# Signal to Noise Ratio of a Coded Slit Hyperspectral Sensor
## 2 Materials and Methods: The Signal to Noise (SNR) Model  
### The Configuration of the Coded Aperture Sensor  
The code aperture (CA) sensor configuration that is considered in this paper utilizes a conventional spectrograph for spectral dispersion, but to replace the entrance slit which originally to facilitate the line-by-line scanning, by a 2-dimensional coded aperture. The configuration is very similar to that of the \"single-disperser coded aperture snapshot spectral imager\" (SD-CASSI) [12], and the schematic layout of the system is illustrated in Figure 1. Unlike the conventional dispersive push-broom hyperspectral imaging system which gives only a single direction of spatial and spectral content, the coded aperture sensing system in here produces images with significant spatial and spectral contents of the scene in a snap-shot. There is no assumption about the particular type of spectrograph in this work: it can be a prism as illustrated in Figure 1, or it could also be a grating type or a combination of both. This coded aperture (CA) sensing arrangement produces the type of measurement as illustrated in Figure 2. It is assumed in here that the resolution of the monochromatic image in the plane of the detector array, is about twice as that of the detector array in the direction of the spectral dispersion. This imposes a constraint on the spatial content of the imaging dimension as defined by field stop in the coded aperture plane. This configuration will ensure the complete spectrum of each pixel could be registered by the detector sensor. The coded aperture (CA) that utilizes in this work is based on a binary Hadamard matrix [11], which has been shown to produce optimal SNR under certain assumptions of noise models for the system [11]. In the next section, the extent to which these conditions to be fulfilled will be considered. Note that the coded aperture modulates the amplitude (not phase) of the optical signal, and the elements of the CA are assumed to be either entirely opaque or entirely transparent; and each of them is imaged by one detector element in the 2D-plane of the detector array. Every column of the aperture con  
Figure 1: Shows the schematic illustration of the proposed coded aperture (CA) multiplexed sensor architecture which replaces the slit of the conventional push-broom HSI by a 2-D coded aperture [1]. that is, the pattern is kept constant in the direction perpendicular to that in which light is dispersed by the spectrograph. This work does not consider the effects of diffraction by small apertures, i.e., the diffraction limited condition. The CA is formulated through the binary Hadamard matrix by replacing the 1\({}^{\prime}\)s with 0\({}^{\prime}\)s and \(-1^{\prime}\)s with 1\({}^{\prime}\)s; subsequently the row and column that contain only 1\({}^{\prime}\)s are removed as shown in Figure 3. The Hadamard matrix is chosen in the way that, the columns of the resulting binary matrix are cyclic shifts of the adjacent columns when its columns are ordered appropriately. The CA is then constrained to a binary matrix of size equal to the dimensions of the detector array in the direction of spectral dispersion. This requirement also defines the number of wavebands that can be detected by the sensor. Note that Hadamard matrices are available only for certain dimensions, thus the choice of the number of wavebands in the present configuration is limited. If \(N\) is the number of wavebands, this CA code contains \(2N-1\) elements and it is created from the row of the matrix by repeating elements in the same sequence up to that length [2]. Similar to that of the pushbroom imager, the field of view of the CA multiplexed sensing is scanned in the direction parallel to the spectral dispersion of the image. This scene scanning method has been shown to improve image quality over a single snapshot, as demonstrated by the CASSI system which exploits compressive sensing for signal recovery [13].