Scattering and Gathering for Spatially Varying Blurs

A spatially varying blur kernel
$h(\mathbf{x},\mathbf{u})$
is specified by an input coordinate
$\mathbf{u} \mathbf{\in} \mathbb{R}^{2}$
and an output coordinate
$\mathbf{x} \mathbf{\in} \mathbb{R}^{2}$
. For computational efficiency, we sometimes write
$h(\mathbf{x},\mathbf{u})$
as a linear combination of spatially invariant basis functions. The associated pixelwise coefficients, however, can be indexed by either the input coordinate or the output coordinate. While appearing subtle, the two indexing schemes will lead to two different forms of convolutions known as
scattering
and
gathering
, respectively. We discuss the origin of the operations. We discuss conditions under which the two operations are identical. We show that scattering is more suitable for simulating how light propagates and gathering is more suitable for image filtering such as denoising.