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Sparse MRI

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Sparse MRI Powered By Docstoc
					    Markus Strohmeier




  Sparse MRI: The Application of
Compressed Sensing for Rapid MRI
    Michael Lustig, David Donoho, John M. Pauly
                         Outline

 Overview of MRI imaging
 Motivation for Compressed Sensing


   Signal constraints for CS, Sparsity, PSF
   Sampling Schemes and Data Processing
   Results of Sparse MRI
   Outlook


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         Overview 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.

                           B x = B0 G x x

   Limiting factors are: Slew rate and amplitude of gradient

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         Overview of MRI imaging (2)




   Magnetic Resonance Imaging samples the frequency space
     of the human body -> Data set consists of Fourier
     Coefficients

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         Overview of MRI imaging (3)




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Motivation for Compressed Sensing
Most 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?



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Motivation for Compressed Sensing

This 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 time


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             Signal 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).



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            Signal Constraints for CS




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            Signal Constraints for CS




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Point Spread Function & Coherence




    The peak side-lobe ratio contains incoherence information .

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Point Spread Function & Coherence




     The peak side-lobe ratio is a measure of the incoherence.

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                   Sampling 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.


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                   Sampling 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 sampling


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            Reconstruction of Images
   Basic image reconstruction algorithm is the following
    minimization problem, based on minimizing the L1-norm:

      minimize ∥ m∥1      such that:    ∥ F u m− y∥2


           = operator, transforming from pixel to sparse representation
       m   = reconstructed image
       F u = undersampled Fourier transform
       y   = measured k-space data
           = parameter, that assures accuracy between
             reconstruction and measured data

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            Reconstruction of Images
Simulated phantom serves as an input
  for the reconstruction algorithms.

                                  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-sampled

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            Reconstruction of Images




                                   Generally, CS gives
                                    the best results:

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            Reconstruction of Images




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            Reconstruction of Images




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            Reconstruction of Images




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            Reconstruction of Images




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            Reconstruction of Images




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            Reconstruction of Images




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               Reconstruction Results
 Blood flow due to bypass is only visible with 5x CS an Nyquist sampling




    Nyquist sampled   Low resolution                        CS
                                         ZF w/dc
     reconstruction   reconstruction
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                   Summary & 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 scanners


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                   Thank you...




                      ... the end!




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