Multidimensional signals are generated in a large area of applications. In health care, such signals are used extensively and are generated with imaging techniques such as computed tomography (CT), magnetic resonance imaging (MRI) and ultra-sound imaging. These techniques produce large data sets that has to be processed and visualized, a process that is computationally complex and time consuming. Improvements in imaging techniques has made it possible to acquire not only ordinary images but e.g. 3-D ’pictures’ of the human body.
The aim of this thesis is to investigate whether genetic algorithms can be used to place coefficients in filter networks. A method is developed and tested on 2-D filters and the resulting filters have lower distortion values while still maintaining the same or lower number of coefficients than filters designed with previously known methods.
Contents
1 Introduction
1.1 Multidimensional Signal Processing
1.2 Fast Filtering
1.3 Genetic Algorithms
1.4 Aims of this Study
1.5 Outline
2 Genetic Algorithms
2.1 Introduction
2.2 Encoding
2.3 Population
2.4 Fitness Function
2.5 Genetic Operators
2.5.1 Selection
2.5.2 Crossover
2.5.3 Mutation
2.5.4 Parameters
2.6 Theoretical Analysis
2.6.1 Schema Theorem
3 Signals and Filtering
3.1 Introduction
3.2 Filters
3.2.1 Smoothness Constraint
3.2.2 Repetitivity Constraint
3.3 Filter Design
3.3.1 DFT in Operator Form
3.3.2 Fourier Weighting Function
3.3.3 Representation Spaces
3.3.4 Filter Optimization
4 Filter Networks
4.1 Introduction
4.2 Filter Network Optimization
5 Fast Multidimensional Filtering
5.1 Background
5.1.1 Weighted Low Rank Approximation
5.1.2 Generalized Convolution
5.1.3 Sparse Array Optimization
5.2 Method
5.2.1 Encoding
5.2.2 Population, Selection and Fitness
5.2.3 Crossover
5.2.4 Mutation
5.2.5 Parameters
6 Results
6.1 Gaussian Low-pass Filter
6.1.1 Optimization Results
6.2 Diagonal Quadrature Filter
6.2.1 Weighted Low-Rank Approximation
6.2.2 Coefficient Placement by Hand
6.2.3 Genetic Algorithm
6.2.4 Optimization Results
6.3 Tridirectional Quadrature Filter Network
6.3.1 Optimization Results
7 Conclusions and Future Work
7.1 Conclusions
7.1.1 A Note on Optimality
7.2 Future Work
7.2.1 Filter Optimization
7.2.2 Genetic Algorithm
7.2.3 Parallelization
Author: Langer, Max
Source: Linköping University
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