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

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

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!

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”

MR Imaging Process Imagine a camera that takes pictures row by row – A few seconds to create the image Cartesian sampling

k-space representation Average intensity

MRI Motion artifacts Good patient Bad patient

Automatic Creation of Subtraction Images for MR Angiography

Magnetic Resonance Angiography Angiography = imaging blood vessels “Video” of MRI’s as dye is injected InputDesired output

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

Contrast agent arrival Mask images (Before contrast) Arterial phase images (After contrast)

MRA + Motion = Trouble - Subtraction in MRA magnifies effects of motion =

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 didn’t work

arterial 1arterial 2arterial 3arterial 4arterial 5arterial 6arterial 7arterial 8 masks 1 masks 2 masks 3 masks 4 masks 5

Deep Blue analogy Evaluation function isn’t very smart – Doesn’t know any anatomy – But if it thinks an image is great, it’s usually right We consider a lot of different pairs – Skip ones that are unlikely to give good images

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

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

FFT P 1 : amp-restrict bad image ref image IFFT P 3 : parenchyma P 4 : bkgnd-correct P 2 : phase-correct K- space Image space POCS Algorithm

Evaluation criterion Expert RadiologistComputer

Another example Expert RadiologistComputer

How much better is the expert? Computer much better Computer better Same Computer worse Computer much worse Statistically significant at p= % 47% 13% 34% 0%

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

Motion by lines Image 1Image 2 Motion1 Motion2

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

Parallel Imaging

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 Coils Region imaged

Combiner Reconstructed image  Each coil sees a different image  Different multiplicative factors  “spatial sensitivity”  Can use this to overcome aliasing introduced by undersampling Imaging target

kyky kxkx Reconstructed k-space Under-sampled k-space kyky kxkx Parallel Imaging Reconstruction

Parallel Imaging Model (Noiseless) y1y1 y2y2 y3y3 y4y4 y1y2y3y4y1y2y3y4 =Hx Image to be reconstructed Coil outputs (observed) System matrix, obtained from coil sensitivities x

Parallel Imaging Models y = H x (1) [noiseless] y = H x + n (2) [instrumentation noise only] y = (H + ΔH) 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

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

Structure of system matrix 11 11 LL

Maximum likelihood solution  Assume n, δ are iid Gaussian; n, δ are uncorrelated  Then total noise g(x) = y-Ex = (n+ΔH x) is Gaussian  The ML solution : maximize Pr(y|x)  exp{-½ (y - Ex) R -1 (y - Ex) }  where R=R g (x)= ε {g(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

ML algorithm We have shown that the ML problem reduces to: arg min η ║y – ψη║ 2 1+(σ s /σ n ) 2 ║η║ 2 where η is a collection of aliasing pixels of desired image, and ψ the corresponding collection of pixels from sensitivity maps. A standard LS problem, but with non-linear denominator – ║η║ is slowly-varying as we iterate Converges almost as fast as quadratic minimization

Example results SENSETL-Sense

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

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