Spectro-Perfectionism: An Algorithmic Framework for Photon Noise-Limited Extraction of Optical Fiber Spectroscopy

TL;DR

The paper introduces Spectro-Perfectionism, a forward-modeling approach for extracting 1D spectra from 2D fiber spectroscopy images at the photon-noise limit. It formulates the data as $\mathbf{p} = \mathbf{A} \mathbf{f} + \mathbf{n}$, where $\mathbf{A}$ encodes PSF, trace, and throughput, and recovers $\mathbf{f}$ via $\mathbf{f} = (\mathbf{A}^T \mathbf{N}^{-1} \mathbf{A})^{-1} \mathbf{A}^T \mathbf{N}^{-1} \mathbf{p}$, then re-convolves with a resolution matrix $\mathbf{R}$ to obtain $\tilde{\mathbf{f}} = \mathbf{R} \mathbf{f}$ with uncorrelated errors and the same native resolution as the 2D data. The method robustly handles non-separable PSFs, cross-talk between fibers, and sky foregrounds by integrating sky-object decomposition into a single 2D model, validated by end-to-end tests showing $\chi^2$ near unity after reconvolution. The approach yields statistically valid model testing against spectral hypotheses and promises substantial improvements for current and upcoming large multi-fiber spectroscopic surveys, provided calibration is accurate and computational challenges are managed with sparse, iterative solvers.

Abstract

We describe a new algorithm for the "perfect" extraction of one-dimensional spectra from two-dimensional (2D) digital images of optical fiber spectrographs, based on accurate 2D forward modeling of the raw pixel data. The algorithm is correct for arbitrarily complicated 2D point-spread functions (PSFs), as compared to the traditional optimal extraction algorithm, which is only correct for a limited class of separable PSFs. The algorithm results in statistically independent extracted samples in the 1D spectrum, and preserves the full native resolution of the 2D spectrograph without degradation. Both the statistical errors and the 1D resolution of the extracted spectrum are accurately determined, allowing a correct chi-squared comparison of any model spectrum with the data. Using a model PSF similar to that found in the red channel of the Sloan Digital Sky Survey spectrograph, we compare the performance of our algorithm to that of cross-section based optimal extraction, and also demonstrate that our method allows coaddition and foreground estimation to be carried out as an integral part of the extraction step. This work demonstrates the feasibility of current- and next-generation multi-fiber spectrographs for faint galaxy surveys even in the presence of strong night-sky foregrounds. We describe the handling of subtleties arising from fiber-to-fiber crosstalk, discuss some of the likely challenges in deploying our method to the analysis of a full-scale survey, and note that our algorithm could be generalized into an optimal method for the rectification and combination of astronomical imaging data.

Figures (4)

• Figure 1: Left: from left to right: simulated noise-free emission-line image, 2D extraction model of simulated image, and row-wise extraction model of simulated image. Color scaling is in units of the base-10 logarithm of the pixel value, with the pixel values themselves scaled to have an average value of unity across the entire image. Right: residuals resulting from the subtraction of the 2d extraction model (left) and row-wise extraction model (right) from the simulated emission-line image. Color scale is in (non-logarithmic) units of residual counts. Image regions are 31$\times$101 pixels in size in all cases.
• Figure 2: Extracted spectra of simulated noise-free emission-line image. Thin black line: deconvolved spectrum from the 2D modeling extraction method. Note significant ringing. Thick blue line (upper of the two thick lines, rendered with steps): deconvolved spectrum from 2D modeling extraction reconvolved to the native spectrograph resolution using the resolution matrix $\mathbf{R}$ defined in the text. Thick green line (lower of the two thick lines, rendered with steps): 10$\times$ the difference between the reconvolved 2D-model extracted spectrum and the row-model extracted spectrum. "Upward-peakiness" at the position of the emission lines indicates that the 2D-extracted spectrum has higher resolution than the row-extracted spectrum.
• Figure 3: Simulated multifiber, multi-exposure spectroscopic data, including noise, flexure, a non-uniform PSF, and "sky". Left: Three "exposures" of four fiber spectra, including simulated flexure and sky-spectrum variability. Object spectra are included in central two fibers in each set, but are too faint to see directly. Center: same as left, after subtraction of extraction model for the sky component in each exposure, with gray-scale stretched by a factor of 40 to reveal the traces of the object spectra. Right: As in center, but after subtraction of extraction model for the object spectra as well. When scaled by the pixel errors, these residuals are consistent with a reduced $\chi^2$ of unity. Each "exposure" is 51$\times$101 pixels in size.
• Figure 4: Extracted spectra of two simulated objects from multiple exposures as described in §\ref{['test4x3']} and depicted in Figure \ref{['modelframe']}, together with extracted "sky" spectra. Black lines (rendered with steps) show the extracted spectra, while blue lines (rendered smoothly and tracing the black lines) show the input spectra convolved with the 1D resolution. Green lines of varying shades (thinner, with higher peak values, and identical in both plots) indicate the extractions of the three different realizations of the "sky" spectrum in the three individual exposures, divided by a factor of 20 for display purposes.