壓縮感應（Compressive sensing），一種新的抽樣壓縮理論，在2004年一經提出便引起了很大的關注。該理論指出：當某信號在某個變換域內是稀疏的，也就是說這個信號在某個正交變換基下的係數大部份為零或者接近零，那麼這個信號就可以用很少（遠小於奈奎斯特頻率需要的數量）的非自適應線性投影精確的重構。

在這篇論文中，我們首先簡單介紹了壓縮感應（CS）的理論以及解決它的各種演算法，比如匹配追蹤（MP）, 線性規劃（LP）等等。然後我們對基於壓縮感應的單像素數字相機提出了一個新的架構，來完成它對自然圖片的感測和重構。這個架構最基本的概念是以一種漸進的方式，對圖片的每一列像素依次感測，並且依次重構。由於自然圖片中各列的稀疏度不同，在感測過程中，我們採用了一種“折叠”的方式組合一些固定位置的列，使得組合后每一列的稀疏度近似相等。在重構過程中，我們用一種基於混雜轉換的迭代合作式的重建方法，來使得重構圖保持局部平滑性，並且引進了一種基於非局部平均的3D轉換域協作濾波器，來保持自然圖片的非局部自相似性。

運用Matlab中的CVX工具包，結果顯示迭代合作式的重建在峰值信噪比（PSNR）上有$4$到$5$dB的增益。另外，我們提出的整個重構策略複雜度很低，但保持了很好的品質，無論是數值上的PSNR還是視覺效果。比起整張圖一次性重構，PSNR的值同樣好甚至超過它一點點，但計算的時間遠小於它。

Compressive sensing (CS), a new mathematical theory for sampling and compression, has drawn quite much attention since it was proposed in 2004. It tells that a signal can be reconstructed accurately using a small number (much less than suggested by Nyquist rate) of nonadaptive linear projections of this signal, as long as it exhibits sparsity in some domain: the most coefficients under some orthogonal basis are zero or near-zero.

In this thesis, we make a brief introduction to CS theory and various algorithms solving this problem, such as linear programming (LP), belief propagation (BP) and so on. Then we propose a CS scheme to sense and recover natural images, which can be applied to the single-pixel digital camera. The basic idea of this framework is sensing and recovering column by column, in a progressive way, for the sake of reducing the size of sensing matrix and recovering complexity. As the fact that the columns of a natural image have different sparsity, we take a “folding” action to make the sparsity of each column approximately equal when sensing. At the recovering processing, we employ iteratively cooperative reconstructions based on a hybrid transform maintaining the local smoothness and introduce $3$D transform-domain collaborative filtering based on non-local means (NLM), remaining the nonlocal self-similarity of natural images.

With the help of Matlab package CVX, the results say that the iteratively cooperative reconstruction has a gain of $4\sim 5$ dB on PSNR. Also, the recovery strategy we proposed has low complexity but maintaining good quality both on PSNR and visual sense. The value of PNSR is a little better than which recovering the whole image once, but the computing time is less than it.

Furthermore, in order to reduce the computational time, we proposed a novel algorithm based on belief propagation (BP), which quickly solved the problem above.

1. Introduction
2. Reviews
3. Image Sensing and Recovery
4. Belief Propagation Algorithm
5. Conclusions

[1] Emmanuel J. Candes, and Michael B.Wakin, “An Introduction To Compressive Sampling,” IEEE Signal Processing Magazine, March, 2008.

[2] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol.52, no. 4, pp. 1289-1306, Apr. 2006.

[3] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489-509, Feb. 2006.

[4] E. Candes, and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Trans. Inform. Theory, vol. 52, no. 12, pp. 5406-5425, Dec. 2006.

[5] E. Candes, and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005.

[6] M. Figueiredo, R. Nowak, and S.Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Topic Signal Process., vol. 1, no. 4, pp. 568-597, Dec. 2007.

[7] W.Dai, and O. Milenkovic, “Subspace pursuit for compressive sensing: Closing the gap between performance and complexity,” IEEE Trans. Inf. Theory, vol. 55, pp. 2230-2249, May 2009.

[8] D. Needell, and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comp. Harmon. Anal., vol. 26, pp. 301-321, 2008.

[9] R. Berinde, P. Indyk, and M. Ruzic, “Practical near-optimal sparse recovery in the l1 norm,” presented at the Allerton Conf. Comm., Control, Comput., Monticello, IL, Sep. 2008.

[10] S. Sarvotham, D. Baron, and R. Baraniuk, “Compressed sensing reconstruction via belief propagation,” Electrical and Comput. Eng. Dept.,.Rice Univ., Houston, TX, 2006.

[11] M. Akcakaya, J. Park, and V. Tarokh, “A coding theory approach to noisy compressive sensing using low density frame,” IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5369-5379, Nov. 2011.

[12] M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp.83-91 2008.

[13] M. Lustig, D.L. Donoho, and J.M. Pauly, “Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories,” in Proc. 14th Ann. Meeting ISMRM, Seattle, WA, May 2006.

[14] S. S. Vasanawala, M T. Alley, and B. A. Hargreaves, R. A. Barth, J. M. Pauly, and M. Lustig, “Improved pediatric MR imaging with compressed sensing,” Radiology, vol. 256, no. 2, pp. 607-616, 2010.

[15] X.-Y. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregular progressive edge-growth tanner graphs,” IEEE Trans. Inf. Theory, vol. 51, pp. 386-398, Jan. 2005.

[16] A.Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image denoising,” in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2005, pp. 60-65.

[17] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative ltering,” IEEE Trans. Image Process., vol. 16, no. 8, pp. 2080-2095, Aug. 2007.

[18] J. Zhang, D. Zhao, C. Zhao, R. Xiong, S. Ma, and W. Gao, “Image compressive sensing recovery via collaborative sparsity,” IEEE J. Emerg. Sel. Topic Circuits Syst., 2 (3) (2012), pp. 380-391.

[19] CVX: Matlab Software for Disciplined Convex Programming, [Online]. Available: http://cvxr.com/cvx/

[20] Disciplined Convex Programming, [Online]. Available:
http://cvxr.com/dcp/

[21] M. J. Wainwright, “Sharp thresholds for noisy and high-dimensional recovery of sparsity using l1-constrained quadratic programming (Lasso),” IEEE Trans. Inf. Theory, vol. 55, pp. 2183-2202, May 2009.

[22] T. K. Moon, “The EM algorithm in signal processing,” IEEE Signal Process. Mag., vol. 13, pp. 47-60, Nov. 1996.

[23] D. Baron, M.B. Wakin, M.F. Duarte, S. Sarvotham, and R.G. Baraniuk, “Distributed compressed sensing,” 2005, Preprint.

[24] E. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inform. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005.

[25] M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process., vol. 19, no. 9, pp. 2345-2356, Sep. 2010.

[26] G. Coluccia, S. K. Kuiteing, A. Abrardo, M. Barni, and E. Magli, “Progressive compressed sensing and reconstruction of multidimensional signals using hybrid transform/prediction sparsity model,” IEEE J. Emerg. Sel. Topics Circuits Syst., vol. 2, no. 3, pp. 340-352, Sep. 2012.

[27] D. Baron, S. Sarvotham, and R. Baraniuk, “Bayesian compressive sensing via belief propagation,” IEEE Trans. Signal Process., vol. 58, pp. 269-280, Jan. 2010.

[28] J. Kang, H.-N. Lee, and K. Kim, “Bayesian hypothesis test for sparse support recovery using belief propagation,” Proc. in IEEE Statistical Signal Processing Workshop (SSP), pp. 45-48, Aug. 2012.