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Brief Introduction to Measurement Matrix

Brief Introduction to Measurement Matrix. Presenter : Yumin ( 林祐民 ) Advisor : Prof. An- Yeu Wu Date : 2014/04/08. Outline. Compressive Sensing Construct Sensing Matrix Criteria of RIP Matrices Random Sensing Deterministic Sensing Application of Compressive Sensing Medical Imaging

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Brief Introduction to Measurement Matrix

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  1. Brief Introduction to Measurement Matrix Presenter : Yumin (林祐民) Advisor : Prof. An-Yeu Wu Date : 2014/04/08

  2. Outline • Compressive Sensing • Construct Sensing Matrix • Criteria of RIP Matrices • Random Sensing • Deterministic Sensing • Application of Compressive Sensing • Medical Imaging • Compressive Imagine • Midterm Presentation • Information • Paper Survey

  3. Compressive Sensing

  4. Compressive Sensing(1/2) [1][2][3] • Traditional digital data acquisition • Sample data with Nyquist rate • Compress data • Compressive sensing • Main idea: compression within sampling

  5. Compressive Sensing (2/2) • Measure what should be measured

  6. Construct Sensing Matrix- Criteria of RIP Matrices- Random Sensing- Deterministic Sensing

  7. Measurement • Fundamental questions in compressive sensing • How to construction suitable sensing matrices Φ • How to recovery signal • From orthogonal basis sensing to non-linear sensing X =(x0, x1, x2, x3,∙∙∙∙∙∙∙, xn) V0=(1, 0, 0, 1, ∙∙∙∙∙∙, 0) V1=(0, 1, 0, 0,∙∙∙∙∙∙, 0) ⁞ Vm=(0, 0, 1, 0, ∙∙∙∙∙∙, 1) Y = V∙X X =(x0, x1, x2, x3,∙∙∙∙∙∙∙, xn) V0=(1, 0, 0, 0, ∙∙∙∙∙∙, 0) =δ [k] V1=(0, 1, 0, 0,∙∙∙∙∙∙, 0) =δ [k-1] ⁞ Vn=(0, 0, 0, 0, ∙∙∙∙∙∙, 1) =δ [k-n] Y = V∙X y0 = x0+x3 y1 = x1 +x8 ⁞ ym = x2+xn CS y0=x0 y1=x1 ⁞ yn = xn Non-deterministic Polynomial-time problem Full rank

  8. How Can It Work • Projection Φ not full rank • M<N • Loses information in general • Interested in K-sparse signal • Design Φ so that each of it’s MxKsubmetrices are full rank • Pseidoinverse to recover the nonzero coefficient of x yMx1 xNx1 yMx1 xNx1 yMx1 xNx1 ΦMxN ΦMxN K columns K-sparse

  9. Restricted Isometry Property • Signal Sparsity • S-parse • Restricted Isometry Property • Nearly orthonormal when operation on sparse vector • Random constructions exist δ with high probability x α JPEG2000 < 0.1

  10. Criteria of Good Matrices • Good matrices satisfied • Columns vector of Φ is small linear dependent • Columns vector of Φ is low coherence, which means like randomness • Random matrices satisfied RIP with high probability • Nearly orthonormal when operation on sparse vector • Random matrix: Gaussian random matrix • Partial random matrices: random Fourier matrix δ ≤ , , spark()>2K → [2007’ Donoho D]

  11. Gaussian Random Matrix • Fill out the entries of Φ with i.i.d. samples form Gaussian distribution • Project on to a “random subspace” M: measurement S: non-zero number N: signal dimension M=O(Slog(N/S)) << N

  12. Random Fourier Matrix • Partial Random Measurement Matrixes • Generate NxN matrix Φ0and choose M rows to construct MxN measurement matrix Φ NxN matrix Φ0 : Random set : MxN matrix Φ0 : M=O(Slogp(N/S)) << N

  13. From Random to Deterministic Random Sensing • Non-mainstream of signal processing: Worst Case • Less efficient recovery time • Larger storage • Less measurements for K-sparse signals • Mainstream of signal processing: Average Case • More efficient recovery time • Efficient/compact storage • More measurements for K-sparse signals [4] DeterministicSensing

  14. Issuesfor Simplifying Measurement Matrices • From complex to sparse to: • Structurally-simplified • Numerically-simplified • Steady recovery performance Simplifying Existing Sampling Matrices Becomes Prominent

  15. Deterministic Simplification(1/2) • Structurally-simplified • Numerically-simplified • Devore’s binary (0/1) • BCH-bipolar (±1) • Combinatorial-ternary (±1/0) Generation Complexity = O(kn) Sampling Complexity = O(kn) Generation Complexity = O(n) Sampling Complexity = O(n*logn) 5 5 4 4 3 3 2 2 1 1 … … … …

  16. Deterministic Simplification(2/2) • Structural Simplification • Numerically-simplified Empirical Probability of Success Successful Recovery Rate (SNRrec≥100dB) • Steady Recovery Performance !! Number of Non-Zero Entries Signal Length n

  17. Application of Compressive Sensing- Medical Imaging- Compressive Imagine

  18. Applications of Compressive Sensing • Compressive sensing leads to data acquisition revolution Compressive MIMO Radar • Object Recognition Electronic Gate Random Modulator • Analog-to-Information Conversion Modulated Wideband Converter Ultrasound • Medical Imaging Electrocardiography Single-pixel Camera • Compressive Imaging Lensless Camera High Speed Periodic Video ⁞

  19. Portable ECG [12] • Reduce data rate in bio-signal acquisition system • Sampling rate 256Hz, resolution 12bit • Bandwidth = 256*12 = 3072bit/s = 3Mb/s • CS can provide up to 16X compression rate • Ultra-low-power performance • Bio-signal acquisition devices are usually portable

  20. Compare Two Approaches • Adaptive sampling • Sampling rate is variable • Additional computationcircuit • Compressive sensing • Lower effective sampling rate • Threshold circuit to make signal sparse

  21. Ultrasound System Imagine • Portable ultrasound device • Low power • Less memory • High image quality Use less transmitters for beamforming Trade off !! Use more transmitters for beamforming • How to use less transmitters to obtain high performance ultrasound image?

  22. Reconstruction of Ultrasound Imaging [15] • Spatial Sampling • Frequency sampling

  23. Single-Pixel CS Camera [16] • Rice University, 2008 random pattern on DMD array Single photon detector Image reconstruction 1 2 3 4 5 6 7 8 9 1 2 3 1 1 1 A/D conversion x y = 1 1 1 1 1 1

  24. Single-Pixel CS Camera [16] • Image reconstruction

  25. Midterm Presentation- Information- 查資料的方法

  26. Information • Date: 4/29 (Tue.) 6:30~8:30 • Location: EE2-225 • 兩人一組，每組報告12分鐘，提問3分鐘 Mentor: 林祐民(Yumin，yumin@access.ee.ntu.edu.tw)黃乃珊(NHuang，nhuang@access.ee.ntu.edu.tw) 劉嘉琛(Jiachen，tyrliu@access.ee.ntu.edu.tw) 陳奕(Chris，chris@access.ee.ntu.edu.tw)

  27. 附錄四：查資料的方法 (1) Google 學術搜尋 (不可以不知道) http://scholar.google.com.tw/ (太重要了，不可以不知道) 只要任何的書籍或論文，在網路上有電子版，都可以用這個功能查得到 再按「搜尋」，就可找到想要的資料 輸入關鍵字，或期刊名，或作者

  28. (2) 尋找 IEEE 的論文 http://ieeexplore.ieee.org/Xplore/guesthome.jsp 註：除非你是 IEEE Member，否則必需要在學校上網，才可以下載到 IEEE 論文的電子檔 (3) Google (4) Wikipedia (5) 數學的百科網站 http://eqworld.ipmnet.ru/index.htm 有多個 tables，以及對數學定理的介紹 (6) 傳統方法：去圖書館找資料 台大圖書館首頁 http://www.lib.ntu.edu.tw/ 或者去 http://www.lib.ntu.edu.tw/tulips

  29. (7) 查詢其他圖書館有沒有我要找的期刊 台大圖書館首頁 其他聯合目錄 全國期刊聯合目錄資料庫 如果發現其他圖書館有想要找的期刊，可以申請「館際合作」，請台大圖書館幫忙獲取所需要的論文的影印版 台大圖書館首頁 館際合作 (8) 查詢其他圖書館有沒有我要找的書 「台大圖書館首頁」 「其他圖書館」 (9) 找尋電子書 「台大圖書館首頁」 「電子書」 或「免費電子書」

  30. (10) 中文電子學位論文服務 http://www.cetd.com.tw/ec/index.aspx 可以查到多個碩博士論文 (尤其是 2006年以後的碩博士論文) 的電子版 (11) 想要對一個東西作入門但較深入的了解: 看書會比看 journal papers 或 Wikipedia 適宜 如果實在沒有適合的書籍，可以看 “review”， “survey”， 或 “tutorial” 性質的論文 (12) 有了相當基礎之後，再閱讀 journal papers (以 Paper Title， Abstract， 以及其他 Papers 對這篇文章的描述， 來判斷這篇 journal papers 應該詳讀或大略了解即可)

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