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In this paper, we study the Light Transport Operator
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This operator transports radiance fields in a scene
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to the next surface, where it combines them with material properties
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Applying the operator to a light source as shown here
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gives direct illumination
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Applying it a second time gives the next bounce, etc
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It is known that the light transport equilibrium in a scene
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can be obtained by adding all bounces produced by this operator
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When the scene is Lambertian, this operator can be
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re-formulated in the space of radiant exitance distributions
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Both operators T and Tb act on the infinite dimensional
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spaces of light distributions over the scene
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Consequently, many light transport problems are solved numerically
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by projecting radiance, or radiant exitance,
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onto some finite dimensional spaces of functions
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and express T (or Tb) in this space, as a matrix Tn
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These functions can be piecewise constant functions, as in old radiosity methods
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wavelets, spherical harmonics, polynomials,
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or any combination of these.
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Many methods rely on such finite rank approximations
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Finite element global illumination
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in which the implicit assumption is that
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for the actual solution L, the matrix approaches the operator
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Precomputed radiance transfer methods
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where the assumption is stronger
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The matrix should indeed give a uniform error
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across an entire subspace of light distributions
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Pretty much the same is expected for neural rendering
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where the learning phase implicitly builds a finite rank approximation of T
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Inverse lighting. In this case a finite rank approximation of T
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is used to approximate the inverse of T
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...and dimensional analysis where the eigenvalues of Tn
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are supposed to converge to the eigenvalues of T
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While most analyses in the literature have been done with discretisation,
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in this paper we study the original operators
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and show that some of the above assumptions are not at all straightforward
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Generally speaking, linear operators in infinite dimensions behave badly
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for instance they may not have a countable set of eigenvalues
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contrary to a matrix
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and even if they do
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these eigenvalues may have infinite multiplicity
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A consequence of that
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is that the spectrum of a matrix approximating the operator
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may not converge to the spectrum of the operator itself
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Another consequence is that
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finite rank approximations are not guaranteed to be uniform
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In other words, there is no matrix approximation
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that will guaranty a maximum error across all light distributions
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A subclass of infinite dimensional linear operators however behave better:
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these are the so-called compact operators
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Compact operators are the operators with
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countable eigenvalues that converge to 0
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In this case, the spectrum of a matrix approaching the operator
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is going to converge to the spectrum of the operator
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and finite rank approximations of the operator
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come with a uniform error bound across all distributions
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So, now, the question is:
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Are the light transport operators compact?
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In this paper we show that T is never compact
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However, T still shares some key properties with compact operators
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In particular we show that in closed scenes
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T has a Schmidt expansion
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which stands for a SVD in infinite dimensions
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We also show that Tb is not compact either
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But still Tb keeps some interesting properties of compact operators
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First of all, it is not invertible
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and it acts as a low pass filter almost everywhere
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We also prove that the reflectance operator Kx
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which combines the incident illumination with the material at point x
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is compact, which makes it not invertible
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Finally in the paper we connect these findings
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to the various choices that characterize low rank approximations in the literature
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Now I'll quickly explain how we prove that Tb is not compact
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Compact linear operators map bounded sequences
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into sequences with converging subsequences
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So, to prove non-compactness
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we just need to find one sequence
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of radiosity functions
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that is mapped to a sequence which elements
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are always far away from each other
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In a finite dimensional space
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this would obviously not be possible
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for a bounded operator such as Tb
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But in infinite dimensions, we can
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create such a sequence
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by leveraging the increase of frequency in its elements
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so as to keep each element away from all other elements
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This is precisely what happens next to an abutting edge
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We choose each radiosity distribution Bn
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to be constant over a disk
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which radius decreases with n
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Keeping the Bn sequence bounded is easy
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we increase the radiant exitance over the disk
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so as to keep the norm of B_n constant with n
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The sequence of transported energy distributions
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Tb times Bn is of course bounded as well
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since the total energy received will always be less than the emitted power
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However, we show in our paper that
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the frequency of transported radiosity increases in this case
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keeping the transported distributions far away from each other
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Consequently, no subsequence of Tb times Bn
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can ever converge
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Of course a much more detailed proof
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is provided in our paper
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The first consequence of light transport operators not being compact
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is that there is no uniform finite rank approximations of these operators
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In other words,
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For any finite rank approximation
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There is no guarantee on the error
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across all light distributions of unit norm
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This is why adaptive methods are needed
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when solving for global illumination,
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this is also why they require guidance
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Another consequence is that
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neither precomputed radiance transfer methods
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nor deep learning methods
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can give error guarantees beyond their learning space
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Similarly, spectral analysis based on finite rank approximations
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need some additional justification
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Another consequence is that
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approaching light transport by operators with bounded kernels
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is not going to converge uniformly to the real light transport operator
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That explains somehow the bias
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that shows up when bounding the weight
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of connecting close points in path tracing
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Since reflectance is compact, it is not invertible
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Consequently, solving for inverse reflectance
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Using a finite rank approximation
Consequently, solving for inverse reflectance
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Using a finite rank approximation
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is probably not a good idea
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The same happens to Tb
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that is not invertible either
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If we try to for instance find
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the radiosity distribution in the top square that
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produces a step function on the bottom square
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the more we mesh, the higher the values
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and the higher the frequencies
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it looks like the solution somehow tries to escape
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the space in which we look for it
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It should be noted however that
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inverting multi-bounce transport is trivial.
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Our conclusion is that the light transport operators
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are generally not well behaved
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and one needs to be very cautious when using
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finite rank approximations
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We also prove that T_b in the Lambertian case
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is not invertible, which was not obvious to us at first
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Interestingly the cause of these deficiencies
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is not visibility
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Usually visibility is...what causes problems
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Here T is not compact because T is a partial integral operator
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and Tb is not compact because of abutting edges
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So, that concludes my presentation
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Thanks you for your attention