APPLICATION OF INTEGRAL EQUATION FOR SOME PROBLEMS IN COMPUTERIZED IMAGE RECONSTRUCTION

Abstract

There are various applied problems which may be formulated in terms of Radon's . integral equation. Since one of these problems appears in Tomography, so we are concerned with the solution of this equation. The problem is known to be ill• conditioned, thus the methods of regularization and singular value decomposition were introduced for the solution of this equation . Some problems of Tomography within an ellipse with centre at x-axis, circle with centre at x-axis and ellipse with centre at the origin are also discussed. In chapter 1, we give a brief survey of Tomography, the method of reconstruc• tion of images and a short description about the working way of the apparatus ray. Also, in this chapter a new approach for the derivation of the known function which represents the right hand side of Radon's integral equation is introduced. This deriva• tion contains a generalization for the distance between any two points on an ellipse with centre at any position. The examples introduced by Mahmoud (1977) were gen• eralised. In the rest of this chapter a case of some concentric ellipses was introduced. Chapter 2, deals with the product integration methods for solving Radon's integral.

equation, namely the piecewise constant method and the piecewise linear method. The results obtained using the piecewise constant method, Chebyshev points and our recurrence relation for various large values of the order n of Chebyshev polynomial are given. For comparisons, values obtained by using our recurrence relation and Che• byshev points for various values of the order n of Chebyshev polynomial and for each

t of the piecewise constant method and the piecewise linear method are shown. A com-

parison of the results obtained by Mahmoud (1977) and Alliney & S gallari ( 1984) re• currence relations for each of the piecewise constant method and the piecewise linear method is presented. Also, the behaviour of the diagonal matrix elements for each of the piecewise constant and the piecewise linear methods for large and small values of the order n of Chebyshev polynomial are depicted graphically.