# A Nonlinear Optimization Design Algorithm for Nearly Linear-Phase 2D IIR Digital Filters
## 1 Introduction  
Two-dimensional (2D) digital filters have found many signal processing applications, such as image processing, video signal filtering, beamforming systems, and remote sensing. Digital filters are broadly classified into two main categories, namely, finite impulse response (FIR) filters and infinite impulse response (IIR) filters [1]. An FIR filter is one whose impulse response is of finite duration. The output of such a filter is calculated solely from the current and previous input values. This type of filter is hence said to be non-recursive. On the other hand, an IIR filter is one whose impulse response (theoretically) continues forever in time. They are also termed as recursive filters. In addition to the current and previous input values, the current output of such a filter depends on the previous output values. FIR filters are generally easier to implement, as they are non-recursive, always stable (by definition), and can be designed to have a linear phase characteristic. They are, however, not well-suited for very sharp (narrow transition band) frequency responses. IIR filters, on the other hand, are much more difficult to design due to their non-linear phase and their stability constraints. However, they can effectively accommodate very narrow transition band frequency responses, which makes them suitable for a broad range of applications. Moreover, selective IIR filters are much more efficient than FIR filters, as the latter require higher orders (i.e., more multiplications) [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. The efficiency of IIR filters stems from their ability to satisfy the required filter specifications with a much lower filter order, thereby reducing the computational requirements. In addition, IIR filters can have a much smallergroup delay compared with FIR filters [1, 11, 12, 13]. Due to these advantages, IIR filters find extensive applications in the domains of de-noising of digital images, biomedical imaging and digital mammography, X-ray image enhancement, and seismic data processing. Since many signal processing applications are very sensitive to phase non-linearity, the design of IIR filters needs to take into consideration not only the amplitude requirements but also the phase constraints. Therefore, for most of these applications, the designed IIR filters should have a nearly linear-phase response. Moreover, hard constraints on the location of the poles of IIR filters must be enforced to ensure the stability of the obtained filters [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. These and other design requirements usually lead to highly nonlinear constrained optimization problems that require highly sophisticated optimization methods [1, 12, 29, 30, 31, 32]. Research on 2D IIR filters has produced many constrained and non-constrained optimization problems, depending on the researchers' design objectives. Moreover, each researcher has designed specialized algorithms and techniques to handle their optimization problems. Most of these methods can be classified into linear and nonlinear techniques. In linear design techniques, the nonlinear optimization problem is approximated by a sequence of linear optimization problems [1, 11], which are then solved using linear programming. Starting from an initial filter that satisfies most of the design specifications, the algorithm linearizes the nonlinear optimization around this initial filter to produce a linear optimization problem. Solving this problem leads to an improved filter, around which the linearization is carried out. This iterative process is repeated until an acceptable filter is achieved. Although the linear methods can be very fast, and the designer might be able to find an initial filter satisfying most of the design specifications, there is no guarantee that the will converge or produce accurate solutions. As such, nonlinear techniques are preferred. Some researchers have attempted to develop design methods based on modern optimization algorithms.