Outline Problem: creating good MR images MR Angiography – Simple methods outperform radiologists...
-
Upload
reynold-bates -
Category
Documents
-
view
219 -
download
0
Transcript of Outline Problem: creating good MR images MR Angiography – Simple methods outperform radiologists...
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
50 100 150 200 250
20
40
60
80
100
120
140
160
180
200
220
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
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
Contrast agent arrival
Mask images (Before contrast)
Arterial phase images (After contrast)
16 17 18 19 20
MRA + Motion = TroubleMRA + Motion = Trouble
-
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
m
asks
1
m
asks
2
m
asks
3
m
asks
4
m
asks
5
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