  In this paper we consider the fundamental operations dilation and erosion of
mathematical morphology. Many powerful image filtering operations are based on
their combinations. We establish homomorphism between max-plus semi-ring of
integers and subset of polynomials over the field of real numbers. This enables
to reformulate the task of computing morphological dilation to that of
computing sums and products of polynomials. Therefore, dilation and its dual
operation erosion can be computed by convolution of discrete linear signals,
which is efficiently accomplished using a Fast Fourier Transform technique. The
novel method may deal with non-flat filters and incorporates no restrictions on
shape or size of the structuring element, unlike many other fast methods in the
field. In contrast to previous fast Fourier techniques it gives exact results
and is not an approximation. The new method is in practice particularly
suitable for filtering images with small tonal range or when employing large
filter sizes. We explore the benefits by investigating an implementation on
FPGA hardware. Several experiments demonstrate the exactness and efficiency of
the proposed method.
