Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Markus StrohmeierSparse MRI: The Application of Compressed Sensing for Rapid MRIMichael Lustig, David Donoho, John M. PaulyMarch 29th, 2011 M. Strohmeier 2Outline Overview of MRI imaging Motivation for Compressed Sensing Signal constraints for CS, Sparsity, PSF Sampling Schemes and Data Processing Results of Sparse MRI OutlookMarch 29th, 2011 M. Strohmeier 3Overview of MRI imaging (1)The sample is exposed to a static magnetic field B0 which polarizes the protons along a certain direction.In the B0-field, the protons show a resonance behavior when excited by a microwave which can be seen by a receiver coil.By applying a spatial gradient to the static B-field, one changes the resonance frequency as a function of the spatial coordinate.Limiting factors are: Slew rate and amplitude of gradientB  x= B0GxxMarch 29th, 2011 M. Strohmeier 4Overview of MRI imaging (2)Magnetic Resonance Imaging samples the frequency space of the human body -> Data set consists of Fourier CoefficientsMarch 29th, 2011 M. Strohmeier 5Overview of MRI imaging (3)March 29th, 2011 M. Strohmeier 6Motivation for Compressed SensingMost images can be compressed with some transform algorithm (JPEG or JPEG2000), as the most important information is carried by only a fraction of the Fourier coefficients.Neglecting the high frequency coefficients (they carry only little energy) doesn't degrade the image noticeable enough for the human eye.QUESTION:If we throw away "most" of the image information anyway, why do we have to acquire it at all in the first place?March 29th, 2011 M. Strohmeier 7Motivation for Compressed SensingThis approach does not work for images captured in the spatial domain: Which and how much pixels should be omitted?However, since MRI captures frequency information, CS has the potential to reduce the necessary amount of acquired data to reconstruct the image.→ Reduced acquisition time makes a scan shorter and less stressful for the patient.→ MRI scanners would be able operate more economically since more patients can be examined in the same timeMarch 29th, 2011 M. Strohmeier 8Signal Constraints for CS Signal has to be sparse in a domain, that is it has to be compressible by a transform algorithm. Under-sampling artifacts must be incoherent. Then they appear in the reconstructed data like noise and can be thresholded. Knowing the Point-Spread-Function is a measure of the incoherence. The image needs to be reconstructed by a non-linear algorithm in order to enforce sparsity and keep the consistency of the acquired samples with the reconstructed image (see later).March 29th, 2011 M. Strohmeier 9Signal Constraints for CSMarch 29th, 2011 M. Strohmeier 10Signal Constraints for CSMarch 29th, 2011 M. Strohmeier 11Point Spread Function & CoherenceThe peak side-lobe ratio contains incoherence information .March 29th, 2011 M. Strohmeier 12Point Spread Function & CoherenceThe peak side-lobe ratio is a measure of the incoherence.March 29th, 2011 M. Strohmeier 13Sampling Schemes Incoherence has to be preserved when sampling the k-space. → No equispaced under-sampling, but random under-sampling!! "Randomness is too important to be left to Chance!" → The (random) sampling is controlled in the sense that different regions of the k-space are sampled with different densities. Monte-Carlo Incoherent Sampling Design is an approach to try to optimize the random under-sampling. → Iterative procedure in order to avoid "bad" point spread functions which would destroy incoherence.March 29th, 2011 M. Strohmeier 14Sampling Schemes For simplicity reasons, mostly Cartesian coordinates to sample the k-space were used up to now. However, w.r.t. variable density sampling, spiral or radial trajectories have been successfully tested. Those schemes are just slightly less coherent compared to random 2D samplingMarch 29th, 2011 M. Strohmeier 15Reconstruction of Images Basic image reconstruction algorithm is the following minimization problem, based on minimizing the L1-norm:∥ m∥1∥Fum− y∥2minimizesuch that:= operator, transforming from pixel to sparse representationm= reconstructed imageFu= undersampled Fourier transformy= measured k-space data= parameter, that assures accuracy between reconstruction and measured dataMarch 29th, 2011 M. Strohmeier 16Reconstruction of Images Image size: 100x100 pixels. 5.75 % of the pixels are non zero, 18 objects with 3 distinct intensities and 6 different sizes: → Sparse image, similar to angiogram or brain scan. Interested in how the artifacts evolve as the data is under-sampledSimulated phantom serves as an input for the reconstruction algorithms.March 29th, 2011 M. Strohmeier 17Reconstruction of ImagesGenerally, CS gives the best results:March 29th, 2011 M. Strohmeier 18Reconstruction of ImagesMarch 29th, 2011 M. Strohmeier 19Reconstruction of ImagesMarch 29th, 2011 M. Strohmeier 20Reconstruction of ImagesMarch 29th, 2011 M. Strohmeier 21Reconstruction of ImagesMarch 29th, 2011 M. Strohmeier 22Reconstruction of ImagesMarch 29th, 2011 M. Strohmeier 23Reconstruction of ImagesMarch 29th, 2011 M. Strohmeier 24Reconstruction ResultsNyquist sampledreconstructionCSZF w/dcLow resolutionreconstructionBlood flow due to bypass is only visible with 5x CS an Nyquist samplingMarch 29th, 2011 M. Strohmeier 25Summary & Outlook It was shown that for an appropriate data set, compressed sensing has the capability to perform a "random" sub-Nyquist sampling and still recover the image to a large extent without noticeable visual artifacts. Depending on the respective demands, a extreme sub-sampling is possible without losing significant amounts of information. With increasing computing power and code optimization, it might be possible in the (near) future to implement CS into commercially available scannersMarch 29th, 2011 M. Strohmeier 26Thank you...... the