Pierri R, Liseno A, Soldovieri F, Solimene R
Seconda Università di Napoli, Dipartimento di Ingegneria dell'Informazione, Aversa, Italy.
J Opt Soc Am A Opt Image Sci Vis. 2001 Feb;18(2):352-9. doi: 10.1364/josaa.18.000352.
The problem of determining the achievable resolution limits in the reconstruction of a current distribution is considered. The analysis refers to the one-dimensional, scalar case of a rectilinear, bounded electric current distribution when data are collected by measurement of the radiated field over a finite rectilinear observation domain located in the Fresnel zone, orthogonal and centered with respect to the source. The investigation is carried out by means of analytical singular-value decomposition of the radiation operator connecting data and unknown, which is made possible by the introduction of suitable scalar products in both the unknown and data spaces. This strategy permits the use of the results concerning prolate spheroidal wave functions described by B. R. Frieden [Progress in Optics Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam 1971), p. 311.] For values of the space-bandwidth product much larger than 1, the steplike behavior of the singular values reveals that the inverse problem is severely ill posed. This, in turn, makes it mandatory to use regularization to obtain a stable solution and suggests a regularization scheme based on a truncated singular-value decomposition. The task of determining the depth-resolving power is accomplished with resort to Rayleigh's criterion, and the effect of the geometrical parameters of the measurement configuration is also discussed.
本文考虑了在电流分布重建中确定可实现分辨率极限的问题。该分析针对一维标量情况，即直线型、有界电流分布，此时数据是通过在位于菲涅尔区且相对于源正交并居中的有限直线观测域上测量辐射场来收集的。通过对连接数据和未知量的辐射算子进行解析奇异值分解来开展研究，这通过在未知量空间和数据空间中引入合适的标量积得以实现。该策略允许使用由B. R. 弗里登所描述的关于长椭球波函数的结果[《光学进展》第九卷，E. 沃尔夫编辑（北荷兰，阿姆斯特丹，1971年），第311页]。对于远大于1的空间带宽积值，奇异值的阶梯状行为表明反问题是严重不适定的。这进而使得必须使用正则化来获得稳定解，并提出了一种基于截断奇异值分解的正则化方案。借助瑞利准则完成了确定深度分辨能力的任务，并且还讨论了测量配置的几何参数的影响。
