Atkinson, D

Batchelor, PG

Menys, A

Tremoulheac, B

UCL Discovery

Decomposition of dynamic MR images with low-rank and sparse matrix separationB. Trémoulhéac1, P. G. Batchelor2, A. Menys3 and D. Atkinson31Centre for Medical Image Computing, University College London, 2Imaging Sciences and Biomedical Engineering,  King's College London, 3Centre for Medical Imaging, UCL Division of MedicineBackground.  A recent mathematical optimisation problem investigates the topic of decomposing a given matrix into its  low-rank and sparse components [1, 2, 3]. While these papers mostly focus on theoretical aspects – in particular, whether  it is possible to exactly recover the unobserved low-rank and sparse components from the observed matrix – Candès et  al. [3] have shown practical use of the decomposition in video surveillance and face recognition. In medical imaging, Gao et  al. [4]  have  recently  incorporated  this  separation  model  into  the  conventional  inverse  problem  in  Computed Tomography and were able to reconstruct better images with fewer measurements than state of the art techniques.   Objectives.  We investigate the potential usefulness of this separation process in dynamic Magnetic Resonance Imaging  (MRI). More specifically,  motivated by experiments on real data, we are interested to know if this method (a)  could separate the global (or respiratory) motion from the local motion, and (b) could detect local changes of intensity.Methods.   Mathematically,  the decomposition process  can  be posed as  the following convex optimisation problem,  minL,S ||L||*+λ||S||1 subject  to  X=L+S,  where  ||L||* and ||S||1 denote respectively  the nuclear  norm of  L (the sum of the singular values) and the l1 norm of S defined as the sum of the absolute values of all entries (i.e. l1 norm of S viewed as a vector). λ is a trade-off parameter between low-rank and sparse components. X is the observed data matrix, X=[x1...xM], X∈RN×M,  with  temporal  image  i of  the  M-sequence  as  its  column vectors  xi.  Minimisation  of  the  problem can  be performed by various optimisation techniques but the Alternating Direction Method (ADM) appears to be the best choice, especially when it comes to large-scale problems. ADM, an improvement of the Augmented Lagrangian method, exploits the favorable structure appearing in both the objective function and the constraint [2, 3]. Both MATLAB implementations of the algorithm in [2, 3] have been made available online at  perception.csl.uiuc.edu/matrix-rank/. Although λ is set to max(N,M)-1/2  in [3],  justified  by  a theoretical  analysis  and  a  no  tuning parameter approach, it is in our case a parameter  we  tune  by  visual inspection. Based on the Shepp-Logan phantom,  we  generate  a  20  frames sequence  with  time  varying  intensity and  a  mixture  of  global  and  local motion  (figure 1) to understand how the separation works in this context.Results.  We show in figure 2 the resulting decomposition of the entire Shepp-Logan phantom sequence for a choosen value of the trade-off parameter equal to 2.max(N,M)-1/2. We first note that  local  motion is clearly  reflected  in every  sparse frame yielding  a  straight  forward  segmentation  of  the  moving  local ellipses. We also remark the respiratory-like motion appears only in the low-rank frames, although edges of the outer ellipse do appear in  the  sparse  frames.  Perhaps  more  interestingly  is  the  intensity varying ellipse which is distinctly represented in the sparse matrix in  the  last  few frames  of  the  whole  sequence.  This  could  yield applications in Dynamic Contrast Enhanced MRI since this method gives  a  direct  segmentation  of  the  varying  local intensity change. Figure 3 shows the sensitivity of the method to isolating peristalsing small bowel motion in a healthy  individual  undergoing  small  bowel enterography. (A cine MRI sequence of 20 frames with both large-scale respiratory motion and local motion of the  small  bowel.) This  could  be  useful  for  clinicians since  motion  detection  is  becoming  an  increasingly useful diagnostic parameter for assessing disease status non-invasively. Finally, we think it is also important to notice that processing the Shepp-Logan sequence or the real  data (both of size 512x512x20) takes only a few seconds on a simple desktop computer using algorithms [2] or [3] thanks to the use of the ADM approach.Conclusions. We have proposed to apply the low-rank and sparse separation problem to dynamic MRI sequences as an image post-processing method. Using a simple simulation and a specific value of the trade-off parameter, we have shown  this method can detect local motion and local intensity changes in the sparse frames, as well as isolating the respiratory motion in the low-rank frames. Through application to real data, we hope to have convinced the reader that this method could lead to a better understanding and/or interpretation of the images from a clinical perspective.References.  [1] V. Chandrasekaran  et  al.  Rank-sparsity  incoherence  for  matrix  decomposition.  SIAM J.  Optim.  (to  appear), 2009.  [2] X. Yuan and J. Yang. Sparse and low-rank matrix decomposition via alternating direction methods. Pac. J. Optim. (to appear), 2009. [3] E. Candès et al. Robust principal component analysis? J. ACM 58, 2011. [4] H. Gao et al. Robust principal component analysis-based four-dimensional computed tomography. Phys. Med. Biol. 56, 2011.Fig.3:  Decomposition on small  bowel  MR data.  Respiratory  motion appears largely in the low-rank time frames whereas  peristaltic motion appears in the sparse frames.  (Left)  Data (Middle) Low-rank component. (Right) Sparse component.Fig.1: Global (respiratory) motion is simulated by slightly and slowly changing  size of the main ellipse. Local motion is simulated by three small ellipses at the  bottom. Intensity change from 0 to 1 is visible on the right side ellipse. (Left) y-t  profile  of  the  generated  sequence.  (Right) Extracted  frames  from the  whole  sequence at time {1, 4, 7, 12, 16, 20}.Fig.2: Low-rank and sparse components resulting  from  the  decomposition.  (Left)  y-t  profile  of  the  whole sequence  (Right) Frame 20.

Decomposition of dynamic MR images with low-rank and sparse matrix separation

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Decomposition of dynamic MR images with low-rank and sparse matrix separation

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