MUSE (多單位分光探測器) 是一個擁有廣域視野的積分場光譜儀 (integral field spectrograph)，目前正在建設於歐洲南方天文台的超大望遠鏡上，這個光學儀器可以獲得在二維視場內天文物體的光譜訊息；因此，MUSE 將會提供三維的高光譜數據立方體 (hyperspectral data cube)，其中包含了兩個空間軸與一個波長軸。然而，由於從 MUSE 儀器收到的數據會含有大量雜訊 (noise) 且會被隨著平移改變 (translation variant) 的三維模糊函數 (blur function) 所影響，所以想利用這些數據來偵測與分析天體物理訊號源是一件非常艱鉅的事。為了對付這個深具挑戰性的工作，在本論文中，我們著重於如何準確地估測出離地球非常遙遠的星系光譜 (galaxy spectra) 與相對應的豐度圖 (abundance maps)，豐度圖是指每個星系被模糊函數影響過後在二維視場上的含量比例分布圖。為此，我們首先利用一些對 MUSE 的真實假設將原本的捲積 (convolution) 訊號模型重新公式化成為數個基於不重疊之頻段框架 (spectral frame) 的線性混和 (linear mixing) 模型。根據這些線性混合模型，我們提出一個偵測星系光譜的演算法，稱之為空間-頻譜分離 (spatial-spectral unmixing, SSU)。在每個頻段框架中， SSU 演算法可以藉由理論上的證明鑑別出只由單一星系所構成的純星系區域 (pure galaxy regions)，接著利用對星系光譜的稀疏逼近假設 (sparse approximation assumption) 估測出在此頻段框架中的星系光譜。而後，根據已估測到的星系光譜，我們可以使用倒置程序 (inversion process) 去找到相對應的豐度圖。一旦得到在所有頻段框架中估測到的星系光譜，我們先適當地調整在每個頻段框架中星系光譜的排列順序，接著將所有在不同頻段框架下估測到的星系光譜堆疊以還原完整的星系光譜。最後，我們利用電腦模擬驗證 SSU 演算法的優良效能。

MUSE (Multi-Unit Spectroscopic Explorer) is a wide-field integral field spectrograph at the Very Large Telescope (VLT) for the European Southern Observatory (ESO) under construction and it is an optical instrument to be used to obtain spectra of astronomical objects over a two-dimensional field of view. Hence, MUSE will provide three-dimensional hyperspectral data cube with two spatial axes and a wavelength axis. However, due to the high noise level and the three-dimensional translation variant blur function, detection and analysis of astrophysical sources from the forthcoming MUSE instrument is of greatest challenge. In this thesis, we tackle this challenging task by studying how to accurately estimate the spectra of very distant galaxies and the corresponding abundance maps (or proportional contribution of each galaxy over the field of view affected by spatial blur). We first use some realistic hypotheses of MUSE to reformulate the data convolution model into a set of linear mixing models corresponding to different, disjoint spectral frames. Based on the linear mixing models, we propose a spatial-spectral unmixing (SSU) algorithm to detect and characterize the galaxy spectra. In each spectral frame, the SSU algorithm identifies the pure galaxy regions with a theoretical guarantee, and estimate galaxy spectra based on a sparse approximation assumption. Then, the inversion process is applied to estimate the abundances associated with the spectra estimates. The full galaxy spectra can finally be recovered by concatenating the spectra estimates associated with all the spectral frames through an advisable permutation. Some computer simulations are performed to demonstrate the efficacy of the proposed SSU algorithm.

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