  Graph filtering is a fundamental tool in graph signal processing. Polynomial
graph filters (PGFs), defined as polynomials of a fundamental graph operator,
can be implemented in the vertex domain, and usually have a lower complexity
than frequency domain filter implementations. In this paper, we focus on the
design of filters for graphs with graph Fourier transform (GFT) corresponding
to a discrete trigonometric transform (DTT), i.e., one of 8 types of discrete
cosine transforms (DCT) and 8 discrete sine transforms (DST). In this case, we
show that multiple sparse graph operators can be identified, which allows us to
propose a generalization of PGF design: multivariate polynomial graph filter
(MPGF). First, for the widely used DCT-II (type-2 DCT), we characterize a set
of sparse graph operators that share the DCT-II matrix as their common
eigenvector matrix. This set contains the well-known connected line graph.
These sparse operators can be viewed as graph filters operating in the DCT
domain, which allows us to approximate any DCT graph filter by a MPGF, leading
to a design with more degrees of freedom than the conventional PGF approach.
Then, we extend those results to all of the 16 DTTs as well as their 2D
versions, and show how their associated sets of multiple graph operators can be
determined. We demonstrate experimentally that ideal low-pass and exponential
DCT/DST filters can be approximated with higher accuracy with similar runtime
complexity. Finally, we apply our method to transform-type selection in a video
codec, AV1, where we demonstrate significant encoding time savings, with a
negligible compression loss.
