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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 - PowerPoint PPT Presentation
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ACCESS IC LAB
Graduate Institute of Electronics Engineering, NTU
Brief Introduction to Measurement Matrix
Presenter : Yumin ( 林祐民 )Advisor : Prof. An-Yeu Wu
Date : 2014/04/08
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page2
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
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page3
COMPRESSIVE SENSING
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page4
Compressive Sensing(1/2) Traditional digital data acquisition
Sample data with Nyquist rate Compress data
Compressive sensingMain idea: compression within sampling
[1][2][3]
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Compressive Sensing (2/2)
Measure what should be measured
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page6
Construct Sensing Matrix- Criteria of RIP Matrices- Random Sensing- Deterministic Sensing
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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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, 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
Full rank
y0=x0
y1=x1
⁞yn = xn
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
y0 = x0 + x3
y1 = x1 + x8
⁞ym = x2+xn
Non-deterministic Polynomial-time problem
CS
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page8
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 MxK submetrices are full rank Pseidoinverse to recover the nonzero coefficient of x
yMx1 xNx1ΦMxN
K-sparse
yMx1 xNx1ΦMxN yMx1
K columns
xNx1
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page9
Restricted Isometry Property Signal Sparsity
S-parse
Restricted Isometry Property Nearly orthonormal when operation on sparse vector Random constructions exist δ with high probability
xJPEG2000
α< 0.1
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page10
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
[2007’ Donoho D]
δ ≤ , , spark()>2K →
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page11
Gaussian Random Matrix Fill out the entries of Φ with i.i.d. samples form
Gaussian distribution Project on to a “random subspace”
M=O(Slog(N/S)) << N
M: measurementS: non-zero numberN: signal dimension
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page12
Random Fourier Matrix Partial Random Measurement Matrixes
Generate NxN matrix Φ0 and choose M rows to construct MxN measurement matrix Φ
NxN matrix Φ0 :
Random set :
MxN matrix Φ0 :
M=O(Slogp(N/S)) << N
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page13
From Random to Deterministic
Random Sensing
Non-mainstream of signal processing: Worst CaseLess efficient recovery timeLarger storageLess measurements for K-sparse signals
Deterministic Sensing
Mainstream of signal processing: Average Case More efficient recovery time Efficient/compact storage More measurements for K-sparse signals [4]
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page14
Issues for Simplifying Measurement Matrices
From complex to sparse to: Structurally-simplified Numerically-simplified Steady recovery performance
Simplifying Existing Sampling Matrices Becomes Prominent
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page15
Deterministic Simplification(1/2) Structurally-simplified
Numerically-simplified Devore’s binary (0/1) BCH-bipolar (±1) Combinatorial-ternary (±1/0)
1 2 3 4 5
… …… …
2 3 4 5 1
Generation Complexity = O(kn)
Sampling Complexity = O(kn)
Generation Complexity = O(n)
Sampling Complexity = O(n*logn)
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page16
Deterministic Simplification(2/2) Structural Simplification Numerically-simplified
Signal Length n
Empi
rical
Pro
babi
lity
of S
ucce
ss
Number of Non-Zero Entries
Succ
essf
ul R
ecov
ery
Rat
e(S
NR
rec≥
100d
B)
Steady Recovery Performance !!
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page17
Application of Compressive Sensing- Medical Imaging- Compressive Imagine
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page18
Applications of Compressive Sensing
Compressive sensing leads to data acquisition revolution
Object RecognitionCompressive MIMO Radar
Electronic Gate
Analog-to-Information ConversionRandom Modulator
Medical ImagingUltrasound
Electrocardiography
Compressive ImagingSingle-pixel Camera
Lensless Camera
High Speed Periodic Video
⁞
Modulated WidebandConverter
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page19
Portable ECGReduce data rate in bio-signal acquisition
systemSampling rate 256Hz, resolution 12bitBandwidth = 256*12 = 3072bit/s = 3Mb/sCS can provide up to 16X compression rate
Ultra-low-power performanceBio-signal acquisition devices are usually portable
[12]
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page20
Compare Two ApproachesAdaptive sampling
Sampling rate is variable
Additional computationcircuit
Compressive sensingLower effective sampling rateThreshold circuit to make signal sparse
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page21
Ultrasound System ImaginePortable ultrasound device
Low powerLess memoryHigh image quality
68%
32%
Power Consumption in single channel
TransmitterReceiver
Use less transmitters for beamforming
Use more transmitters for beamforming Trade off !!
How to use less transmitters to obtain high performance ultrasound image?
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page22
Spatial Sampling Frequency sampling
Reconstruction of Ultrasound Imaging [15]
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page23
Single-Pixel CS Camera Rice University, 2008
randompattern onDMD array
Single photon detector
Image reconstruction
A/D conversion y = 1
11
11
1
11
1
1 2 3 4 5 6 7 8 9
x
1 2 3
[16]
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Single-Pixel CS CameraImage reconstruction
[16]
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Midterm Presentation- Information- 查資料的方法
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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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
Number Subject1. Image via Compressive Sensing2. Medical Application via Compressive Sensing3. Reconstruction Algorithm: Orthogonal Matching Pursuit (OMP)4. Reconstruction Algorithm: Iterative Thresholding5. Hardware Implementation of Reconstruction Algorithm6. Sampling Algorithm: Structured Matrices
Mentor: 林祐民 (Yumin , [email protected]) 黃乃珊 (NHuang , [email protected]) 劉嘉琛 (Jiachen , [email protected]) 陳奕 (Chris , [email protected])
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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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
輸入關鍵字,或期刊名,或作者再按「搜尋」,就可找到想要的資料
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
page28
(2) 尋找 IEEE 的論文http://ieeexplore.ieee.org/Xplore/guesthome.jsp
(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
(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
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
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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(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
台大圖書館首頁 館際合作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」 「其他圖書館」(8) 查詢其他圖書館有沒有我要找的書
「台大圖書館首頁」 「電子書」 或「免費電子書」 (9) 找尋電子書
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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http://www.cetd.com.tw/ec/index.aspx
(10) 中文電子學位論文服務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
(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 )
(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