Upload
hashim
View
92
Download
1
Embed Size (px)
DESCRIPTION
An Introduction to Compressive Sensing. Speaker: Ying- Jou Chen Advisor: Jian-Jiun Ding. Compressive Compressed. Sensing Sampling. CS. Outline. Conventional Sampling & Compression Compressive Sensing Why it is useful? Framework When and how to use Recovery Simple demo. Review… - PowerPoint PPT Presentation
Citation preview
An Introduction to Compressive Sensing
Speaker: Ying-Jou ChenAdvisor: Jian-Jiun Ding
Compressive
Compressed
SensingSamplingCS
Outline
• Conventional Sampling & Compression• Compressive Sensing• Why it is useful?• Framework• When and how to use• Recovery• Simple demo
Review…
Sampling and Compression
Nyquist’s Rate
• Perfect recovery
Transform Coding
• Assume: signal is sparse in some domain…• e.g. JPEG, JPEG2000, MPEG…1. Sample with frequency . Get signal of length N2. Transform signal K (<< N) nonzero
coefficients3. Preserve K coefficients and their locations
Compressive Sensing
Compressive Sensing
• Sample with rate lower than !!
• Can be recovered PERFECTLY!
Comparison
Nyquist’s Sampling Compressive Sensing
Sampling Frequency
Recovery Low pass filter Convex Optimization
Some Applications
• ECG• One-pixel Camera• Medical Imaging: MRI
Framework
Φ¿𝑦 𝑓
NM
N
M
N: length for signal sampled with Nyquist’s rateM: length for signal with lower rate Sampling matrix
𝐲=𝚽𝐟
When? How?
Two things you must know…
When….
• Signal is compressible, sparse…
Ψ
𝑥Φ¿
𝑦 𝑓
NM
N
M
Example… ECG
: 心電圖訊號: DCT (discrete cosine transform)
How…
• How to design the sampling matrix?• How to decide the sampling rate
(M)?
Ψ
𝑥Φ¿
𝑦M
N
Sampling Matrix
• Low coherence
Ψ
𝑥Φ¿
𝑦 Low coherence
Coherence
• Describe similarity
– High coherence more similar Low coherence more different
Example: Time and Frequency
0 10 20 30 40 50 60 70 80 90 100-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
• For example, • ,
Fortunately…
• Random Sampling– iid Gaussian N(0,1)–Random
• Low coherence with deterministic basis.
More about low coherence…
Random Sampling
Sampling Rate
• Can be exactly recovered with high probability.
Theorem
C : constant S: sparsityn: signal length
Recovery
Ψ
𝑥Φ¿
𝑦
BUT….
M
N
M
N
𝑓
N
Recovery
• Many related research…– GPSR (Gradient projection for sparse reconstruction)– L1-magic– SparseLab– BOA (Bound optimization approach)…..
Total Procedure
f Find an incoherent matrix e.g. random matrix
Sample signal
𝒂𝒓𝒈 �̂�𝒎𝒊𝒏‖�̂�‖𝟏𝑠 . 𝑡 . 𝐲=𝚯�̂� �̂�=𝐇�̂�
Sampling (Assume f is spare somewhere)
Recovering
已知 :
Demo Time
Reference• Candes, E. J. and M. B. Wakin (2008). "An Introduction To Compressive Sampling."
Signal Processing Magazine, IEEE 25(2): 21-30.• Baraniuk, R. (2008). Compressive sensing. Information Sciences and Systems, 2008.
CISS 2008. 42nd Annual Conference on.• Richard Baraniuk, Mark Davenport, Marco Duarte, Chinmay Hegde. An
Introduction to Compressive Sensing.• https://sites.google.com/site/igorcarron2/cs#sparse• http://videolectures.net/mlss09us_candes_ocsssrl1m/
Thanks a lot!