Numerical reconstruction of apodized acoustical holograms

• Published in: Journal of sound and vibration 1976-12, Vol.49 (3), p.327-344
• Main Authors: Kramer, D.B., Anand, D.K.
• Format: Article
• Language: English
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/0022-460X(76)90424-7

Apodization is a process that was first used to improve the resolving power of imaging systems involving diffraction patterns. It originally signified a reduction in the energy in the side lobes, but it can also be used in a broader sense, as will be done here, to include either a reduction or an enlargement of the side lobes in an acoustical hologram. Since apodization can produce desired characteristics in a reconstructed image of an acoustical hologram, it adds flexibility to the numerical reconstruction process. This flexibility may help to solve one of the most formidable problems facing numerical holography—the large demands for computer memory and computer processing time. It is the objective of the study to show that this is so. Apodization cannot usually produce an overall improvement in the quality of a reconstructed image of an acoustical hologram: although one aspect of the image is improved, another is degraded. However, apodization can be helpful since it can improve the characteristic of interest, even at the expense of other aspects. In this paper, an example is given in which apodization is applied by a computer during the numerical reconstruction of an acoustical hologram. Numerical holography was chosen because the type of apodization described here is readily applied and, as far as could be determined, has not yet been used in numerical holography, although it has been applied with great effectiveness to the shading of sonar arrays. In order to understand the application, a rectangular object formed by illuminating a slot is taken as the first object to be viewed by means of holography. A mathematical description of the resulting hologram is determined by means of the Huygens-Fresnel diffraction formula. Examination of the expression reveals a pattern to which apodization can be applied. The acoustic pattern consists of a large main lobe with monotonically decreasing minor lobes (side lobes) on either side. This paper describes and evaluates a method of apodization in which all side lobes are held at a specified level and the main lobe assumes the minimum width possible for the side lobe level. After demonstrating the method with the rectangle, a triangle is used as the second object, this time expressing it as a summation of rectangles. When this object also worked well, it raised the possibility that any two- or three-dimensional object could be used after expressing it as a summation of rectangular prisms. This proved to be the case