壓縮感測(CS)為信號處理的新領域，它可以用比Shannon-Nyquist rate更低的取樣頻率來完美重建信號。本論文分為三個部份，首先我們在超大影像重建上提出了一個透過樹狀結構結合稀疏性來求凸規劃的快速演算法。其可用來降低計算複雜度以及維持良好的回復品質。透過實驗比較可以驗証其有效性。
第二個部份則是在分散式壓縮感測架構下討論信號彼此間的差異性和重建效能分析。由於之前的理論並沒有考慮信號間的關係，使得其理論結果皆退化回傳統架構的理論結果，但這樣和一般實驗模擬有出入。本研究理論推導出當信號彼間歐式距離越近，則效能會明顯提升。我們理論推導是建立在SOMP演算法上，但其概念可用於其他CS的貪婪算法上。實驗証實了當信號間歐式距離很短時，其效能有明顯地提升，而這與我們所提的理論相符。 在SOMP演算法中，我們也修改了某些步骤更進一步提升效能。
最後應用的部份，我們則提出了在合作式頻譜偵測(CSS)中基於CS貪婪演算法的停止條件。我們分析且推導了與先驗知識(例如稀疏性)無關的停止條件。實驗証實了在CSS中，我們提出的停止條件對於偵測效能有顯著地提升。

Compressive sensing (CS) has emerged as a new framework in signal processing which states that one may achieve an exact signal reconstruction from sufficient CS measurements even lower than the well-known Shannon-Nyquist theorem tells us. Cost-efficient compressive sensing of large-scale images with quickly reconstructed high-quality results is very challenging. We present an algorithm to solve convex optimization via the tree structure sparsity pattern, which can be run in the operator to reduce computation cost and maintain good quality, especially for large-scale images. The feasibility of our method is verified through simulations and comparison with state-of-the-art algorithms.

On the other hand, distributed compressive sensing is a framework considering jointly sparsity within signal ensembles along with multiple measurement vectors (MMVs). The current theoretical bound of performance for MMVs, however, is derived to be the same with that for single MV (SMV) because the characteristics of signal ensembles are ignored. In this work, we introduce a new factor called ``Euclidean distances between signals' for the performance analysis of a deterministic signal model under MMVs framework. We show that, by taking the size of signal ensembles into consideration, MMVs indeed exhibit better performance than SMV. Although our concept can be broadly applied to CS algorithms with MMVs, the case study conducted on a well-known greedy solver called simultaneous orthogonal matching pursuit (SOMP) will be explored in this thesis. We show that the performance of SOMP, when incorporated with our concept by modifying the steps of support detection and signal estimations, will be improved remarkably, especially when the Euclidean distances between signals are short. The performance of modified SOMP is verified to meet our theoretical prediction.

In application part, we proposed stopping criteria for greedy algorithm based on CS in cooperative spectrum sensing (CSS). We analyze and derive oracle stopping bounds that are independent of prior information such as sparsity for greedy algorithms. Simulations are provided to confirm that, in compressive cooperative spectrum sensing, the proposed stopping criteria for greedy algorithms can remarkably improve detection performance.

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