Performance analysis of a Hadamard Transform Spectral Imaging system

TL;DR

This paper analyzes Hadamard Transform Spectral Imaging (HTSI) as a multiplexing approach for spectral imaging under low-photon flux, using inverse Hadamard reconstruction $\psi = (1/n) H_n^{\mathrm{T}}\eta$ to recover spectra from masked observations. It demonstrates that HTSI can boost SNR when detector read noise dominates, while showing no average gain under purely Poisson noise, with emission lines benefiting in both regimes. Through simulations on a 1D artificial dataset and a 2D NGC 7009 data cube, the study quantifies SNR improvements and RMSE reductions, illustrating the SNR gain follows models like $\mathrm{SNR\,gain} = \sqrt{\frac{n(\langle r\rangle + \sigma^2)}{n\langle r\rangle + \sigma^2}}$ and highlighting the dependence on read-to-shot noise ratio. The work discusses HTSI’s relevance for future MEMS-based spectrographs and space missions (e.g., SASAFRAS, CASTOR, HWO), and outlines directions for future exploration, including handling variable conditions and data missingness in HTSI acquisitions.

Abstract

Hadamard Transform Spectral Imaging (HTSI) is a multiplexing technique used to recover spectra via encoding with multi-slit masks, and is particularly useful in low photon flux applications where signal-independent noise is the dominant noise source. This work focuses on the procedure that is used to recover spectra encoded with multi-slit masks generated from a Hadamard matrix; the decoding process involves multiplying the output encoded spectral images by the inverse of the Hadamard matrix, which separates any spectra that were overlapping in the target object. The output from HTSI is compared to direct measurement methods, such as single-slit scanning, to evaluate its performance and identify under which conditions it can provide an advantage or disadvantage. HTSI resulted in an increase in the average signal-to-noise (SNR) ratio of spectra when signal-independent noise, such as detector read noise, is present, and has no average net effect when signal dependent-noise, such as Poisson photon noise, is the only noise source present. The SNR of emission lines was found to be greater with HTSI than with single-slit scanning under both signal-independent and signal-dependent noise, and increases as the ratio of read-to-shot noise increases.

Figures (15)

• Figure 1: In its most simple application, Hadamard transform spectroscopy can be performed by replacing the exit slit of a monochromator with a series of slit masks. Thus, the spectral image is sampled by cycling through the slit masks, rather than by scanning the spectral image along the exit slit. This is very similar to the concept of a multiplexed single-pixel camera. See Harwit and Sloane (1979)Harwit_Sloane_1979 for an excellent review.
• Figure 2: In HTSI, rows of a Hadamard matrix are used to generate "bar code" style slit masks to multiplex the spectral image; however, the negative elements cannot be created directly, each row must be decomposed into two complimentary binary masks, whose difference recreates a Hadamard row. Here, an example for the 2nd row of a rank 4 Hadamard matrix (Eq.\ref{['eq:h4']}) is shown. This procedure is repeated for all rows of the matrix. Ideally, an HTSI spectrograph can acquire the images multiplexed by the binary masks simultaneously.
• Figure 3: Six examples of artificial spectra are displayed. Each spectrum contains 256 spectral points, and several emission lines with pseudo-random means, widths, and amplitudes. The first line is set to be located in the left half of each spectrum, and the second line in the right half, such that they do not overlap and remain distinct. Axes are in arbitrary units; the x-axis is the equivalent to "counts" or relative intensity, and the x-axis is relative wavelength.
• Figure 4: An example of a single set of simulated observations of emission lines with intrinsic SNR of 2, 3, 4, 5, and 6. In the case of a perfect detector with zero additional noise, HTSI under-performs when observing the faintest signals, due to the extra noise introduced due to the multiplexing. This is known as Fellget's disadvantage.
• Figure 5: Three simulations with the same input spectra as in Fig. \ref{['fig:fellgetDisadvantage']}, but with detector read noise of 1, 3, and 5 counts rms. Note that the noise floor in the slit scan measurements increases as one might expect, while the HTSI measurements show a significantly suppressed noise floor - a key goal of multiplexing.