Outline

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Outline

• Problem creating good MR images
• MR Angiography
• Simple methods outperform radiologists
• Parallel imaging
• Maximum likelihood approach
• MAP via graph cuts?
• An application of scheduling

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MR is incredibly flexible

• CT and X-ray can only measure tissue opacity
• MR can image a variety of tissue properties

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Image construction problem

• MR requires substantial cleverness in image
  formation
• Unique among image modalities
• Under-appreciated part of what Radiologists do
• Huge field involving software, algorithms and
  hardware
• Easy to validate algorithms!

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Challenge time versus accuracy

• The imaging process is slow
• Few body parts can hold still for very long
• MR images are vulnerable to motion artifacts
• Consequence of a very strange camera

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MR Imaging Process

• Imagine a camera that takes pictures row by row
• A few seconds to create the image

Cartesian sampling
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k-space representation
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MRI Motion artifacts
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Automatic Creation of Subtraction Images for MR
Angiography
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Magnetic Resonance Angiography

• Angiography imaging blood vessels
• Video of MRIs as dye is injected

Input
Desired output
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Subtraction

• Select a before (pre-contrast) image and an
  after (post-contrast) image
• Easy problem if there is no motion
• Currently done by hand
• Radiologist finds a pair where the difference
  image allows them to see what they are looking for

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Mask images (Before contrast)
Contrast agent arrival
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Arterial phase images (After contrast)
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MRA Motion Trouble
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Simple but effective algorithm

• Divide the images into before and after
• Image processing to detect contrast arrival
• Find the pair whose difference is most
  artery-like
• Evaluation function looks for long, thin
  structures
• Arteries are predominantly vertical
• More complex methods didnt work

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masks 1 masks 2 masks 3 masks
4 masks 5
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Deep Blue analogy

• Evaluation function isnt very smart
• Doesnt know any anatomy
• But if it thinks an image is great, its usually
  right
• We consider a lot of different pairs
• Skip ones that are unlikely to give good images

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Projection onto Convex Sets (POCS)

• POCS algorithm is widely used, but not for MRA
• Method to impose constraints on a candidate
  solution
• Repeatedly project a candidate onto convex sets
• Good performance when sets are orthogonal
• Most data is good use it to fix bad data
• Nudge each input towards a reference image
• Define desirable properties as convex projections

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POCS Projections

• Reference frame
• Projection P1 small change in k-space magnitude
• Projection P2 similar to P1, for phase
• Projection P3 flesh should stay constant
• Projection P4 background should be black

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POCS Algorithm
K- space
Image space
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Evaluation criterion
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Another example
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How much better is the expert?
Statistically significant at p0.016
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Need a better approach

• Simple methods are surprisingly effective
• They consider the input to be images
• Which is wrong, even for Cartesian sampling
• Input comes one line (row) at a time
• Motion occurs at a set of lines

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Motion by lines
Motion1
Image 1
Image 2
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Spiral imaging

• Asymmetry of cartesian sampling is still a
  problem
• Motion in the middle of k-space destroys the
  image
• Solution spiral sampling of k-space

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Parallel Imaging
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Basics of Parallel Imaging

• Used to accelerate MR data acquisition
• k-space is under-sampled, aliased
• De-aliased using multiple receiver coils
• In MR, speed saves lives (literally)
• This is the hot topic in MR over the last 5 years

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Reconstructed image
Imaging target

• Each coil sees a different image
• Different multiplicative factors
• spatial sensitivity
• Can use this to overcome aliasing
• introduced by undersampling

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Parallel Imaging Reconstruction
Under-sampled k-space
Under-sampled k-space
ky
kx
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Parallel Imaging Model (Noiseless)
y1
y2
y3
y4
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Parallel Imaging Models

• y H x (1) noiseless
• y H x n (2) instrumentation
  noise only
• y (H ?H) x n (3) system and
  instrumentation noise
• For noise model (2) with iid Gaussian noise,
  least squares computes the maximum likelihood
  estimate of x
• Famous MR algorithm called SENSE
• What about noise model (3)? TL-SENSE

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TL-SENSE

• With noise model (3) and iid instrumentation
  Gaussian noise, TLS finds the maximum likelihood
  estimate
• Well-known method of Golub Van Loan
• Unfortunately, system noise is not iid!
• Need to derive a maximum likelihood estimator
• Based on a reasonable noise model

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Structure of system matrix
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Maximum likelihood solution

• Assume n, d are iid Gaussian n, d are
  uncorrelated
• Then total noise g(x) y-Ex (n?H x) is
  Gaussian
• The ML solution maximize
• Pr(yx) ? exp-½ (y - Ex) R-1 (y - Ex)
• where RRg(x)eg(x)g(x)H is the total noise
  cov. matrix
• ML estimate depends on x (data), hence non-linear
• Note that there is no dependence between
  neighboring pixels

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ML algorithm

• We have shown that the ML problem reduces to
• arg min? y ??2
  
• 1(ss/sn)2
  ?2
• where ? is a collection of aliasing pixels of
  desired image, and ? the corresponding collection
  of pixels from sensitivity maps.
• A standard LS problem, but with non-linear
  denominator
• ? is slowly-varying as we iterate
• Converges almost as fast as quadratic
  minimization

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Example results
SENSE
TL-Sense
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Beyond TL-SENSE

• Gaussian noise for sensitivity maps (TL-SENSE) is
  much more realistic than no noise (SENSE)
• However, the real noise will have structure
• Coil positioning differences, e.g.
• Can we estimate sensitivity maps from patient
  data?
• Can we use priors instead of ML?
• Medical imaging has stronger priors than vision

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Priors via Graph Cuts

• Consider equations of the form
• Image denoising if H is identity matrix
• No D for non-diagonal H

Noise
Unknown image
Observed image