和傳統取樣技術相比，壓縮感知能利用更少的點數去壓縮訊號。而一個重要的問題便是如何去重建原始的訊號。有許多著名的演算法像是基追蹤、匹配追蹤、正交匹配追蹤、壓縮採樣匹配追蹤。而這些演算法都需要事先知道訊號的稀疏度。這篇論文我們提出一個基於子空間追蹤不斷更新的演算法。我們提出的論文命名為稀疏度更新子空間追蹤。這篇論文主要的貢獻在於處理稀疏度未知的問題。而這個特性讓此演算法可以處理許多實際上稀疏度未知的應用。而此演算法不單處理稀疏度未知的問題，在稀疏度已知的情況下表現也好於許多演算法。

Compared with traditional data acquisition, compressive sensing can reconstruct signal from far fewer samples than the traditional method. One of the critical problem is to reconstruct the original signal from the compressed measurement. There are many popular reconstruction algorithms such as Basis pursuit(BP), Matching Pursuit(MP), Orthogonal Matching Pursuit(OMP), Compressive Sampling Matching Pursuit(CoSaMP); Despite their good performance, these algorithms require the sparse level as a prior information for reconstruction. This thesis demonstrates a recursively updating function based on Subspace Pursuit algorithm, which is a famous algorithm for signal recovery. Our proposed algorithm is called Sparsity Updating Subspace Pursuit(SUSP). The main contribution is that the proposed SUSP can solves the sparse signal problem when the sparsity is not available. This property makes the algorithm to solve many practical compressive sensing problem when the sparse level, the number of non-zero elements of a signal is not given. While SUSP can deal with unknown sparsity problem, results show that it outperforms many existing famous greedy algorithm when the sparsity is pre-known.

Abstract i
Contents ii
1 Introduction 1
2 Signal Recovery Algorithms for Reconstruction 5
2.1 Basis Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Greedy Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Backtracking Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Subspace Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Compressive Sampling Matching Pursuit . . . . . . . . . . . . . . . . 15
2.3.3 Sparsity Adaptive Matching Pursuit . . . . . . . . . . . . . . . . . . 16
3 Proposed Sparsity Updating Subspace Pursuit for Reconstruction 20
3.1 Proposed Sparsity Updating Subspace Pursuit Algorithm . . . . . . . . . . . 20
3.1.1 Sparsity Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.2 Correlation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.3 Stage Change Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.4 Tail Biting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Procedure of SUSP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 SUSP Algorithm for Reconstruction with known Sparsity . . . . . . . 26
3.2.2 SUSP Algorithm for Reconstruction with Unknown Sparsity . . . . . 28
4 Simulation 31
4.1 Performance of SUSP when Sparsity-Preknown . . . . . . . . . . . . . . . . 31
4.1.1 Comparison of SP, OMP and SUSP vs. the signal sparsity . . . . . . 31
4.1.2 Comparison of SP, OMP and SUSP vs. the number of measurement . 32
4.1.3 Simulations for Gaussian Sparse Signals . . . . . . . . . . . . . . . . 33
4.1.4 Comparison of Compression Ratio . . . . . . . . . . . . . . . . . . . . 34
4.1.5 Performance of SUSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.6 Robust Signal Recovery from Noisy Data . . . . . . . . . . . . . . . . 37
4.2 Performance of SUSP when Sparsity-Unknown . . . . . . . . . . . . . . . . . 38
4.2.1 Performance of Sparsity Estimation . . . . . . . . . . . . . . . . . . . 39
4.2.2 Performance of SUSP vs. Tail Biting Rules . . . . . . . . . . . . . . . 39
4.2.3 Comparison of Existing Algorithm Dealing with Unknown Sparsity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Conclusions 44

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