Multidim Syst Sign Process DOI 10.1007/s11045-016-0458-4 Analytical design methods for directional Gaussian 2D FIR filters Radu Matei 1 Received: 16 December 2015 / Revised: 18 June 2016 / Accepted: 22 September 2016 © Springer Science+Business Media New York 2016 Abstract This paper proposes two analytical design methods in the frequency domain for directional Gaussian 2D FIR filters, with a straight directional or an elliptically-shaped fre- quency response and with a specified selectivity and orientation in the frequency plane. One method relies on the substitution of a frequency mapping into the factored polynomial approximation of the Gaussian, while the other one is based on decomposing the frequency response into Gaussian components along three properly chosen directions in the frequency plane. The frequency response of the 2D directional filter results directly in a factored form, which is a major advantage in implementation. The filters are accurate, efficient and they eliminate the necessity of interpolation. With the mapping substitution method, they result also adjustable in orientation and aspect ratio. Several design examples are given for various specifications, and simulation results of directional filtering on test images are provided, to prove their applicability in image processing. Keywords 2D FIR filters · Analytical design methods · Approximations · Frequency transformations 1 Introduction The field of two-dimensional filters has known a steady development, and their design meth- ods are well founded Najim (2006). Various aspects of two-dimensional IIR or FIR filter design have been approached by many authors. Most of the analytical design methods rely on 1D prototype low-pass filters and spectral transformations from s to z plane via bilinear approximation, in order to obtain 2D filters with desired frequency response, as approached in early papers like Harn and Shenoi (1986). A convenient tool for 2D FIR filter design is also the McClellan transform Chen and Lee (1994), Yeung and Chan (2002). Many authors have B Radu Matei rmatei@etti.tuiasi.ro 1 Faculty of Electronics, Telecommunications and Information Technology, “Gheorghe Asachi” Tech- nical University, Bd. Carol I No. 11, 700506 Iasi, Romania 123