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