  We develop the pruned continuous Haar transform and the fast continuous
Fourier series, two fast and efficient algorithms for rectilinear polygons.
Rectilinear polygons are used in VLSI processes to describe design and mask
layouts of integrated circuits. The Fourier representation is at the heart of
many of these processes and the Haar transform is expected to play a major role
in techniques envisioned to speed up VLSI design. To ensure correct printing of
the constantly shrinking transistors and simultaneously handle their
increasingly large number, ever more computationally intensive techniques are
needed. Therefore, efficient algorithms for the Haar and Fourier transforms are
vital. We derive the complexity of both algorithms and compare it to that of
discrete transforms traditionally used in VLSI. We find a significant reduction
in complexity when the number of vertices of the polygons is small, as is the
case in VLSI layouts. This analysis is completed by an implementation and a
benchmark of the continuous algorithms and their discrete counterpart. We show
that on tested VLSI layouts the pruned continuous Haar transform is 5 to 25
times faster, while the fast continuous Fourier series is 1.5 to 3 times
faster.
