Single-projection radiography for noncircular symmetries: Generalization of the Abel transform method

We present a new method which extends the use of the single projection radiographic Abel method, hitherto applicable only to objects of circular and elliptical cross sections, to objects having general, noncircular symmetries. This is done by developing a new integral equation that is similar in applications to Abel's equation, and includes it as a special case. The use of the new equation is discussed for objects having a smooth and convex cross-section boundary (e.g., elliptic), a piecewise smooth convex boundary (e.g., bi-parabolic), and a boundary with regions of zero curvature (e.g., polygons). Specific examples are given for each of these three classes, and analytic inverses are calculated for these cases. Also, numerical inversion of the integral equation is given, showing satisfactory results. We show that in contrast to Abel's equation in many cases the kernel of the integral equation is non-singular. Consequently, fairly simple inversion techniques are sufficient. Finally, the azimuthal variation of the transmitted intensity is employed to provide a convenient and fast nondestructive evaluation test of the deviation of the radiographed object from a prescribed symmetry.