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In computer vision, object recognition involves segmentation of the image into separate components. One way to do this is to detect the edges of the components. Several algorithms for edge detection exist and one of the most sophisticated is the Canny edge detector.Canny [2] designed an optimal edge detector for images which are corrupted with noise. He suggested that a Gaussian filter be applied to the image and edges be sought in the smoothed image. The directional derivative of the Gaussian is obtained, then convolved with the image. The direction, n, involved is normal to the edge direction. Edges are assumed to exist where the result is a local extreme, i.e., where∂2 (g * f) = 0.(0.1)_____∂n2In the above, g(x, y) is the Gaussian, f (x, y) is the image function and The direction of n is an estimate of the direction of the gradient of the true edge. In this thesis, we discuss the computational algorithm of the Canny edge detector and its implementation. Our experimental results show that the Canny edge detection scheme is robust enough to perform well over a wide range of signal-to-noise ratios. In most cases the Canny edge detector performs much better than the other edge detectors. / Department of Mathematical Sciences

The thesis develops two families of numerical methods, based upon new rational approximations to the matrix exponential function, for solving second-order parabolic partial differential equations. These methods are L-stable, third- and fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by new difference approximations. Then parallel algorithms are developed and tested on one-, two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions with discontinuities between initial and boundary conditions. The schemes are seen to have high accuracy. A family of cubic polynomials, with a natural number dependent coefficients, is also introduced. Each member of this family has real zeros. Third- and fourth-order methods are also developed for one-dimensional heat equation subject to time-dependent boundary conditions, approximating the integral term in a new way, and tested on a variety of problems from the literature.

Previously obtained results from the study of homoclinic bifurcations in ordinary differential equations are presented. The standard technique of analysis involves the construction of a Poincaré map on a surface near to the homoclinic point. This map is the composition of an inside map, with behaviour linearized about the homoclinic point, together with an outside map, with behaviour linearized about the homoclinic orbit. The Poincaré map is then reduced to a one-dimensional map, involving the return time between successive visits to the Poincaré surface. These standard techniques in the contemplation of homoclinic systems are then extended to a class of partial differential equations, on unbounded domains. This follows a method introduced by Fowler [Stud. Appl. Math. 83 (1990), pp. 329–353]. This extension involves more technicalities than in the case of ordinary differential equations. The method of Fowler is extended to cover the case of vector-valued partial differential equations, and to consider the consequences of symmetry invariances. A Poincaré map is derived, and then is reduced to a finite-dimensional map. This map has dimension equal to the number of symmetry invariances of the system. Some simple examples of this finite-dimensional map are studied, in isolation. A number of interesting bifurcation pictures are produced for these simple examples, involving considerable variation with the values of coefficients of the map. Partial differential equations on finite domains are then considered, yielding similar results to the ordinary differential equation case. The limit as domain size tends to infinity is examined, yielding a criterion for distinction between the applicability of finite and infinite domain results. Finally, these methods are applied to the Ginzburg-Landau system. This involves the numerical calculation of coefficients for the finite-dimensional map. The finite-dimensional map thus derived supports an interesting interlocked isola structure, and moreover correlates with numerical integration data.

This thesis is concerned with the asymptotic behaviour of solutions of the differential-difference equation:- w'(s) = g(s)[w(s - 1) - w(s)] where g(s) is a continuous real-valued function. g(s) is assumed to have one of the following asymptotic behaviours <ul><li> algebraic</li><li>exponential algebraic</li><li>constant</li><li>zero</li><li>periodic</li></ul> Chapter 2 covers the behaviour of solutions as s→ + ∝ for g(s) ≥ 0. Chapter 3 covers the behaviour of solutions as s→ + ∝ for g(s) ≤ 0. Chapter 4 covers s→ - ∝ and g(s) ≥ 0. Chapter 5 covers s→ - ∝ and g(s) ≤ 0.

This thesis investigates the convergence of numerical schemes for the computation of equilibrium solutions. These are solutions of evolutionary PDEs that arise from (bounded, non-decaying) boundary forcing after the dissipation of any (initial data dependent) transients. A rigorous definition of the term 'equilibrium solution' is given. Classes of evolutionary PDEs for which equilibrium solutions exist uniquely are identified. The uniform well-posedness of equilibrium problems is also investigated. Equilibrium solutions may be approximated by evolutionary initialization: that is, by finding the solution of an initial boundary value problem, with arbitrary initial data, over a period of time t ϵ [0,T]. If T is chosen large enough, the analytic transient will be small, and the analytic solution over t ϵ [T, T + T₀] will be a good approximation to the analytic equilibrium solution. However, in numerical computations, T must be chosen so that the analytical transient is small in comparison with the numerical error E_h, which depends on the fineness of the grid h. Thus T = T_h, and, in general, T_h→∞ as h→0. Convergence is required over t ϵ [T_h,T_h + T₀]. The existing Lax-Richtmyer and GKS convergence theories cannot ensure convergence over such increasing periods of time. Furthermore, neither of these theories apply when the forcing does not decay. Consequently, these theories are of little help in predicting the convergence of finite difference methods for the computation of equilibrium solutions. For these reasons, a new definition of stability - uniform stability — is proposed. Uniformly stable, consistent, finite difference schemes, for uniformly well posed problems, converge uniformly over t ≥ 0. Uniformly convergent schemes converge for bounded and nondecaying forcing. Finite difference schemes for hyperbolic PDEs may admit waves of zero group velocity, even when the underlying analytic problem does not. Such schemes may be GKS convergent, provided that the boundary conditions exclude these waves. The deficiency of the GKS theory for equilibrium computations is traced to this fact. However, uniform stability finds schemes that admit waves of zero group velocity to be (weakly) unstable, regardless of the boundary conditions. It is also shown that weak uniform instabilities are the result of time-dependent analogues of the 'spurious modes' that occur in steady-state calculations. In addition, uniform stability theory sheds new light on the phenomenon of spurious modes.