ee123 digital signal processingee123/sp16/notes/...lab + frequency challenge • lab 4 – make sure...
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M. Lustig, EECS UC Berkeley
EE123Digital Signal Processing
Lecture 26Compressed Sensing
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M. Lustig, EECS UC Berkeley
Lab + Frequency challenge
• Lab 4– Make sure you get good signal -- like the one I recorded– Think of detecting bursts -- a robust method will lead to
good results in the last part• Frequency challenge
– Beacon in 5th floor, around 144.280MHz using 1ppb accurate GPSDO. Accurate up to 1/100 Hz.
– Transmits my callsign in morse code 5 times then 2 minutes break.
– Submit frequency on bcourses by Thursday 04/07– You can only use the rtl-sdr to participate -- no cheating!– Closest submission will win a radio!
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M. Lustig, EECS UC Berkeley
Radios
• https://inst.eecs.berkeley.edu/~ee123/sp16/radio.html
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Compressive Sampling
Q: What is the rate you need to sample at?A: At least Nyquist!
0 TAnything
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Q: What is the rate you need to sample at?A: Maybe less than Nyquist....
Compressive Sampling
0 TSomething
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Images are compressibleStandard approach: First collect, then compress
M. Lustig, EECS UC Berkeley
Image Compression
100110100110100010011101010100110100010001010110101010101011001011101110111010101011011011010100111111
1010001101001101011 Compression
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Medical images are compressibleStandard approach: First collect, then compress
M. Lustig, EECS UC Berkeley
Image Compression
100110100110100010011101010100110100010001010110101010101011001011101110111010101011011011010100111111
1010001101001101011 Compression
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Medical images are compressibleStandard approach: First collect, then compress
M. Lustig, EECS UC Berkeley
Compressed Sensing
100110100110100010011101010100110100010001010110101010101011001011101110111010101011011011010100111111
1010001101001101011 Compression
Sensing
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Medical images are compressibleStandard approach: First collect, then compress
M. Lustig, EECS UC Berkeley
Compressed Sensing
100110100110100010011101010100110100010001010110101010101011001011101110111010101011011011010100111111
1010001101001101011 Compression
Sensing
*Courtesy, M. Uecker, J Frahm Max Planck
*
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Example I: Audio
Raw audio: 44.1Khz, 16bit, stereo = 1378 Kbit/sec
MP3: 44.1Khz, 16bit, stereo = 128 Kbit/sec
10.76 fold!
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Example II: Images
Raw image ( RGB ): 24 bit/pixel
JPEG : 1280x960, normal = 1.09 bit/pixel
22 fold!
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Example III: Videos
Raw Video: (480x360)p x 24b/p x 24fps + 44.1Khz x 16b x 2 = 98,578 Kb/s
MPEG4 : 1300 Kb/s
75 fold!
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CompressionSignal
S p a r s eTransform
Quantization
Entropyencoding
Signal
Almost all compression algorithm use transform coding
mp3: DCT
JPEG: DCT
JPEG2000: Wavelet
MPEG: DCT & time-difference
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Sparse TransformSignal
S p a r s eTransform
Quantization
Entropyencoding
Signal
DCT
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Sparse TransformSignal
S p a r s eTransform
Quantization
Entropyencoding
Signal
DCT
sorted coefficients
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Sparse TransformSignal
S p a r s eTransform
Quantization
Entropyencoding
Signal
Difference
What sparsifying transform would you use here?
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M. Lustig, EECS UC Berkeley
S p a r s i t y &
Compressibility
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not sparsesparse
M. Lustig, EECS UC Berkeley
Sparsity and Noise
*image courtesy of J. Trzasko
*
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M. Lustig, EECS UC Berkeley
Sparsity and Noise
*image courtesy of J. Trzasko
sparse not sparse
*
denoise/separate by threshold
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M. Lustig, EECS UC Berkeley
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M. Lustig, EECS UC Berkeley
Transform Sparsity
not sparse Sparse Edges
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M. Lustig, EECS UC Berkeley
Transform Sparsity and Denoising
sparsenot sparsewavelet transform
low-frequency
high frequency
denoised
DL Donoho, I Johnstone Biometrika 1994;81(3):425-55
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M. Lustig, EECS UC Berkeley
Transform Sparsity and Denoising
sparsenot sparsewavelet transform
denoised
low-frequency
high frequency
DL Donoho, I Johnstone Biometrika 1994;81(3):425-55
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M. Lustig, EECS UC Berkeley
Transform Sparsity and Denoising
wavelet denoising
DL Donoho, I Johnstone Biometrika 1994;81(3):425-55
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M. Lustig, EECS UC Berkeley
More Sparse Transforms
time
posi
tion
posi
tion
temporal frequency
Sparse
not Sparse *Video courtesy of Juan Santos, Heart Vista
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M. Lustig, EECS UC Berkeley
Sparsity and Compression
• Only need to store non-zeros
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M. Lustig, EECS UC Berkeley
From Samples to Measurements
• Shanon-Nyquist sampling–Worst case for ANY bandlimited data
• Compressive sampling (CS)! “Sparse signals statistics can be recovered from a
small number of non-adaptive linear measurements”
–Integrated sensing, compression and processing.–Based on concepts of incoherency between signal and
measurements
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M. Lustig, EECS UC Berkeley
• x∈ℜN is a signal • Make N linear measurements
N N
Traditional Sensing
xy
=
Φ
sensing matrix
NxN
Desktop scanner/ digital camera sensing
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M. Lustig, EECS UC Berkeley
• x∈ℜN is a signal • Make N linear measurements
N N
Traditional Sensing
xy
=
Φ
sensing matrix
NxN
MRI Fourier Imaging
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M. Lustig, EECS UC Berkeley
• x∈ℜN is a signal • Make N linear measurements
N N
Traditional Sensing
xy
=
Φ
sensing matrix
NxN
A “good” sensing matrix is orthogonal
Φ* Φ = I
Arbitrary sensing
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M. Lustig, EECS UC Berkeley
sensing matrix
Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006)
• x∈ℜN is a K-sparse signal (K<<N)• Make M (K<M<<N) incoherent linear projections
x
=
Φ
MxNK
sensing matrix
A “good” compressed sensing matrix is incoherent i.e, approximately orthogonal
Φ* Φ ≈ I
Incoherency can preserve information
M
y
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M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
=
Φ xy
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M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
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M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
minimize ||x||2 s.t. y = Φx
WRONG!
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M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
minimize ||x||0 s.t. y = Φx
HARD!
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M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
minimize ||x||1 s.t. y = Φx
need M ≈ K log(N) <<NSolved by linear-programming
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M. Lustig, EECS UC Berkeley
minimum ||x||1 minimum ||x||2
Geometric Interpretation
domain of sparse signals