compressed sensing
DESCRIPTION
Compressed sensing. 3D Digitization Course. Carlos Becker, Guillaume Lemaître & Peter Rennert. Outline. Introduction and motivation Compressed sensing and reconstruction workflow Applications: MRI and single-pixel camera. What is compressed sensing? Signal sparsity. - PowerPoint PPT PresentationTRANSCRIPT
COMPRESSED SENSING3D Digitization Course
Carlos Becker, Guillaume Lemaître & Peter Rennert
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OUTLINE
• Introduction and motivation
• Compressed sensing and reconstruction workflow
• Applications: MRI and single-pixel camera
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WHAT IS COMPRESSED SENSING?SIGNAL SPARSITY
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WHAT IS COMPRESSED SENSING?WHY DO WE CARE ABOUT SPARSITY?
Original 1 Megapixel image
Non-sparse values
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WHAT IS COMPRESSED SENSING?WHY DO WE CARE ABOUT SPARSITY?
But, in the wavelet domain we get these coefficients:
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WHAT IS COMPRESSED SENSING?WHY DO WE CARE ABOUT SPARSITY?
And the histogram of those coefficients is:
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The image is a nearly sparse in the wavelet domain…
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What happens if we only keep the 50000 highest coefficients in the wavelet domain, set the rest to zero and reconstruct the image ?
Original imageReconstructed image
(only 50k highest wavelet coefficients)
95% of the original image data was discarded
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WHAT IS COMPRESSED SENSING?• Classic approach to compression:
– Measure everything (ie: all pixels)– Apply some compression algorithm (ie: JPEG2000)
– But, why would we sample 1 million pixels if we are going to throw away 90% of image data when compressing the image in JPEG?
• Compressed sensing approach: if signal is sparse in some domain– Sample M << N random measurements– Reconstruct original signal by L1 minimization
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Full-resolution image(N pixels/measurements) Lossy compression
Random sampling(M << N measurements)
Image reconstruction
Candès et al. showed that it is possible to subsample a signal if it is sparse in some domain, being able to obtain a perfect
reconstruction if certain conditions are met.
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COMPRESSED SENSINGMOTIVATION FROM MRI
• 2004, Candes came to results that people of his time could not believe
• For a simple phantom (a) its possible to sample at only 22 radial lines (b) (equal to a sampling rate of π / 22, about 50 times below the Nyquist rate of 2 π) to retrieve a perfect reconstruction (d)
• What does the trick? Simply setting the unknown Fourier coefficients to 0 leads to a very bad result (c)
Candès, E.J.; Romberg, J.; Tao, T.: “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information” (2004)
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COMPRESSED SENSINGRECONSTRUCTION WORKFLOW
• Sparse signal gets randomly sampled in another non-sparse domain (k-space)
• Reconstruction leads to noisy non-sparse signal with significant peaks where original signal was high
• After thresholding of significant peaks the strongest components of the original signal are detected
• Using the noisy reconstruction of the newly sampled strongest components in k-space, the impact of this strongest components on the first reconstruction are determined and subtracted, leaving peaks of weaker components
• With this iterative strategy weaker and weaker components can be retrieved
Michael Lustig, David Donoho, John M. Pauly: “Sparse MRI: The application of compressed sensing for rapid MR imaging” (2007)DL Donoho, I Drori, Y Tsaig : Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit” (2006)
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COMPRESSED SENSINGRAPID MRI – NON-SPARSE SIGNAL
• Non- sparse signal sampled in sparse domain
• That means: reconstruction of samples will produce no significant peaks (since there are no outstanding peaks in the signal domain)
• Solution: use other sparse domain of signal for “reconstruction” and filtering of significant peaks
Michael Lustig, David Donoho, John M. Pauly: “Sparse MRI: The application of compressed sensing for rapid MR imaging”
(Signal here means the underlying image that is sensed in the Fourier space)
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SINGLE PIXEL CAMERAGENERAL PRINCIPLE
M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, "Single-Pixel Imaging via Compressive Sampling," IEEE Signal Processing Magazine, Vol. 25, No. 2, pp. 83-91, March 2008
ΣObject
DMD
Photodiode
MemorySeveral
Measurements
Reconstruction
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SINGLE PIXEL CAMERARESULTS
M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, "Single-Pixel Imaging via Compressive Sampling," IEEE Signal Processing Magazine, Vol. 25, No. 2, pp. 83-91, March 2008
Original:16384 pixels
10 % measurements 20 % measurements
2 % measurements 5 % measurements
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COMPRESSED SENSINGCONCLUSION
• Compressed sensing lets us sub-sample a signal w.r.t. Nyquist rate and reconstruct it perfectly, if this signal is known to be sparse in some domain and some conditions are met
• Compressed sensing is promising for a wide range of future technologies, specially for high-frequency signals– Speeds up acquisition process: specially interesting for MRI– Cheaper hardware (ie: IR cameras with only a few sensors)
• Sparsity can also be exploited for classification and image processing tasks[Huang, K. and Aviyente, S., Sparse representation for signal classification]
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