L1 Magic & MR Imaging
2
Magic and Magnetic Resonance Imaging Pradosh K Roy Asia Pacific Institute of Information Technology Panipat , 132103 India [email protected] In the year 2009, Shreyas Vasanawala a paediatric radiologist at the Lucile Packard Children’s Hospital at Stanford University didn’t know for sure what was wrong with a 2- year old boy and hoped the MRI [1] would reveal the answer. The child had earler received a portion of a donor’s liver to replace his own failing organ. For a while, he did well. ‘But his latest lab tests were alarming. Something was going wrong — there was a chance that one or both of the liver’s bile ducts were blocked’. Vasnawa la needed a high resolution scan , but that ‘would take a full two minutes for a standar d MRI to capture the image’, which would had been medically catastrophi c for the child. Together with his colleague Michael Lusitig , Vasnawala used a new computational framewok which goes against the common wisdom of data acuisition and processing according to the Shannon Theorem . 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 blood oxygenation 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, gently clearing the blockages and installing tiny tubes that allowed the bile to drain properly. And with that — a bit of math and a bit of medicine – the child’s lab test results headed back to n ormal’ [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 in Medicine , Journal of Cardiovascular Magnetic Resonance ,IEEE Signal Processing Magazine [4] reporting similar achievements. As another example , a fully sampled MR angiograph has been compared with a six times undersampled and sparse reconstructed image using - minimization in the adjoining figure. Compressed Sensing , the emerging sampling paradigm , asserts that if the signals has a sparse representation in some orthonormal basis, then one can recover the signals from far few er non-adaptive linear 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 ℓ 1 - minimization problem . The nonadaptive measurements have t he character of “random” linear combinations of the basis/frame coefficients. Gaussian and Bernoulli Random Matrices are preferred because they are maximally incoherent with known bases e.g. Wavelets , Cosine , Fourier. Fig. Reproduced from ‘Emmanuel Candès , Compresive Sensing : A 25 Minute Tour , First EU-US Frontiers of Engineering Symposium, Cambridge, September 2010. The fact that a signal can be recovered using only a few incoherent sparse measurements proportional to its information level has implications that are far reaching and leads to a number of possible applicatio ns in Data Compress ion , Channel Coding and Data Acquisition. The last of these applications suggest that CS could have an enormous impact in areas where conventional hardware design 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 Institu te of Technolo gy , Haifa , I srael , as ‘removing the Nyquist barrier in the re solution limited applications .. could make a prominent impact on the analog-digital world surrounding us’. REFERENCES [1] http://en.wikipedia.org/wki/Magnetic_resonance_imaging [2] Jordan Ellenberg , Fill in the Blanks : Using Math t o Turn Lo-Res Datasets Into Hi-Res Samples , http://www.wired.com/magazine/2010/02/ff_algorithm [3] S. Vasanawala, M. All ey, R. B arth, B. Harg reaves, J. Paul y, and M. Lustig. Faster pediatric MRI via compressed sensing. In Proc. Annual Meeting Soc. Pediatric Radiology (SPR), Carlsbad, CA, Apr. 2009. [4] Michael Lus tig ,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 F roniers of Enginering Symposium , Cambridge, Spetember, 2010. .
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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 [email protected]
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.
.
