New Design Methods for Two-Dimensional Filters Based on 1D Prototypes and Spectral Transformations
Radu P. Matei · InTech eBooks · 2011
Digital Filters 92The Bamberger directional filter bank (Bamberger & Smith, 1992), is a purely directional decomposition that provides excellent frequency domain selectivity with low computational complexity.This family of filter banks has been successfully used for image denoising, character recognition, image enhancement etc. Diamond filters are currently used as antialiasing filters for the conversion between signals sampled on the rectangular sampling grid and the quincunx sampling grid.Some design techniques, mainly for FIR diamond filters were developed (Lim & Low, 1997;Low & Lim, 1998).Stability of the two-dimensional recursive filters is also an important issue and is more complicated than for 1D filters.For 2D filters, in general, it is quite difficult to take stability constraints into account during the stage of approximation (O'Connor, 1978).For this reason, various techniques were developed to separate the stability from the approximation problem.If the designed filter becomes unstable, some stabilization procedures are needed (Jury, 1977).Unlike 1D filters, in 2D filters the numerator can affect the filter stability and can sometimes stabilize an otherwise unstable filter.The design methods in the frequency domain described in this chapter are also based on spectral transformations, or frequency transformations, a term more often used in text.Starting from an 1D prototype filter with a desired characteristics, for instance low-pass maximally-flat, selective low-pass or band-pass etc., some specific spectral transformations will be applied in order to obtain the 2D filter with a desired shape.Various types of 2D filters will be approached: directional selective filters, oriented wedge filters, fan filters, diamond-shaped filters etc.All these filters have already found specific applications in image processing.The general case will be approached, when we start from a 1D prototype which is a common digital filter, either maximally-flat or equiripple (Butterworth, Chebyshev, elliptic etc.) given by a transfer function in variable z, which is decomposed into a product of elementary functions of first or second order.In this case the design consists in finding the specific complex frequency transformation from the variable z to the complex plane 1 2 (z ,z ) .Once found this mapping, the 2D filter function results directly through substitution.The case of zero-phase 2D filters will be treated as well, since they are very useful in various image filtering applications due to the absence of phase distortions.This method is at the same time simple, efficient and versatile, since once found the adequate frequency transformation, it can be applied to different prototype filters obtaining the 2D filter.The latter inherits the selectivity properties of its 1D counterpart (bandwidth, flatness, transition band etc.).Changing the prototype filter parameters will change the properties of the obtained 2D filter.All the proposed design techniques are mainly analytical but also involve numerical optimization, in particular rational approximations (Padé or Chebyshev-Padé).Since the design starts from a factorized transfer function, the 2D filter function will also result directly factorized, which is a major advantage in its implementation.For each specified shape of the 2D filter, a particular frequency transformation is derived.Some proposed methods involve the bilinear transform as an intermediate step.Depending on their shape, the designed filters may present non-linearity distortions towards the margins of the frequency plane, due to the frequency warping effect.In order to compensate for these errors, a pre-warping may be applied, which increases the filter order.Other proposed methods avoid from the start the use of bilinear transform and the filter coefficients result through a change of frequency variable and a bivariate Taylor or www.intechopen.comwww.intechopen.comDigital Filters 94 the form:1 2 F( , ) .The elementary transfer functions (2) and (3) can be put into the form of a complex frequency response: How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following: