A3T7 Environment Lighting

The final task of this assignment will be to implement a new type of light source: an infinite environment light. An environment light is a light that supplies incident radiance (really, the light intensity $\frac{d\Phi}{d\Omega}$) from all directions on the sphere. Rather than using a predefined collection of explicit lights, an environment light is a capture of the actual incoming light from some real-world scene; rendering using environment lighting can be quite striking.

The intensity of incoming light from each direction is defined by a texture map parameterized by phi and theta, as shown below.

In this task you will get Environment_Lights::Sphere working by implementing Samplers::Sphere::Uniform and Samplers::Sphere::Image in src/pathtracer/samplers.cpp. You'll start with uniform sampling to get things working, and then move onto a more advanced implementation that uses importance sampling to significantly reduce variance in rendered images.

Note that for the purposes of this task, (0,0) is actually the bottom left of the HDR image, not the top left. Think about how this will affect your calculation of the $\theta$ value for a pixel.

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Step 0: Know your customers

First, check out the interface and implementation of Environment_Lights::Sphere in src/scene/env_light.h/.cpp. Particularly, path attention to how it initializes and uses its member importance to sample directions from an HDR_Image, which it also passes to the constructor of importance. (The HDR_Image interface may be found in util/hdr_image.h.)

Step 1: Uniformly sampling the environment map

Implement Sphere::Uniform::sample in src/pathtracer/samplers.cpp.

Now make an implementation of Sphere::Image::sample and Sphere::Image::pdf that call Sphere::Uniform::sample and Sphere::Uniform::pdf.

This should be sufficient to get environment maps working in the renderer (albeit in a high-variance / slow-convergence way).

Since high dynamic range environment maps can be large files, we have not included them in the Scotty3D repository. You can download a set of sample environment maps here or -- for more interesting environment maps -- check out poly haven.

To use a particular environment map with your scene, select layout -> Create Object -> Environment Light Instance, then set the underlying Light type to Sphere, add a new Texture, set the texture type to Image and, finally, press Change and select your file.

Step 2: Importance sampling the environment map

Much like light in the real world, most of the energy provided by an environment light source is concentrated in the directions toward bright light sources. Therefore, it makes sense to prefer sampling directions for which incoming radiance is the greatest. For environment lights with large variation in incoming light intensities, good importance sampling will significantly reduce the variance of your renderer.

The basic idea of importance sampling an image is assigning a probability to each pixel based on the total radiance coming from the solid angle it subtends.

A pixel with coordinate $\theta = \theta_0$ subtends an area $\sin\theta d\theta d\phi$ on the unit sphere (where $d\theta$ and $d\phi$ are the angles subtended by each pixel as determined by the resolution of the texture). Thus, the flux through a pixel is proportional to $L\sin\theta$. (Since we are creating a distribution, we only care about the relative flux through each pixel, not the absolute flux.)

Summing the flux for all pixels, then normalizing each such that they sum to one, yields a discrete probability distribution over the pixels where the probability one is chosen is proportional to its flux.

The question is now how to efficiently get samples from this discrete distribution. To do so, we recommend treating the distribution as a single vector representing the whole image (row-major). In this form, it is easy to compute its CDF: the CDF for each pixel is the sum of the PDFs of all pixels before it. Once you have a CDF, you can use inversion sampling to pick out a particular index and convert it to a pixel and a 3D direction.

The bulk of the importance sampling algorithm will be found as Samplers::Sphere::Image in src/pathtracer/samplers.cpp. You will need to implement the constructor, the inversion sampling function, and the PDF function, which returns the value of your PDF at a particular direction.

Be sure your Samplers::Sphere::Image::pdf() function takes into account the fact that different elements of your computed pdf_ take up different areas on the surface of the sphere (so need to be weighted differently).

Or, to say that more verbosely: the PDF value that corresponds to a pixel in the HDR map should be multiplied by the Jacobian below before being returned by Samplers::Sphere::Image::pdf.

The Jacobian for transforming the PDF from the HDR map sampling distribution to the unit sphere sampling distribution can be thought of as two separate Jacobians: one to a rectilinear projection of the unit sphere, and then the second to the unit sphere from the rectilinear projection.

The first Jacobian scales the $w \times h$ rectangle to a $2\pi \times \pi$ rectangle, going from $(dx, dy)$ space to $(d\phi, d\theta)$ space. Since we have a distribution that integrates to 1 over $(w,h)$, in order to obtain a distribution that still integrates to 1 over $(2\pi, \pi)$, we must multiply by the ratio of their areas, i.e. $\frac{wh}{2\pi^2}$. This is the first Jacobian.

Then in order to go from integrating over the rectilinear projection of the unit sphere to the unit sphere, we need to go from integrating over $(d\phi, d\theta)$ to solid angle $(d\omega)$. Since we know that $d\omega = \sin(\theta) d\phi d\theta$, if we want our new distribution to still integrate to 1, we must divide by $\sin(\theta)$, our second Jacobian.

Altogether, the final Jacobian is $\frac{wh}{2\pi^2 \sin(\theta)}$.

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Tips

• When computing areas corresponding to a pixel, use the value of theta at the pixel centers.

• Compute the PDF and CDF in the constructor of Samplers::Sphere::Image, storing them values in fields _pdf and _cdf. See src/pathtracer/sampler.h.

• Spectrum::luma() returns the luminance (brightness) of a Spectrum. The weight assigned to a pixel should be proportional both its luminance and the solid angle it subtends.

• For inversion sampling, use std::upper_bound: it's a binary search. Read about it here.

• If you didn't use the ray log to debug area light sampling, start using it now to visualize what directions are being sampled from the environment map.

• src/scene/shapes.h/.cpp declare/define Sphere::uv which converts from directions to lat/lon space.

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Reference Results

Extra Credit

• Implement a version of Samplers::Sphere::Image that uses the alias method to do constant-time sampling (as opposed to the inversion sampling approach described above). Allow the methods to be switched with a constexpr bool global flag (along the lines of, e.g., SAMPLE_AREA_LIGHTS in pathtracer.cpp). Report the speedup (if any) of your alias method sampler as part of your write-up.