基於超定線性方程式觀點之光譜重建研究

在近年來，為了使光譜儀成為可以方便攜帶並能廣泛運用的檢測儀器，許多人都朝著微型光譜儀的方向來做研究。在本篇論文中，我們探討在光感測器陣列上實現微型光譜儀之性能，並透過將此系統表示成一組線性方程式來重建光譜；由於所得之線性方程式的條件數(condition number)非常大，導致系統方程式為病態(ill-condition)，為了能夠有效且精確地重建光譜，在此提出樣板選取以及樣板優化設計的概念，大幅降低系統之條件數，使得系統經過降維後能正確的解回目標光譜。另外，在本文中我們提出以非負最小平方演算法(Non-negative least square, NNLS)搭配廣義交叉驗證法(generalized cross validation, GCV)實現參數自適應性來解光譜重建的問題。此外，我們亦利用常見於求解最小平方之演算法如：虛擬反矩陣(pseudo inverse)、NNLS、Tikhonov正規化非負最小平方(TNNLS)、NNLS搭配停止規範(NNLSSC)以及多輸入多輸出(MIMO)系統中常見的偵測信號法如：最小均方差估計(MMSE)和MMSE搭配V-Blast演算法，來求解光譜重建之超定線性方程式，並驗證低成本光感測器重建光譜的能力，使其能達到與一般傳統光譜儀之性能。實驗數據顯示，使用TNNLS演算法搭配優化設計之樣板可達到相關誤差(correlation error)小於0.005之光譜重建效能。

關鍵字

光感測器陣列；樣板選取；光譜重建

並列摘要

Recently, miniature spectrometers have gained a significant attention from both academia and industries. In this work, we aim to provide ultra-low-cost miniature spectrometers by applying filter-array spectrum sensors. Therefore, we represented the system of filter-array spectrum sensors as a set of over-determined linear equations and investigated the performance of reconstructed spectrum. Due to the large condition number of the system, the process of spectrum reconstruction is a challenge. In order to reconstructed spectrum effectively and accurately, first we tend to reduce the condition number of the system by applying the template selection and the points of optimal template designing. Second, we propose an adaptive-parameter algorithm by combining non-negative least square algorithm (NNLS) with a generalized cross validation (GCV) method. Besides, we compare with algorithms for solving least squares such as pseudo inverse, NNLS, Tikhonov regularized nonnegative least square (TNNLS) and NNLS with stop criterion (NNLSSC). In addition, algorithms for detecting signals in multi-input multi-output (MIMO) detectors such as minimum mean squared error (MMSE) and MMSE with V-Blast algorithms are applied for solving the over-determined linear equations of spectrum reconstruction. Simulation results show that using TNNLS algorithm with the designed optimal template for spectrum reconstruction can reach correlation errors less than 0.005.

並列關鍵字

Filter-array spectrum sensors ； template selection ； spectrum reconstruction

參考文獻

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基於超定線性方程式觀點之光譜重建研究

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