Numerical transforms.

Numerical computation of transforms is now widely practiced in science and industry and has been revolutionized by the development of fast transforms that make feasible computing projects that once could not be contemplated. The article discusses the significance of transforms in numerical work, defines the modern forms of several common transforms and their inverses, gives examples, and describes and gives references to methods of numerical evaluation.

Blurring in tomograms made with x-ray beams of finite width.

Tomographic reconstruction has ordinarily assumed that the measurement data can be regarded as line integrals, but the finite width of the X-ray beam invalidates this assumption. The data can however be expressed in the form of integrals over a strip rather than a line. The strip integral kernel is calculated allowing for extended source and detector, as well as for nonuniform photon emission and detector sensitivity. Strip eccentricity, which occurs in practice, is also taken into account. Even if the measurement data were to cover all scanning angles, there would be imperfect reconstruction expressible as a space-variant point spread function deducible from the strip integral kernel. To deal with this it is convenient to introduce the concepts of generalized projection and generalized Radon transform. Point-spread functions are given for cases involving piecewise-uniform symmetrical source distributions and uniform detectors.

Correction for collimator width (restoration) in reconstructive X-ray tomography.

X-ray measurements made by scanning in a plane, in successive position angles, involve a degradation (smoothing, or blurring) associated with the nonzero width of the strip occupied by the X-ray beam. The collimating holes that define the X-ray beam cannot be made indefinitely small without increasing the X-ray source power indefinitely. However, making a tomogram from the strip integrals is usually handled as though the data consisted of line integrals. The result is that each infinitesimal element of the tomogram is spread out into a density distribution resembling a cone with a central crater. Restoration of the tomogram to correct for the blurring associated with the strip width is warranted and will receive greater attention as the precision of measurements increases and as demand for resolution approaching that of conventional radiography begins to push instrumental design against various limits. Restoration is a well-developed subject, but in each field of application has its own peculiarities. Systematic and statistical properties of tomograms, the character of their errors, and the desired information will influence the future development of restoration in tomography. X-ray tomography is also special in that diffraction of the rays is negligible and that sharp shadow edges may be formed.