Hyperspectral imaging systems based on multiple-beam interference (MBI), such as Fabry-Perot interferometry, are attracting interest due to their compact design, high throughput, and fine resolution. Unlike dispersive devices, which measure spectra directly, the desired spectra in interferometric systems are reconstructed from measured interferograms. Although the response of MBI devices is modeled by the Airy function, existing reconstruction techniques are often limited to Fourier-transform spectroscopy, which is tailored for two-beam interference (TBI). These methods impose limitations for MBI and are susceptible to non-idealities like irregular sampling and noise, highlighting the need for an in-depth numerical framework. To fill this gap, we propose a rigorous taxonomy of the TBI and MBI instrument description and propose a unified Bayesian formulation which both embeds the description of existing literature works and adds some of the real-world non-idealities of the acquisition process. Under this framework, we provide a comprehensive review of spectroscopy forward and inverse models. In the forward model, we propose a thorough analysis of the discretization of the continuous model and the ill-posedness of the problem. In the inverse model, we extend the range of existing solutions for spectrum reconstruction, framing them as an optimization problem. Specifically, we provide a progressive comparative analysis of reconstruction methods from more specific to more general scenarios, up to employing the proposed Bayesian framework with prior knowledge, such as sparsity constraints. Experiments on simulated and real data demonstrate the framework's flexibility and noise robustness. The code is available at https://github.com/mhmdjouni/inverspyctrometry.

翻译：基于多光束干涉（MBI）的高光谱成像系统，例如法布里-珀罗干涉仪，因其紧凑的设计、高光通量和精细的分辨率而备受关注。与直接测量光谱的色散器件不同，干涉系统中的目标光谱需从测量的干涉图中重建。尽管MBI器件的响应由艾里函数建模，但现有的重建技术通常局限于为双光束干涉（TBI）量身定制的傅里叶变换光谱学。这些方法对MBI施加了限制，并且容易受到不规则采样和噪声等非理想因素的影响，这突显了对深入数值框架的需求。为填补这一空白，我们提出了TBI和MBI仪器描述的严格分类法，并提出了一种统一的贝叶斯公式，该公式既嵌入了现有文献工作的描述，又增加了采集过程中一些现实世界的非理想因素。在此框架下，我们对光谱学正向和逆向模型进行了全面综述。在正向模型中，我们提出了对连续模型离散化及问题不适定性的透彻分析。在逆向模型中，我们扩展了现有光谱重建解决方案的范围，将其构建为一个优化问题。具体而言，我们提供了从更特定到更一般场景的重建方法的渐进式比较分析，直至采用所提出的、结合先验知识（如稀疏性约束）的贝叶斯框架。在模拟和真实数据上的实验证明了该框架的灵活性和噪声鲁棒性。代码可在 https://github.com/mhmdjouni/inverspyctrometry 获取。