7/27/2019 L1 Magic & MR Imaging

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  Magic and Magnetic Resonance Imaging

Pradosh K Roy 

Asia Pacific Institute of Information Technology

Panipat , 132103 Indiaroypkin@yahoo.com

In the year 2009, Shreyas Vasanawala a paediatric radiologist atthe Lucile Packard Children’s Hospital at Stanford University  didn’t

know for sure what was wrong with a 2- year old boy and hoped theMRI [1] would reveal the answer. The child had earler received a

 portion of a donor’s liver to replace his own failing organ. For awhile, he did well. ‘But his latest lab tests were alarming. Somethingwas going wrong —  there was a chance that one or both of the liver’s bile ducts were blocked’. Vasnawala needed a high resolution scan ,

 but that ‘would take a full two minutes for a standard MRI to capturethe image’, which would had been medically catastrophic for thechild. Together with his colleague Michael Lusitig , Vasnawala useda new computational framewok which goes against the common

wisdom of data acuisition and processing according to the ShannonTheorem . The technique , known as Compressed Sensing (CS) or compressive sampling dates back to 2004 and is due to Emmanuel

Candés of Stanford University.

Vasanawala and Lustig took an incomplete scan for 40 seconds only,during which the child had suffered no appreciable loss in bloodoxygenation level and ‘later that day, the Compressed Sensing

algorithm was able to produce a sharp image from the brief scan,good enough for Vasanawala to see the blockages in both bile ducts.

An interventional radiologist snaked a wire into each duct, gentlyclearing the blockages and installing tiny tubes that allowed the bileto drain properly. And with that — a bit of math and a bit of medicine

 –  the child’s lab test results headed  back to normal’ [3].

The success of Vasanawal and Lustig is not an isolated one.Since 2007 research papers had started appearing in journals e.g.

IEEE Transactiona on Medical Imaging , Magnetic Resonance inMedicine , Journal of Cardiovascular Magnetic Resonance ,IEEE

Signal Processing Magazine [4] reporting similar achievements.

As another example , a fully sampled MR angiograph has beencompared with a six times undersampled and sparse reconstructed

image using - minimization in the adjoining figure.

Compressed Sensing , the emerging sampling paradigm , assertsthat if the signals has a sparse representation in some orthonormal basis, then one can recover the signals from far fewer non-adaptivelinear measurements . The accuracy of the recovered signal is as good

as that attainable with Transform Coding i.e. direct knowledge of 

sparsity , the most important coefficients and its locations. Moreover,a good approximation to those important coefficients is extracted

from the measurements by solving a ℓ1- minimization problem . The

nonadaptive measurements have the character of “random” linear 

combinations of the basis/frame coefficients. Gaussian and BernoulliRandom Matrices are preferred because they are maximally

incoherent with known bases e.g. Wavelets , Cosine , Fourier.

Fig. Reproduced from ‘Emmanuel Candès , Compresive Sensing : A 25Minute Tour , First EU-US Frontiers of Engineering Symposium, Cambridge,September 2010. 

The fact that a signal can be recovered using only a fewincoherent sparse measurements proportional to its information levelhas implications that are far reaching and leads to a number of 

 possible applications in Data Compression , Channel Coding and

Data Acquisition. The last of these applications suggest that CS couldhave an enormous impact in areas where conventional hardwaredesign has significant limitations. This could lead to a very efficient

method of data acquisition and storage in future.

The potential pay-offs of CS are huge, comments Yonina Eldar ,

a researcher at Israel Institute of Technology , Haifa , Israel , as‘removing the Nyquist barrier in the resolution limited applications ..

could make a prominent impact on the analog-digital worldsurrounding us’.

REFERENCES

[1]  http://en.wikipedia.org/wki/Magnetic_resonance_imaging

[2]  Jordan Ellenberg , Fill in the Blanks : Using Math to Turn Lo-Res Datasets Into Hi-ResSamples, http://www.wired.com/magazine/2010/02/ff_algorithm 

[3] S. Vasanawala, M. Alley, R. Barth, B. Hargreaves, J. Pauly, and M. Lustig. Faster pediatric

MRI via compressed sensing. In Proc. Annual Meeting Soc. Pediatric Radiology (SPR),

Carlsbad, CA, Apr. 2009.[4] Michael Lustig ,David L. Donoho , Juan M. Santos , John M. Pauly , Compressed Sensing MRI ,

IEEE Signal Processing Magazine , March 2008 , pp 72- 82.

[5] Emmanuel Candes , Compressive Sampling : A 25 Minute Tour , FirstEU-US Froniers of Enginering Symposium , Cambridge, Spetember, 2010.

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