OutlineOutline

Problem: creating good MR imagesMR Angiography

– Simple methods outperform radiologists

Parallel imaging– Maximum likelihood approach– MAP via graph cuts?

An application of scheduling

MR is incredibly flexibleMR is incredibly flexible

CT and X-ray can only measure tissue opacityMR can image a variety of tissue properties

Image construction problemImage 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 accuracyChallenge: time versus accuracy

The imaging process is slowFew body parts can hold still for very longMR images are vulnerable to motion artifacts

– Consequence of a very strange “camera”

MR Imaging ProcessMR Imaging Process

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

Cartesiansampling

k-space representationk-space representation

Averageintensity

MRI Motion artifactsMRI Motion artifacts

Good patient

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Bad patient

Automatic Creation of Subtraction Automatic Creation of Subtraction Images for MR AngiographyImages for MR Angiography

Magnetic Resonance AngiographyMagnetic Resonance AngiographyAngiography = imaging blood vessels“Video” of MRI’s as dye is injected

Input Desired output

SubtractionSubtraction

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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Contrast agent arrival

Mask images (Before contrast)

Arterial phase images (After contrast)

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MRA + Motion = TroubleMRA + Motion = Trouble

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Subtraction in MRA magnifies effects of

motion=

Simple but effective algorithmSimple 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 1 arterial 2 arterial 3 arterial 4 arterial 5 arterial 6 arterial 7 arterial 8

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asks

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asks

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asks

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asks

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asks

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Deep Blue analogyDeep 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)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 ProjectionsPOCS Projections

Reference frame:

Projection P1: small change in k-space magnitudeProjection P2: similar to P1, for phase

Projection P3: flesh should stay constantProjection P4: background should be black

FFTP1 : amp-restrict

bad image

ref image

IFFT

P3 : parenchyma

P4 : bkgnd-correct

P2 : phase-correct

K-space

Imagespace

POCS AlgorithmPOCS Algorithm

Evaluation criterionEvaluation criterion

Expert RadiologistComputer

Another exampleAnother example

Expert RadiologistComputer

How much better is the expert?How much better is the expert?

Computer much better

Computer better

Same

Computer worse

Computer much worse

Statistically significant at p=0.016Statistically significant at p=0.016

6%

47%

13%

34%

0%

Need a better approachNeed a better approach

Simple methods are surprisingly effectiveThey 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 linesMotion by lines

Image 1 Image 2

Motion1Motion2

Motion2

Spiral imagingSpiral 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 ImagingParallel Imaging

Basics of Parallel ImagingBasics 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 Reconstructedimage

Each coil sees a different image Different multiplicative factors

“spatial sensitivity” Can use this to overcome aliasing introduced by undersampling

Imaging target

k y

kx

Reconstructed k-spaceUnder-sampled k-spacek y

kx

Under-sampled k-space

Parallel Imaging ReconstructionParallel Imaging Reconstruction

Parallel Imaging Model (Noiseless)Parallel Imaging Model (Noiseless)

y1 y2

y3y4

y1

y2

y3

y4

= H x

Image to be reconstructed

Coil outputs(observed)

System matrix, obtained from coilsensitivities

x

Parallel Imaging ModelsParallel 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-SENSETL-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 matrixStructure of system matrix1

1

L

Maximum likelihood solutionMaximum 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=Rg(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 algorithmML algorithm

We have shown that the ML problem reduces to: arg minη ║y – ψη║2

1+(σs/σn)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 resultsExample resultsSENSE TL-Sense

Beyond TL-SENSEBeyond 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 CutsPriors via Graph Cuts

Consider equations of the form

Image denoising if H is identity matrix– No D for non-diagonal H

NoiseUnknownimage

Observedimage