Brief Introduction to Measurement Matrix

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


page3

COMPRESSIVE SENSING


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]


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Compressive Sensing (2/2)

Measure what should be measured


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


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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


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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


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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


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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 →


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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


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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


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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]


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Issues for Simplifying Measurement Matrices

From complex to sparse to: Structurally-simplified Numerically-simplified Steady recovery performance

Simplifying Existing Sampling Matrices Becomes Prominent


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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)


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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 !!


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Application of Compressive Sensing- Medical Imaging- Compressive Imagine


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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


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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]


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Compare Two ApproachesAdaptive sampling

Sampling rate is variable

Additional computationcircuit

Compressive sensingLower effective sampling rateThreshold circuit to make signal sparse


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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?


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Spatial Sampling Frequency sampling

Reconstruction of Ultrasound Imaging [15]


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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]


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Single-Pixel CS CameraImage reconstruction

[16]


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Midterm Presentation- Information- 查資料的方法


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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])

mailto:[email protected]





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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


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(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


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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


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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

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Brief Introduction to Measurement Matrix