpler to implement.

Again, given a disk centered origin (0,0) with radius r

// Draw two uniform random numbers in the range [0,1)
R1 = RAND(0,1);
R2 = RAND(0,1);

// Avoid a potential divide by zero
if (R1 == 0 && R2 == 0) {
    x = 0; y = 0;
    return;
}

// Initial mapping
phi = 0; radius = r;
a = (2 * R1) - 1;
b = (2 * R2) - 1;

// Uses squares instead of absolute values
if ((a*a) > (b*b)) { 
    // Top half
    radius  *= a;
    phi = (pi/4) * (b/a);
}
else {
    // Bottom half
    radius *= b;
    phi = (pi/2) - ((pi/4) * (a/b)); 
}

// Map the distorted Polar coordinates (phi,radius) 
// into the Cartesian (x,y) space
x = cos(phi) * radius;
y = sin(phi) * radius;

This gives a uniform distribution of samples over the disk in Cartesian space. The result of the mapping applied to the same set of uniform square samples is shown above. Notice how we now get full coverage of the disk using just as many samples, and that each point has (relatively) equal distance to all of it's neighbors, meaning no bunching at the poles, and no under-sampling at the fringe.

I've applied this sampling technique to my path tracer as a means of sampling the aperture of the virtual camera when computing depth of field. Convergence to the true out-of-focus light distribution is now much faster and more accurate than it was with Polar sampling which, due to oversampling at the poles, cause a disproportionate number of rays to be fired along trajectories very close to the true ray.