在壓縮性感知中，最重要的步驟是如何將一個壓縮後的訊號恢復成它原本的樣子，也就是所謂的稀疏訊號重建問題。
　　本論文主要的貢獻在於提出一個能不斷更新與更正追蹤匹配所製造出解的演算法，追蹤匹配是用於做稀疏訊號重建的一個著名演算法。我們的演算法命名為漸進式正交追蹤匹配演算法，因為它不斷地更新欲重建的訊號，並漸進式地找出正確的解。這篇論文主要的貢獻在於提升重建的效能，同時不增加所需要的取樣數，換句話說，利用較多的運算時間來達到節省硬體成本的目的。
　　除此之外，本論文也提出漸進式過度偵測正交追蹤匹配演算法，用來處理實際應用中，當稀疏性未知的重建問題，這同時也是一個尚未被完美處理的追蹤匹配問題。本論文提出的演算法能夠快速地解出稀疏訊號重建問題，並同時保有相當好的重建表現。

Contents
Abstract i
Contents ii
1 Introduction 1
2 Progressive Orthogonal Matching Pursuit for Reconstruction with Known
Sparsity 5
2.1 Conventional Compressive Sensing Reconstruction Algorithm . . . . . . . . . 5
2.1.1 Matching Pursuit [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Largest Inner Product and Orthogonal Projection . . . . . . . . . . . 7
2.1.3 Orthogonal Matching Pursuit [3] . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Orthogonal Matching Pursuit with Over-Detection . . . . . . . . . . 11
2.2 Proposed Progressive Orthogonal Matching Pursuit Algorithm . . . . . . . . 12
2.2.1 Beginning with an estimation of x . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Dierence between actual and estimated x . . . . . . . . . . . . . . . 12
2.2.3 Eliminating the wrongly selected atoms . . . . . . . . . . . . . . . . . 13
2.2.4 Orthogonal Projection with Over-Detection . . . . . . . . . . . . . . 14
ii
2.2.5 End of ProOMP Algorithm . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.6 Progressive OMP with Over-Detection . . . . . . . . . . . . . . . . . 15
2.2.7 Multiple Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Early Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Progressive Over-Detected OMP Algorithm for Reconstruction with Un-
known Sparsity 19
3.1 Proposed Progressive Over-Detected OMP Algorithm . . . . . . . . . . . . . 20
3.1.1 Solution-Candidate Set . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.2 Property of Orthogonal Projection in OMP . . . . . . . . . . . . . . 22
3.1.3 Progressive Orthogonal Matching Pursuit Algorithm for Compensation 23
3.1.4 Procedure of ProODOMP Algorithm . . . . . . . . . . . . . . . . . . 24
3.2 Further improvement on Progressive Over-Detected OMP Algorithm . . . . . 25
4 Simulations 27
4.1 Comparison of OMP and OMP with Over-Detection . . . . . . . . . . . . . . 27
4.2 Comparison of Number of Renements in Progressive OMP Algorithm . . . 29
4.3 Comparison of Number of p in Progressive OMP Algorithm . . . . . . . . . 30
4.4 Comparison of Existing Algorithms Dealing with Known Sparsity Problem . 32
4.5 Validity of Solution-Candidate Set . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 The Estimated Sparsity by Progressive Over-Detected OMP Algorithm . . . 34
4.7 Comparison of Compression Ratio . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8 Performance of Progressive Over-Detected OMP . . . . . . . . . . . . . . . . 36
iii
4.9 Comparison of Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Conclusions 40

[1] S. Mallat and Z. Zhang, "Matching pursuits with time-frequency dictionaries.," IEEE Trans. Sign. Process., vol. 41, no. 12, pp. 3397{3415, 1993.
[2] S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Journal on Scientic Computing, vol. 20, pp. 33{61, Jan. 1998.
[3] J. Tropp and A. Gilbert, "Signal recovery from random measurements via orthogonal matching pursuit," IEEE Trans. on Information Theory, vol. 53, no. 12, pp. 4655{4666,
2007.
[4] E. Candes and M. Wakin, "An introduction to compressive sampling," IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21{30, 2008.
[5] M. Mishali and Y. Eldar, "From theory to practice: Sub-nyquist sampling of sparse wideband analog signals," IEEE Journal of Selected Topics in Signal Processing, vol. 4,
no. 2, pp. 375{391, 2010.
[6] Z. Tian, "Compressed wideband sensing in cooperative cognitive radio networks," in IEEE Global Telecommunications Conference, 2008., pp. 1{5, 2008.
[7] D. Donoho, "Compressed sensing," IEEE Trans. on Information Theory, vol. 52, no. 4, pp. 1289{1306, 2006.
[8] Y. Pati, R. Rezaiifar, and P. S. Krishnaprasad, "Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition," in Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Comput-
ers, 1993., pp. 40{44 vol.1, 1993.
[9] D. Donoho, Y. Tsaig, I. Drori, and J.-L. Starck, "Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit," IEEE Trans. on
Information Theory, vol. 58, no. 2, pp. 1094{1121, 2012.
[10] D. Needell and R. Vershynin, "Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit," IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 2, pp. 310{316, 2010.
[11] D. Needell and J. Tropp, "Cosamp: Iterative signal recovery from incomplete and inaccurate samples," Applied and Computational Harmonic Analysis, vol. 26, no. 3, pp. 301
{ 321, 2009.
[12] W. Dai and O. Milenkovic, "Subspace pursuit for compressive sensing signal reconstruction," IEEE Trans. on Information Theory, vol. 55, no. 5, pp. 2230{2249, 2009.
[13] W. Yin, Z. Wen, S. Li, J. Meng, and Z. Han, "Dynamic compressive spectrum sensing for cognitive radio networks," in 45th Annual Conference on Information Sciences and
Systems (CISS), pp. 1{6, 2011.
[14] D. Baby and S. Pillai, "Ordered orthogonal matching pursuit," in 2012 National Conference on Communications (NCC), pp. 1{5, 2012.
[15] T. Do, L. Gan, N. Nguyen, and T. Tran, "Sparsity adaptive matching pursuit algorithm for practical compressed sensing," in Proc. IEEE 42nd Asilomar Conference on Signals,
Systems, and Computers, pp. 581{587, 2008.
[16] E. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies," IEEE Transactions on Information Theory, vol. 52, no. 12,
pp. 5406{5425, 2006.
[17] Y. E. Nesterov and A. Nemirovskii, Interior point polynomial methods in convex programming: theory and algorithms. SIAM Publications. SIAM, Philadelphia, USA, 1993.