الفهرس 
Abstract
Di®erential Geometry of SurfacesOnly 14 pages are availabe for public view
 |  | from | 105 |  |  |
| | | from | 105 | | |
Abstract
Image processing is a system of mathematical transformations of an image, either to modify some characteristics or extract some features. Current practice in the image processing relies on the concepts of Linear Algebra (vectors, Hessian matrix, eigenvalues, eigenvectors etc.) and differential geometry (ridges and ravines are which defined through the principal curvatures and directions).
Edge detection is in the forefront of image processing system, hence
it is a problem of fundamental importance in the image processing. For object detection as edge of surface image, it is crucial to have a good understanding of edge detection algorithms. It is one of the most commonly used operations in image analysis.
Curves detection (straight lines, circles, ellipses etc.) from edge image is one of the most frequent and necessary tasks in the realm of image processing. Circles are important patterns in many automatic image inspection applications.
Ridges and ravines are characteristic curves of surface that mark
the salient intrinsic features of its shape and are therefore valuable for shape matching, surface quality control, visualization and various other applications. Ridges and Ravines are loci of points on surface where one of the principal curvatures attain a critical value in its respective principal direction. Typically, one must calculate second-order differential characteristics of the surfaces such as the maximum, mean, and Gaussian curvatures. This thesis is organized as follows;
Chapter [1] reviews the basic theory of surfaces including such topic as maximum, Gaussian and Mean curvatures, ridges and ravines. Also, this chapter reviews some vector calculus which we use in the thesis including such topic as Gradient, Laplacian, Gaussian distribution and Lie derivative.
In chapter [2] we provide the problem formulation of edge detection of image. We present and analyze the various edge detection techniques. We apply the edge detection techniques of geometric surfaces
(Revolution and Saddle surfaces) with studying geometric properties of
this surfaces and we present the advantages and disadvantages of various edge detection techniques by visual comparisons. Also, we present the parabolic points and minimal surface on the Gaussian distribution surface, Finally, we discuss the conclusions reached by analysis and visual comparison of various edge detection techniques developed using MATLAB 7.11.
In chapter [3] we introduce the circle-point mapping (C-P mapping) of image space to the parameter space (R3+) and we present some geometrical properties for detecting circles in image space. These
results are illustrated by figures and through the main theorem (3.1.1).
In chapter [4] we show how to compute the curvatures, maximum
curvatures, ridges and ravines of the image defined by the isosurface
equation (or the edge points) using the gradient, Hessian matrix and its
derivatives (i.e. the first, second, third and fourth partial derivatives)
of the isosurface equation (or the edge points). Some examples are
introduced and plotted.