RSL/MRSL Journal Club 8.27.2010 VIPR/Radial MRI Kitty Moran. Outline VIPR 2. Aliasing Artifact in Radial MRI. Outline 1. 2. Simulations of Undersampling Artifact. Radial/Projection MRI. History of Radial MRI.
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RSL/MRSL Journal Club
8.27.2010
VIPR/Radial MRI
Kitty Moran
Outline
1.
2.
Radial/Projection MRI
History of Radial MRI
Projection Acquisition was the first MRI k-space trajectory (Lauterbur, 1973)
Bernstein,et al. Handbook of MRI Pulse Sequences2004
History of Radial MRI
Bernstein,et al. Handbook of MRI Pulse Sequences2004
Undersampling Cartesian K-space
Sampling in Polar Coordinates
Sampling in Polar Coordinates
Scheffler, Hennig, Reduced Circular Field-of-View Imaging, 2005
Discrete sampling in kr, continuous sampling in kθ
Discrete concentric rings in k-space
We assume S(kr,kθ) is circularly symmetric then S(kr,kθ) S(kr) then
Discrete sampling in kr, continuous sampling in kθ
Sum over zero order Bessel functions
Lauzon, Rutt, MRM 36:940-949 (1996)
Discrete sampling in kθ, continuous sampling in kr
Discrete sampling in kθ, continuous sampling in kr
Discrete sampling in kθ, continuous sampling in kr
Where anθ = 2πrcos(nθΔkθ – θ)
¼ projections
½ projections
Radial Imaging
* Slide courtesy of Frank Korosec
kx
ky
VIPR
Barger, et al.
VIPR
VIPR
VIPR
VIPR
1,500 projections with 16 signal averages
6,000 projections with 4 signal averages
1,5000 projections with 16 signal averages
12,000 projections with 2 signal averages
24,000 projections with no signal averages
VIPR
VIPR – Time Resolved
VIPR – Time Resolved
VIPR – Isotropic Resolution
VIPR
HYPR
HighlY constrained back Projection
Data are acquired in time frames comprised of interleaved and equally spaced k-space projections
All of the acquired data is combined and used to form a composite image (either through filtered back projection or gridding and inverse fourier transform)
Each projection in a time frame is inverse fourier transformed and back projected in image space
Each back projected time point is weighted by the respective projection of the composite image
The contributions from all projections in a time frame are summed
HYPR
HYPR
HYPR
HYPR
Explain what artifacts look like in an image
Differences between polar and Cartesian undersampling
Polar sampling generally produces streaking and radial aliasing arifacts
These artifacts appear either inside or outside the circular field of view depending on the size of the object
Even for optimal polar sampling, streaks and rings can be observed outside the displayed window
In Cartesian sampling, the Dirichlet-shaped PSF of Cartesian sampling just generates an infinite amount of identical copies of the object separated by the dimensions of the rectangular field of view
In polar sampling the discretization occurs in both the radial and azimuthal directions
We can focus on the radial effects of polar sampling because we effectively have a circularly symmetric ring sampling distribution
The Jacobian is the given area that a discrete sample subtends
In Cartesian sampling, each sample subtends the same constant area (ΔxΔy), however, in equidistant polar sampling, the sample size changes as a function of radius
To normalize the area for each sampling in polar sampling, each sample is weighted by its areal extent which is equivalent to correcting for the sampling density
The Fourier-Bessel (Hankel) transform is for circularly symmetric objects
Main advantages of Projection Acquisition:
Shorter echo times
Reduced sensitivity to motion
Improved temporal resolution for certain applications
Bernstein, pg 898
Trajectory:
3D spherical coordinate system
Readout direction defined by angle θ from the kz-axis and angle Φ from the ky-axis
Resolution in all three dimensions determined by the maximum k-space radius value (kmax)
Diameter of the full field of view is determined
by the radial sample spacing (Δkr)
PA motion artifacts consist of streaks propagating perpendicular to the direction of object motion and are displaced from the image of the moving object itself
Aliasing in Projection Acquisitions:
Since the first lobe ring occurs at 1/Δkr and the object smears circularly, the inner edge of the smeared replicate occurs at (1/Δkr – rmax) where rmax is the maximal radial extent of the object
Thus if rmax is greater than 0.5/Δkr the the replication will overlap the original object
The effective field of view (unaliased region) in polar sampling is a circle of radius 0.5/Δkr
Object size 0.375/Δkr, 3*1/Δkr shown
Object size 0.554/Δkr
In both the polar and Cartesian cases, we have replicates at multiples of 1/Δkr
For Cartesian sampling, the replicates are exact duplicates of the main lobe
For Polar sampling the replications differ from the main lobe in shape and amplitude
Oscillations occur due to the finite sampling extent and are thus often referred to as truncation artifact
In Cartesian sampling, exactly M oscillation periods occur between successive replicates
In polar sampling approximately M oscillation periods occur between successive replicates
In the Cartesian case the complex exponentials are purely periodic while the Bessel function’s non-purely periodic nature leads to destructive interference on summation
In polar sampling, then the radial sampling requirement is the same as in Cartesian sampling, sampling must be above the Nyquist rate. In other words if the radial spacing in k-space is Δkr, the object must be limited to within a circle of diameter 1/Δkr in the image domain to avoid polar aliasing
However, if we assume S(kr,kθ) is circularly symmetric then S(kr,kθ) S(kr) so
Now assume that we sample at the center of k-space (kr = 0) as well as at M equally spaced concentric rings, for radial spacing of Δkr, the ring distribution is given by
The the Fourier-Bessel transform of the sampled pattern is
Sum over zero order Bessel functions weighted by Δkr2
The Cartesian and principal polar PSFs are expressed as summations of complex exponentials and weighted zero-order Bessel functions of the first kind, respectively:
In the Cartesian case the complex exponentials are purely periodic while the Bessel function’s non-purely periodic nature leads to destructive interference on summation
So the FT of S(kr) is
For M = 9, s(r) looks like
VIPR