  The many-light formulation provides a general framework for rendering various
illumination effects using hundreds of thousands of virtual point lights
(VPLs). To efficiently gather the contributions of the VPLs, lightcuts and its
extensions cluster the VPLs, which implicitly approximates the lighting matrix
with some representative blocks similar to vector quantization. In this paper,
we propose a new approximation method based on the previous lightcut method and
a low-rank matrix factorization model. As many researchers pointed out, the
lighting matrix is low rank, which implies that it can be completed from a
small set of known entries.
  We first generate a conservative global light cut with bounded error and
partition the lighting matrix into slices by the coordinate and normal of the
surface points using the method of lightslice. Then we perform two passes of
randomly sampling on each matrix slice. In the first pass, uniformly
distributed random entries are sampled to coarsen the global light cut, further
clustering the similar light for the spatially localized surface points of the
slices. In the second pass, more entries are sampled according to the
possibility distribution function estimated from the first sampling result.
Then each matrix slice is factorized into a product of two smaller low-rank
matrices constrained by the sampled entries, which delivers a completion of the
lighting matrix. The factorized form provides an additional speedup for adding
up the matrix columns which is more GPU friendly. Compared with the previous
lightcut based methods, we approximate the lighting matrix with some signal
specialized bases via factorization. The experimental results shows that we can
achieve significant acceleration than the state of the art many-light methods.

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