Lec 24: April 23, 2020 Compressive Sensingese531/spring2020/handouts/lec24.pdf · ESE 531: Digital...
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ESE 531: Digital Signal Processing
Lec 24: April 23, 2020 Compressive Sensing
Penn ESE 531 Spring 2020 – Khanna
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Today
! Compressive Sampling/Sensing
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Compressive Sampling
! What is the rate you need to sample at? " At least Nyquist
3
t T
0
Anything
Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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Compressive Sampling
! What is the rate you need to sample at? " Maybe less than Nyquist…
4
t T
0
Something
Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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First: Compression
! Standard approach " First collect, then compress
" Throw away unnecessary data
5 Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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First: Compression
! Examples " Audio – 10x
" Raw audio: 44.1kHz, 16bit, stereo = 1378 Kbit/sec " MP3: 44.1kHz, 16 bit, stereo = 128 Kbit/sec
" Images – 22x " Raw image (RGB): 24bit/pixel " JPEG: 1280x960, normal = 1.09bit/pixel
" Videos – 75x " Raw Video: (480x360)p/frame x 24b/p x 24frames/s + 44.1kHz
x 16b x 2 = 98,578 Kbit/s " MPEG4: 1300 Kbit/s
6 Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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First: Compression
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! Almost all compression algorithm use transform coding " mp3: DCT " JPEG: DCT " JPEG2000: Wavelet " MPEG: DCT & time-difference
Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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First: Compression
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! Almost all compression algorithm use transform coding " mp3: DCT " JPEG: DCT " JPEG2000: Wavelet " MPEG: DCT & time-difference
Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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Sparse Transform
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Sparse Transform
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Sparsity
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Signal Processing Trends
! Traditional DSP # sample first, ask questions later
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Signal Processing Trends
! Traditional DSP # sample first, ask questions later ! Explosion in sensor technology/ubiquity has caused
two trends: " Physical capabilities of hardware are being stressed,
increasing speed/resolution becoming expensive " gigahertz+ analog-to-digital conversion " accelerated MRI " industrial imaging
" Deluge of data " camera arrays and networks, multi-view target databases,
streaming video…
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Signal Processing Trends
! Traditional DSP # sample first, ask questions later ! Explosion in sensor technology/ubiquity has caused
two trends: " Physical capabilities of hardware are being stressed,
increasing speed/resolution becoming expensive " gigahertz+ analog-to-digital conversion " accelerated MRI " industrial imaging
" Deluge of data " camera arrays and networks, multi-view target databases,
streaming video...
! Compressive Sensing # sample smarter, not faster
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Compressive Sensing/Sampling
! Standard approach " First collect, then compress
" Throw away unnecessary data
15 Penn ESE 531 Spring 2020 – Khanna Adapted from M. Lustig, EECS Berkeley
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Compressive Sensing
! Shannon/Nyquist theorem is pessimistic " 2×bandwidth is the worst-case sampling rate — holds
uniformly for any bandlimited data " sparsity/compressibility is irrelevant " Shannon sampling based on a linear model, compression
based on a nonlinear model
! Compressive sensing " new sampling theory that leverages compressibility " key roles played by new uncertainty principles and
randomness
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Sensing to Data
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Compressive Sampling
! Sample at lower than the Nyquist rate and still accurately recover the signal, and in most cases exactly recover
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Sparse signal in time Frequency spectrum
Compressive Sampling
! Sample at lower than the Nyquist rate and still accurately recover the signal, and in most cases exactly recover
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Undersampled in time
Compressive Sampling
! Sample at lower than the Nyquist rate and still accurately recover the signal, and in most cases exactly recover
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Undersampled in time
Compressive Sampling
! Sample at lower than the Nyquist rate and still accurately recover the signal, and in most cases exactly recover
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Undersampled in time Undersampled in frequency (reconstructed in time with IFFT)
Compressive Sampling
! Sample at lower than the Nyquist rate and still accurately recover the signal, and in most cases exactly recover
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Undersampled in time
Compressive Sampling
! Sample at lower than the Nyquist rate and still accurately recover the signal, and in most cases exactly recover
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Undersampled in frequency (reconstructed in time with IFFT)
Requires sparsity and incoherent sampling
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Compressive Sampling: Simple Example
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Compressive Sampling
! Sense signal M times ! Recover with linear
program
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Compressive Sampling
! Sense signal M times ! Recover with linear
program
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Example: Sum of Sinusoids
! Two relevant “knobs” " percentage of Nyquist
samples as altered by adjusting number of samples, M
" input signal duration, T " Data block size
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7% 14% 17.5%
20.9% 34.7% 51.9%
Example: Increasing M
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$ ferr,max within 10 mHz $ perr,max decreasing
Example: Increasing M
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T=5 T=10 T=15
T=30 T=60 T=120
Example: Increasing T
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$ ferr,max decreasing $ perr,max decreasing
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Example: Increasing T
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Numerical Recovery Curves
! Sense S-sparse signal of length N randomly M times
" In practice, perfect recovery occurs when M ≈ 2S for N ≈ 1000
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N=256 N=512 N=1024
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A Non-Linear Sampling Theorem
! Exact Recovery Theorem (Candès, R, Tao, 2004): " Select M sample locations {tm} “at random” with
! Take time-domain samples (measurements)
! Solve
! Solution is exactly recovered signal with extremely high probability
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A Non-Linear Sampling Theorem
! Exact Recovery Theorem (Candès, R, Tao, 2004): " Select M sample locations {tm} “at random” with
! Take time-domain samples (measurements)
! Solve
! Solution is exactly recovered signal with extremely high probability
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M > C·μ2(Φ,Ψ)·S·log N
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Biometric Example: Parkinson’s Tremors
! 6 Subjects of real tremor data
" collected using low intensity velocity-transducing laser recording aimed at reflective tape attached to the subjects’ finger recording the finger velocity
" All show Parkinson’s tremor in the 4-6 Hz range.
" Subject 8 shows activity at two higher frequencies
" Subject 4 appears to have two tremors very close to each other in frequency
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Compressive Sampling: Real Data
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$ C=10.5, T=30 $ 20% Nyquist required samples
Biometric Example: Parkinson’s Tremors
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$ Tremors detected within 100 mHz
$ randomly sample 20% of the Nyquist required samples
Biometric Example: Parkinson’s Tremors
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Requires post processing to randomly sample!
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Implementing Compressive Sampling
! Devised a way to randomly sample 20% of the Nyquist required samples and still detect the tremor frequencies within 100mHz " Requires post processing to randomly sample!
! Implement hardware on chip to “choose” samples in real time " Only write to memory the “chosen” samples
" Design random-like sequence generator " Only convert the “chosen” samples
" Design low energy ADC
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CS Theory
Why does it work?
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Sampling
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Compressive Sampling
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How Can It Work?
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How Can It Work?
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How Can It Work?
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How Can It Work?
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RIP
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CS Signal Recovery
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CS Signal Recovery
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L2 Signal Recovery
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L0 Signal Recovery
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L1 Signal Recovery
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x̂
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Universality
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Universality
54 Penn ESE 531 Spring 2020 - Khanna
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Universality
55 Penn ESE 531 Spring 2020 - Khanna
![Page 56: Lec 24: April 23, 2020 Compressive Sensingese531/spring2020/handouts/lec24.pdf · ESE 531: Digital Signal Processing Lec 24: April 23, 2020 Compressive Sensing Penn ESE 531 Spring](https://reader033.fdocuments.in/reader033/viewer/2022051512/60336542a5508a308667274b/html5/thumbnails/56.jpg)
Reference Slide
56 Penn ESE 531 Spring 2020 - Khanna
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Big Ideas
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! Compressive Sampling " Integrated sensing/sampling, compression and processing " Based on sparsity and incoherency
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Admin
! Final Project due – Apr 28th " TA advice – “The report takes time. Leave time for it.”
! Last day of TA office hours – Apr 28th
" Piazza still available " Review session for exam TBD
! Last day of Tania office hours – May 1st ! Final Exam – May 7th
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Final Exam Admin
! Final Exam – 5/7 (3pm-5pm) " In Canvas
" Will have a 2 hr window to complete within a 12 hr time block
" Open course notes and textbook, but cannot communicate with each other about the exam
" Students will have randomized and different questions " Reminder, it is not in your best interest to share the exam
" Old exams posted on old course websites " Covers Lec 1- 20
" Does not include lec 12 (data converters and noise shaping) or IIR Filters
59 Penn ESE 531 Spring 2020 – Khanna