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In compressive sensing, signal sparsity is an important parameter which influences the number of data sampling in reconstruction process and the quantity of the reconstructed result. But in practice, undersampled and oversampled phenomenon will occur because of the unknown sparsity, which may lose the advantages of compressive sensing. So how to determine the image sparsity quickly and accuratly is significant in the compressive sensing process. In this paper, we calculate the image sparsity based on the data acquired during compressive sensing recontruction projection which sparses the origin image in wavelets domain, but we find that its procession is complex, and the final results are seriously influenced by wavelet basis function and the transform scales. We then introduce the principle component analysis (PCA) theory combined with compressive sensing, and establish a linear relationship between image sparsity and coefficient founction variance based on the assumption that PCA is of approximately normal distribution. Multiple sets of experiment data verify the correctness of the linear relationship mentioned above. Through previous analysis and simulation, the sparsity estimation based on PCA has an important practical value for compressive sensing study.
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Keywords:
- compressive sensing /
- sparsity /
- wavelet transform /
- principle component analysis
[1] Donoho D 2006 IEEE Trans. Inform. Theory 52 1289
[2] Zhao X F, Huang S X, Xiang J, Shi W L 2011 Chin. Phys. B 20 099201
[3] Liu Y Y, L Q B, Zeng X R, Huang M, Xiang L B 2013 Acta Phys. Sin. 62 060203 (in Chinese) [刘扬阳, 吕群波, 曾晓茹, 黄旻, 相里斌 2013 物理学报 62 060203]
[4] Jin X L 2010 Acta Phys. Sin. 59 692 (in Chinese) [季小玲 2010 物理学报 59 692]
[5] Wei H Y, Wu Z S, Peng H 2008 Acta Phys. Sin. 57 6666 (in Chinese) [韦宏艳, 吴振森, 彭辉 2008 物理学报 57 6666]
[6] Candés E J, Romberg J, Tao T 2006 IEEE Trans. Signal Process. 52 489
[7] Duarte M F, Baraniuk R G 2012 IEEE Trans. Image Proc. 21 494
[8] Jin L X, Zhang R F 2013 Chin. Phys. B 22 064203
[9] Tsaig Y, Donoho D L 2006 Signal Process. 86 549
[10] Duarte M F 2008 IEEE Signal Proc. Mag. 25 83
[11] Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Liu L Z 2013 Chin. Phys. B 22 078701
[12] He L, Carin L 2009 IEEE Trans. Signal Process. 57 3488
[13] Xue B, Chen X D, Zhang Y, Liu B 2011 Signal Process. 9 1085
[14] Duarte M F, Wakin M, Baraniuk R G 2008 Int. Conf. Acoustics, Speech, and Signal Process. (ICASSP) Las Vegas USA March 30-April 4, 2008 p5137
[15] Hu L Y, Fan H Y 2011 Chin. Phys. B 19 074205
[16] Tropp J A, Gilbert A C 2007 IEEE Trans. Inform. Theory 53 4655
[17] Ronald A D 1998 Acta Numerica 7 51
[18] Kim E, Paul G 1987 Principal Component Analysis (Amsterdam: Elsevier Science Publishers) pp37-52
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[1] Donoho D 2006 IEEE Trans. Inform. Theory 52 1289
[2] Zhao X F, Huang S X, Xiang J, Shi W L 2011 Chin. Phys. B 20 099201
[3] Liu Y Y, L Q B, Zeng X R, Huang M, Xiang L B 2013 Acta Phys. Sin. 62 060203 (in Chinese) [刘扬阳, 吕群波, 曾晓茹, 黄旻, 相里斌 2013 物理学报 62 060203]
[4] Jin X L 2010 Acta Phys. Sin. 59 692 (in Chinese) [季小玲 2010 物理学报 59 692]
[5] Wei H Y, Wu Z S, Peng H 2008 Acta Phys. Sin. 57 6666 (in Chinese) [韦宏艳, 吴振森, 彭辉 2008 物理学报 57 6666]
[6] Candés E J, Romberg J, Tao T 2006 IEEE Trans. Signal Process. 52 489
[7] Duarte M F, Baraniuk R G 2012 IEEE Trans. Image Proc. 21 494
[8] Jin L X, Zhang R F 2013 Chin. Phys. B 22 064203
[9] Tsaig Y, Donoho D L 2006 Signal Process. 86 549
[10] Duarte M F 2008 IEEE Signal Proc. Mag. 25 83
[11] Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Liu L Z 2013 Chin. Phys. B 22 078701
[12] He L, Carin L 2009 IEEE Trans. Signal Process. 57 3488
[13] Xue B, Chen X D, Zhang Y, Liu B 2011 Signal Process. 9 1085
[14] Duarte M F, Wakin M, Baraniuk R G 2008 Int. Conf. Acoustics, Speech, and Signal Process. (ICASSP) Las Vegas USA March 30-April 4, 2008 p5137
[15] Hu L Y, Fan H Y 2011 Chin. Phys. B 19 074205
[16] Tropp J A, Gilbert A C 2007 IEEE Trans. Inform. Theory 53 4655
[17] Ronald A D 1998 Acta Numerica 7 51
[18] Kim E, Paul G 1987 Principal Component Analysis (Amsterdam: Elsevier Science Publishers) pp37-52
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