IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 5 (May. - Jun. 2013), PP 45-55 www.iosrjournals.org www.iosrjournals.org 45 | Page “Trade-Off between Detection and Resolution of Two Point Objects Under Various Conditions of Imaging Situations: Part-I: Mathematical Formulation of the Problem” P. Thirupathi Department of Mathematics, Osmania University, Hyderabad, 500007, A.P, India. Abstract: It is a well-experienced fact that whenever one tries to detect a weak object point in the vicinity of an intense point object, viz., a binary star-SIRUS and its companion weak satellite star, there is always loss of resolution of the optical system. In other words, one wants to improve the defectively of the system, there is always a loss of resolution capabilities of the system. Thus, there is a trade-off between Detection and Resolution of optical systems under various imaging situations. In this first paper of discussion of this trade-off, we have derived the Fourier analytical formulation of this problem. This formulation will be used to find out a compatible trade-off between Detection and Resolution in our further publications. Key words: Fourier Optics, Mathematical Optics, Super-Resolution, etc.- I. Introduction In this paper, the two-point resolution capabilities have been discussed for an optical system with parabolic filters. The Rayleigh and the Sparrow resolution limits are computed and studied as functions of the degree of coherence of illumination, (Incoherent, coherent and partially coherent) of the two point objects. The problem of the definition and determination of an image quality criterion has long been and still is a major one in the field of image evaluation and assessment. Though a number of physical parameters for assessing the quality of an image have been proposed from time to time, none of these measures is completely satisfactory. Some of these parameters are: - Resolving power, Strehl Definition, Optical Transfer Function, Equivalent Pass- band, Relative Structural Content, Correlation Quality. Image Fidelity and peachiness. Historically, one of the first measures established for the evaluation of optical system was to specify how well the system could resolve a two–point object, and the two-point resolution is one of the simplest quality criteria in terms of the impulse response from among several criteria available is chosen. The intensity distribution in the image should satisfy the requirement of the criteria chosen. The limiting separation thus determined gives the imaging system’s response in terms of two-point resolution. In the case of sources of short wave length radiations such as X- rays, gamma rays or sub-atomic particles, conventional method of ray bending i.e., reflection, refraction and diffraction, cannot be used for imaging due to their high penetrating power and rectilinear propagation. In such cases, coded imaging (CI) techniques can play an important role in determining source location and source distribution. According to them “CI”, when reduced to the basics, is a two step process. In the first step, the source information is recorded or encoded by geometrical shadow casting through a coded aperture (no ray bending is involved). In the second step, the image is matched to the coded aperture design. Though the two-point resolution is one of the simplest criterion to assess the performance of optical imaging systems, it has its inherent complexity owing to the fact that the limit of resolution is sensitive to a large number of factors via., nature of the optical system, nature of illumination, object point separation, intensity ratio of the object points degree of coherence, resolution criterion used, etc. Therefore, there has to be some flexibility in the exact quantitative definition of the limiting resolving power achievable. The importance of two- point resolution lies in the fact that it is one of the earliest and simplest physical parameters used to evaluate the performance of optical imaging system in various imaging situations, incoherent, partially coherent and coherent illuminations. It may be noted that the resolving power of an imaging system as determined by the Rayleigh criterion is not the property of the system alone but also of the pair of objects and the coherence condition of illumination. Though the optical transfer function as an assessment parameter is superior to two- point resolution for optical systems operating in incoherent illumination, it should be noted that partially coherent and coherent imaging system become non-linear in both amplitude and intensity, if the detection step is also included in the imaging system. Due to the non-linearity associated with the partially coherent imaging systems, the systems become object-dependent and cannot be completely characterized by a system transfer function as in the linear case.