Compressed sensing

COMPRESSED SENSING3D Digitization Course

Carlos Becker, Guillaume Lemaître & Peter Rennert

Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 2

OUTLINE

• Introduction and motivation

• Compressed sensing and reconstruction workflow

• Applications: MRI and single-pixel camera

8/11/2010


WHAT IS COMPRESSED SENSING?SIGNAL SPARSITY

8/11/2010


WHAT IS COMPRESSED SENSING?WHY DO WE CARE ABOUT SPARSITY?

Original 1 Megapixel image

Non-sparse values

8/11/2010



But, in the wavelet domain we get these coefficients:

8/11/2010



And the histogram of those coefficients is:

8/11/2010

The image is a nearly sparse in the wavelet domain…


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

8/11/2010


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

8/11/2010

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.


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)

8/11/2010


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)

8/11/2010


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)

8/11/2010


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

8/11/2010


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

8/11/2010


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]

8/11/2010

Documents

Compressed sensing