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I'm making a last minute recruiting trip to the bay area this Weds - Sun. I've been lucky to recently cofound a VR content company (8i.com) with an already extraordinary team. We are looking for more talented driven people to help create the future of recorded holographic content with us. Facebook didn't drop 2 billion dollars on Oculus because gaming is the future: the future is much bigger. If you or someone you know is brave enough to pioneer a whole new entertainment medium with us: NOW IS THE TIME. e@8i.com.

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Kind of fxguide to write about my Digipro talk in such detail. Now that I've shifted away from digging through neutron transport papers and into other things light field related, I'll soon be releasing a book 'A Hitchhiker's Guide to Multiple Scattering' as an aid for navigating the transport literature (which is a complete mess). I hope this will save other inspired researchers a ton of time and torture.

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Here's a link to Placzek's lemma in the old text I cited today in the BSSRDF talk.

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Additional info related to today's zero-variance subsurface talk that hopefully clarifies the topic somewhat and that we (clearly) didn't have time to include:

We consider subsurface sub-paths of some full/longer path created during rendering, and treat 'escape' as our source of importance, although this is not a necessary step: the general theory permits knowing (in theory) the full directional radiance distribution inside some volume due to a particular light source(s) outside of it. In this much more complicated case, an exact or approximate internal radiance solution would guide the subsurface sub-paths not only to try to escape the medium, but also tend to positions which permit angle selections that then leave the medium in a direction that tends to hit the light. However, this would be quite complex in practice.

Because we only assumed 'escape' as our goal, the angle selection only involved a cosine: the azimuthal angle selections at each step are chosen uniformly. This is one feature that would change if, say, you knew the source of light outside the medium came from a particular direction. In these cases, more advanced deterministic solutions interior to the medium would be required, and work that Wenzel presents tomorrow and that I will present on Thursday could both be used to produce such importance functions for walk guidance (for the plane-parallel directional light source case, or the point source near a subsurface surface case).

In the case that there is no absorption, the classical subsurface walk is already zero variance: you just keep sampling (with whatever sample weight you had as you entered) until finally you exit: with that same weight no matter where you went, so there is nothing to improve upon in that case (unless, as above, you knew more about where light sources might enter the medium).

Despite the zero variance theory being old, Hoogenboom's recent 2008 NSE article is very important: he corrects some misconceptions about the uniqueness of zero-variance walk construction that have lingered for several decades, and includes discussions about boundary crossing and track-length estimators as well. He describes precisely how to construct a truly zero-variance walk for a half-space that considers escaping from the medium exactly (as opposed to the Dwivedi assumption that the exponential asymptotic term extends outward) which is how we improved the results Jaroslav showed at the very end.

Regarding Wojciech's question: in addition to issues with how the Dwivedi sampling procedure doesn't leave a semi-infinite medium in an optimal way, there are further issues when the medium isn't semi-infinite at all (like a nose/ear). We remember the surface normal at the point of entry and walk around assuming a semi-infinite importance solution aligned to that normal and position, but this is clearly pretty bad for general shapes. This is where the MIS mixing really saves us.

With a 75% mix of Dwivedi and 25% classical, we invariably found lower variance / cpu time for flat or curved surfaces with smooth or rough Fresnel boundaries and with white-sky or general IBL lighting.

If you know a material is thin and know its thickness, you can apply the Dwivedi walk to guide towards the transmission side or both sides of the material. (Case's asymptotic solution then has two exponentials). You can get some incredible variance reductions for transmission through optically thick slabs (as the radiation shielding community also showed). However, this requires knowing when you hit a surface that it is well approximately by some slab of some known thickness.

The exact half-space solutions that we use can be derived in several ways, including what is known as Caseology, named after Kenneth Case who was on the theoretical division at Los Alamos building the first bombs. In 1960 Case studied in detail the 'spectrum' of the transport operator in the plane-parallel case, and looked at the expansion into the discrete asymptotic diffusion mode, and the continuous spectrum of 'singular eigenfunctions' - so named because the angular distributions for the transient terms are singular (yet their superposition cancels out everywhere to produce the right answer). This study of the structure of exact solutions in transport is incredibly insightful and we are the first, to our knowledge, to exploit it directly in a rendering technique.

To clarify one comment Jaroslav made regarding the bounadry conditions and their effect on the interior solution: changing the roughness or Fresnel ratio at the boundary does actually change the magnitude of the asymptotic term, but not it's decay rate. Because we discard the transient singular terms and renormalize, this expansion coefficient goes away, and the Dwivedi scheme is then completely invariant to the direction that incoming light hits the surface, or to the boundary conditions. As the phase function changes away from isotropic, more and more discrete diffusion asymptotic terms appear, and you can pick the largest one and still apply Dwivedi with success (we've tried HG with g = 0.75 and still get a nice variance reduction).

We consider subsurface sub-paths of some full/longer path created during rendering, and treat 'escape' as our source of importance, although this is not a necessary step: the general theory permits knowing (in theory) the full directional radiance distribution inside some volume due to a particular light source(s) outside of it. In this much more complicated case, an exact or approximate internal radiance solution would guide the subsurface sub-paths not only to try to escape the medium, but also tend to positions which permit angle selections that then leave the medium in a direction that tends to hit the light. However, this would be quite complex in practice.

Because we only assumed 'escape' as our goal, the angle selection only involved a cosine: the azimuthal angle selections at each step are chosen uniformly. This is one feature that would change if, say, you knew the source of light outside the medium came from a particular direction. In these cases, more advanced deterministic solutions interior to the medium would be required, and work that Wenzel presents tomorrow and that I will present on Thursday could both be used to produce such importance functions for walk guidance (for the plane-parallel directional light source case, or the point source near a subsurface surface case).

In the case that there is no absorption, the classical subsurface walk is already zero variance: you just keep sampling (with whatever sample weight you had as you entered) until finally you exit: with that same weight no matter where you went, so there is nothing to improve upon in that case (unless, as above, you knew more about where light sources might enter the medium).

Despite the zero variance theory being old, Hoogenboom's recent 2008 NSE article is very important: he corrects some misconceptions about the uniqueness of zero-variance walk construction that have lingered for several decades, and includes discussions about boundary crossing and track-length estimators as well. He describes precisely how to construct a truly zero-variance walk for a half-space that considers escaping from the medium exactly (as opposed to the Dwivedi assumption that the exponential asymptotic term extends outward) which is how we improved the results Jaroslav showed at the very end.

Regarding Wojciech's question: in addition to issues with how the Dwivedi sampling procedure doesn't leave a semi-infinite medium in an optimal way, there are further issues when the medium isn't semi-infinite at all (like a nose/ear). We remember the surface normal at the point of entry and walk around assuming a semi-infinite importance solution aligned to that normal and position, but this is clearly pretty bad for general shapes. This is where the MIS mixing really saves us.

With a 75% mix of Dwivedi and 25% classical, we invariably found lower variance / cpu time for flat or curved surfaces with smooth or rough Fresnel boundaries and with white-sky or general IBL lighting.

If you know a material is thin and know its thickness, you can apply the Dwivedi walk to guide towards the transmission side or both sides of the material. (Case's asymptotic solution then has two exponentials). You can get some incredible variance reductions for transmission through optically thick slabs (as the radiation shielding community also showed). However, this requires knowing when you hit a surface that it is well approximately by some slab of some known thickness.

The exact half-space solutions that we use can be derived in several ways, including what is known as Caseology, named after Kenneth Case who was on the theoretical division at Los Alamos building the first bombs. In 1960 Case studied in detail the 'spectrum' of the transport operator in the plane-parallel case, and looked at the expansion into the discrete asymptotic diffusion mode, and the continuous spectrum of 'singular eigenfunctions' - so named because the angular distributions for the transient terms are singular (yet their superposition cancels out everywhere to produce the right answer). This study of the structure of exact solutions in transport is incredibly insightful and we are the first, to our knowledge, to exploit it directly in a rendering technique.

To clarify one comment Jaroslav made regarding the bounadry conditions and their effect on the interior solution: changing the roughness or Fresnel ratio at the boundary does actually change the magnitude of the asymptotic term, but not it's decay rate. Because we discard the transient singular terms and renormalize, this expansion coefficient goes away, and the Dwivedi scheme is then completely invariant to the direction that incoming light hits the surface, or to the boundary conditions. As the phase function changes away from isotropic, more and more discrete diffusion asymptotic terms appear, and you can pick the largest one and still apply Dwivedi with success (we've tried HG with g = 0.75 and still get a nice variance reduction).

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We are an exciting, funded tech startup in New Zealand with an amazing team and looking for more amazing people to join us, right away. Holographic content has already begun to take off. So look for me at SIGGRAPH this week - I'll be wearing the 8i t-shirts. Send us your resume on our website. We'd love to hear from you. Now is the time!

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Tomorrow at Digipro I'm pleased to have the opportunity to shed some light on my experience over the past 5 years wading through old transport theory literature in search of answers to questions I've had since 2007 (about the dipole, specifically). It ultimately led me into unexpected territory, connecting me with a wealth of new and old ideas that dramatically impacted my own research and also connected me with many amazing people within and outside of graphics who enjoy solving very similar problems, which has been equally rewarding. Thanks Larry Gritz for encouraging me to submit something along these lines, and especially to Joe Letteri, Sebastian Sylwan and Luca Fascione at Weta Digital for trusting me with what must at times have seemed like distractions. The conference is fully booked, but if you can't make it I hope to share my slides shortly after SIGGRAPH.

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If you are interested in our recent BSDF importance sampling paper using the distribution of visible normals please also see Wenzel's note on avoiding the sample space discontinuities (important for QMC).

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Teaser of our upcoming EGSR paper on importance sampling Beckmann and GGX rough conductor and dielectric BSDFs analytically in about the same time as previous methods, but including all terms: even the masking function (the sample weight becomes the shadowing function alone).

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I decided to try WordPress and update my ancient website. It has a blog feature - I'll give it a go for a while. Please let me know if you have any issues with the site. Something new: I made some visualizations of the prefix/suffix analysis of the North American Scrabble dictionary: so now it is much easier to see how the various fixes rank, and what their breakdown is.

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This has been accepted with no additional changes (an updated arxiv will appear in about a day) to the Journal of Computational and Theoretical Transport (formerly TTSP). The reader in computer graphics might find this interesting if you've ever wondered:

a) How are transport theory and the theory of random flights related?

b) There is more than one diffusion approximation!? How, why?

c) How are the various diffusion approximations (like Grosjean's modified diffusion) derived, and what are their various tradeoffs?

d) How would diffusion approximations change if you considered scattering volumes with partially-correlated scattering particles like that EGSR paper with a bowl of glass Buddhas (which leads to non-exponential free path distributions)?

e) How does diffusion theory change if you consider multiple scattering in spaces with number of dimensions other than 3?

f) What is the 'spectrum' of the transport operator, what are these 'singular eigenfunctions' in Caseology, and how does this all relate to diffusion theory?

a) How are transport theory and the theory of random flights related?

b) There is more than one diffusion approximation!? How, why?

c) How are the various diffusion approximations (like Grosjean's modified diffusion) derived, and what are their various tradeoffs?

d) How would diffusion approximations change if you considered scattering volumes with partially-correlated scattering particles like that EGSR paper with a bowl of glass Buddhas (which leads to non-exponential free path distributions)?

e) How does diffusion theory change if you consider multiple scattering in spaces with number of dimensions other than 3?

f) What is the 'spectrum' of the transport operator, what are these 'singular eigenfunctions' in Caseology, and how does this all relate to diffusion theory?

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