# PSF Estimation of Space-Variant Ultra-Wide Field of View Imaging Systems
## 1 Introduction  
Modern photonic imaging systems, used in astronomy and in other areas of science, are required to perform with high imaging quality, a wide aperture (i.e., low F-numbers), and high spatial resolution. These systems are known as UWFOV systems. The wide-angle optical elements used in UWFOV systems introduce a significant shift variance [1], which complicates the modeling task. Many recent research works have focused on finding a model of UWFOV systems that is suitable for reducing the uncertainties in reconstructing image data from wide-field microscopic systems [2; 3; 4], security cameras [5; 6; 7], and all-sky cameras [8; 9; 10; 11; 12; 13; 14; 15]. Any imaging system can be described by its spatial impulse response function, commonly referred to as the Point Spread Function (PSF). PSF is used to represent the aberrations of UWFOV devices, and can be applied either to the entire image or within regions uniquely defined by the isoplanatic angle. However, it can sometimes be complicated to obtain a PSF model of the system. It can be difficult to achieve the desired accuracy using conventional methods--not from the measurement point of view, but because it is difficult to describe the space variance (SV) of the PSF over the Field of View (FOV). This leads to an obvious issue. From the measurement point of view, it is necessary to obtain the PSF for all discrete points of the entire FOV. Then the PSF of the system is described by such a field of individual PSFs. [PERSON] [16] calls this type of PSF partially space-invariant because there are just discrete points (PSFs) in parts of the FOV. However, a space-variant PSF is described by a model in all points of the acquired image. Of course, this bringshigh demands on the precision of the model. Issues associated with sampling, pixel size and the pixel sensitivity profile must be taken into account in the design. It is very difficult to obtain the parameters of the model, especially in image areas with an aberrated impulse response and for objects with a small Full Width at Half Maximum (FWHM) parameter in relation to the pixel size. These objects, captured, e.g., in microscopy using all-sky cameras and/or special optical elements, can hardly be processed using current techniques, which are typically oriented to multimedia content. A widely-used approach for obtaining the PSF is to model wavefront aberrations using Zernike polynomials. These polynomials were introduced by [PERSON] in [17]. This work was later used by [PERSON] [18], who introduced his approach to indexing, which is widely used in the design of optics. Zernike polynomials are usually applied to rotationally symmetric optical surfaces. Zernike polynomials form a basis in optics with [PERSON] [19] field dependent wave aberration. Recently, [PERSON] [20] reformulated the work of [PERSON]. This new formulation is more practical for optical design. A more recent contribution by [PERSON] et al. [21] uses Bi-Zernike polynomials, which is an alternative to the method mentioned in [PERSON]'s article [22]. Zernike polynomials are used in modern interferometers for describing wave aberrations. However, researchers have begun to think about using these polynomials for an overall description of lens optics. When the first all-sky camera projects appeared, the question of how to model aberrations of fisheye lenses arose. [PERSON] [23] proposed the approach of a partially space-invariant model of PSF based on Zernike polynomials. Another all sky project, Pi of the Sky [1], focusing on gamma ray bursts, attempted to make a model of PSF. Their approach was quite different from [PERSON]'s [23]. First, they calculate the dependency of the coefficients of the polynomials on the angle to the optical axis for several points. Then they interpolate these coefficients over the entire FOV. The WILLIAM project [24] (WIde-field aLL-sky Image Analyzing Monitoring system) faces the issue of aberrations at an extremely wide-field FOV (see also [25; 26]).