Physically accurate rendering often calls for taking the wave nature of light into consideration. In computer graphics, this is done almost exclusively locally, i.e. on a micrometre scale where the diffractive phenomena arise. However, the statistical properties of light, that dictate its coherence characteristics and its capacity to give rise to wave interference effects evolve globally: these properties change on, e.g., interaction with a surface, diffusion by participating media and simply by propagation. In this paper, we derive the first global light transport framework that is able to account for these properties of light and, therefore, is fully consistent with Maxwell's electromagnetic theory. We show that our framework is a generalization of the classical, radiometry-based light transport—prominent in computer graphics—and retains some of its attractive properties. Finally, as a proof of concept, we apply the presented framework to a few practical problems in rendering and validate against well-studied methods in optics.
Teaser selected for the back cover of the ACM TOG issue.
In rendering, a pair of approaches are commonly used to describe light:
A radiometric, geometric optics formulation
This implies that we quantify the time-averaged power carried by light (via the radiometric quantities of radiance or irradiance).
Such a description neglects the electromagnetic wave-nature of light, and it can be shown that radiometry is insufficient in formulating a model of light that is consistent with electromagnetism.
However, the wave-nature of light plays a crucial role in a variety of optical phenomena: for example, the colourful appearance of scratches in a metal surface, butterfly wings, oil on water and soap bubbles and the diffraction grating of compact disks and LCD screens.
This makes a radiometric description wholly lacking, and we would like to simulate light behaviour using a more physical description of light.
The importance of accounting for the wave-nature light goes beyond rendering and computer graphics.
Many highly practical applications use electromagnetic radiation — light — for a variety of purposes, e.g., propagation of radar, WiFi and cellular waves, non-optical imaging, and so on.
It is also noteworthy that diffraction effects become more pronounced when dealing with electromagnetic waves of non-optical frequencies, because such waves are typically of significantly longer wavelength compared to visible light.
A deterministic description of the electric field
In computer graphics, a deterministic electromagnetic formulation of light has been successfully used in order to render some material appearance effects, such as thin-film interference and diffractive surface features.
The electric field is assumed to be perfectly coherent and the interaction of this field with matter is described using some simplified optical model (“wave optics”), suitable for the material appearance that each work aims to reproduce.
However, because this approach fails to capture the disorganized nature of partially-coherent light, this formulation of light is only able to adequately model light in a small, local region.
That is, once light has propagated a distance large with respect to the coherent of light, its waveforms lose similarity to the previous waveforms.
These changes are difficult to model deterministically.
The above means that modern rendering techniques (as well as a variety of SBR methods in applied optics and computational electrodynamics) use geometric optics to transport light globally throughout the scene, and physical optics is only used locally, in order to model a particular effect.
Once the local wave-matter interaction has been computed, the fact that light consists of electromagnetic waves is forgotten, and the formalism reverts back to a radiometric description to propagate light away.
Such an approach is inherently flawed.
Virtually all the light that we observe daily and render with is partially-coherent, which means that this light admits a significant degree of random fluctuations.
These statistical wave properties of light — its optical coherence — play a crucial role in quantifying the light-matter interactions and the observable visual responses that arise of these interactions.
For a concrete real-world example, see the soap bubbles video in the research overview page, where light of similar spectrum and intensity is incident upon an object, but difference in its coherence gives rise to strikingly different optical responses.
This makes a light transport formulation that is able to transport these statistical wave-properties of light globally very desirable.
It is also worthy to note that dealing with the underlying electromagnetic fields directly is not only difficult, but also counter-intuitive:
at optical frequencies we do not directly observe and measure the oscillations of the electromagnetic fields, but only their time-averaged values.
Therefore, the impractical detailed analysis of the highly-numerous electromagnetic wave constituents, that compose partially-coherent light, is not even needed:
it is the statistical properties of light that dictate its behaviour and observable properties that arise.
In this paper, we aim to change the current state of affairs in computer graphics and introduce a global physical light transport formalism.
The discussion above prompts us to abandon a deterministic model of light in favour of a purely statistical one, which is consistent with the theory of electromagnetism.
This allows us to treat light as electromagnetic waves globally and propagate the waves’ statistical coherence properties throughout a scene.
Under this formalism, light is modelled as a stochastic process – a wave ensemble.
The primary quantity of interest, which replaces the classical radiance, is the cross-spectral density function (CSD).
The CSD is a function of two points in space, and should be understood as the second-order statistical moment, which quantifies how statistically similar are the light waves (of a given wavelength) that arrive at these pair of points.
High statistical similarity means that the shape of the waveforms that arrive at these pair of points are similar, and therefore constructive- and destructive-wave interference effects may survive time-averaging.
On the other hand, low statistical similarity of light between a pair of points, implies that constructive- and destructive-wave interferences happen with an alike regularity, and such effects cancel out on time-average.
Remember: we only observe time-averaged values.
Clearly, important information about the light’s wave-properties is quantified by that CSD.
A question remains: are second-order statistics sufficient to describe the observable properties of that light?
Yes! We observe light over periods of milliseconds, far longer than the typical polychromatic light’s temporal coherence.
That is, over the course of one observation, many statistically-independent waveforms contribute to the perceived intensity of light, therefore the Central Limit Theorem can be called upon.
The observed properties of polychromatic light are then point-wise Gaussian, and the second-order statistics are sufficient to fully describe these observable properties.
Using the CSD to describe light, we can quantify any observable optical phenomena that occurs with partially-coherent light that can be explained electromagnetically.
One of our primary contributions is the spectral-density transport equation (SDTE), which is derived in the paper.
The SDTE is the physical optics analogue of the classical rendering equations, and provides a computationally-tractable formulation of transport of these CSD function.
The SDTE is a linear integral equation (just like the classical rendering equation), which is what makes it amenable to traditional Monte-Carlo integration methods.
This linearity is a consequence of us dealing with partially-coherent light.
Linearity is of value, because it suggests that we may adapt some of the state-of-the-art rendering work, that aims to solve the classical rendering equation, to physical optics.
The extensive supplemental material includes in-depth discussion and derivation of the relevant mathematics and aspects of optical coherence theory.