## 2 Materials and Methods
If the FPA is used, then it requires only \(N\) camera shots. With coded aperture holography, only a single shot is needed. However, the fork-shaped beam profile precludes the application of non-scanning methods. In this study, in order to extend CAI to FTIRm with the synchrotron beam, CAI is implemented as CASH, where the number of measurements needed is \(M^{2}\), which is the same as scanning-based 2D imaging but can obtain 3D information. In indirect imaging mode, as demonstrated in [1; 2; 3; 4; 5; 6; 7; 8; 9], the imaging process is different from direct imaging mode, consisting of two steps in general. In the first step, the intensity pattern is recorded as \(I_{\rm O}=O\bigotimes I_{\rm PSF}\). When the imaging condition is satisfied, then \(I_{\rm O}\) creates an image of the object as governed by the NA of the system. However, at other planes \(I_{\rm O}\) does not form the image of the object but a blurred intensity distribution. Therefore, in the second step, the image \(I_{R}\) is reconstructed or recovered from \(I_{\rm O}\). This process is also called deconvolution or deblurring. This is achieved by processing \(I_{\rm O}\) with \(I_{\rm PSF}\), where \(I_{\rm R}=I_{\rm O}*I_{\rm PSF}\), which can be rewritten as \(I_{\rm R}=O\bigotimes I_{\rm PSF}*I_{\rm PSF}\), where '\(*\)' is a 2D correlation operator. Depending upon the nature of the autocorrelation function, i.e., \(I_{\rm PSF}*I_{\rm PSF}\), the object information \(O\) is sampled. With scattering masks, it is possible to obtain sharp autocorrelation functions. In [14], it was shown that even with deterministic masks such as COL, it is possible to obtain a sharp autocorrelation function. In this study, it is also possible to obtain sharp autocorrelation functions if the synchrotron beam has a uniform intensity distribution. The blurring with the SOL is not symmetrical and the recording of PSFs at different axial planes was complicated due to the peculiar intensity distribution of the beam. In all the previous studies, the fundamental building block of the object is either \(\sim\)\(\lambda\)/NA or the diameter of the pinhole. It is not possible to achieve resolution better than the size of the pinhole. In this study, to enable 3D imaging, the lateral resolution is set at the size of the pinhole while it is possible to image below that limit with a smaller spot size obtained by the SOL. A larger pinhole creates larger features in the PSF and vice versa, which are transferred to the resolving power through the autocorrelation function \(I_{\rm PSF}*I_{\rm PSF}\) which samples the object function. Therefore, in the proposed system, the pinhole is scanned by the highly focused IR beam from the 36\(\times\) objective lens, and the PSF is formed by the summation of the intensity distributions obtained during scanning, given as \(I_{\rm PSF}=\sum_{p,q=1}^{N}I_{D}\big{(}x_{p},y_{q}\big{)}\), where \(I_{D}\) is the intensity distribution obtained for a point (\(x\),\(y\)) in the sample plane. Consequently, the PSF itself is recorded as a 2D object in a spatially incoherent imaging system by summation of light from different points. This procedure allows recording of the 3D PSFs even with the peculiar intensity distribution and complicated optical configuration. This approach results in decreasing the lateral resolution, as the pinhole is larger than the focal spot size obtained from the SOL. After recording the 3D PSF, the scanning procedure was repeated for a single plane or multiplane object. The 3D image of the object can be reconstructed by processing the PSF library catalogued against the axial location with the object intensity distributions. Unlike optical scanning holography methods [10], where it is necessary to reconstruct at all planes, the use of the SOL as a coded aperture enables direct imaging and holography to co-exist. At the image plane, the CASH technique performs perfectly as a direct imaging system. When defocused, the system behaves as a CAI method and a reconstruction method is required.