G 3 . 1 Genetic programming for stack filters

A range of techniques was used to search for the fittest filter to remove noise from data from a blood flow measurement system. Filter types considered included finite impulse response (FIR), RC (exponential), a generalized FIR form, and stack filters. Techniques used to choose individual filters were heuristic, the genetic algorithm, and genetic programming. The efficacy of filters was assessed by measuring a fitness function, derived from the root mean square error. The fittest filter found was a stack filter, generated by genetic programming. It outperformed heuristically found median filters, and an FIR filter first produced by the genetic algorithm and then improved by genetic programming. Genetic programming proved to be an inexpensive and effective tool for the selection of an optimal filter from a class of filters which is particularly difficult to optimize. Its value in signal processing is confirmed by its ability to further improve filters created by other methods. Its main limitation is that it is, at present, too computationally intensive to be used for on-line adaptive filtering. G3.1.1 Project overview Although there are a number of toolkits and techniques for the selection and design of particular classes of filter, there does not appear to be any general method for the development of a filter intended to remove noise from data. This is particularly true when considering a relatively new class of filter, the stack filter, which offers an almost unlimited range of possibilities, and is easily implemented in hardware. Stack filtering (Wendt et al 1986) consists of three stages. Consider a filter which slides a window of width n across input data with integer values in the range 1–m. Data in a given window are first decomposed into a matrix of boolean values, such that an input value of x = m is represented by a column of m cells containing the value 1 below (n − m) cells containing the value 0, within the matrix. Then, the operations that characterize the filter are applied across the rows of the matrix, resulting in a singlecolumn matrix, which is finally recomposed into the single output value by reversing the decomposition stage (summing the 0 and 1 values of the single column). Any logical and arithmetical operations can be applied in the second stage, including those which result in median filters, a subset of stack filters. For a window of width n = 2a+ 1 containing data values x1–xn, in which the input value x1 is decomposed into a matrix column x11–x1m, a median filter can be represented as applying the operation