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Published byMaude Day Modified over 9 years ago
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EE369C Final Project: Accelerated Flip Angle Sequences Jan 9, 2012 Jason Su
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Overview Originally wanted to further explore view sharing but instead pursued more formal optimization approach particularly because I was interested in applying SPIRiT Tried to replicate the results of Velikina et. al. in “Accelerating Multi-Component Relaxometry in Steady State with an Application of Constrained Reconstruction in Parametric Dimension” ISMRM 2011:2740.
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Goal To accelerate variable flip angle relaxometry sequences like DESPOT1/2 or mcDESPOT by undersampling along the flip angle dimension – 3T, mcDESPOT, 1 mm^3 256x256x160 acquisition, 10 SPGRs and SSFPs with parallel imaging ~40min. Exploit prior knowledge about the signal equation to regularize the reconstruction problem
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The Problem Velikina poses the problem as: – E is the encoding matrix including Fourier terms and coil sensititivies – m is the desired signal for all flip angles (FA, α) – y is the measured k-space data – Hybrid Huber-like norm to “promote sparsity and optimize SNR” – 1 st term enforces data consistency, 2 nd term smoothness in the signal curve
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Regularization To make the reconstruction problem more stable and allow greater undersampling, we use our prior knowledge that the signal curve is smooth – It is near zero for high angles in the 2 nd derivative “space”
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Velikina Results
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Data 3T2, SPGR 1:1:13 degrees – This is very different from Velikina, where up to 25 deg. was used, but a subset of 10 angles was taken – Note that the SPGR 2 nd deriv. is only 0 for 15+ deg. Nova 32ch head coil 110x110x40 matrix TR = 4.5ms
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Alternate SPIRiT Problem This requires knowledge of the coil sensitivities, instead I posed it as a regularized-SPIRiT problem: – x is the desired k-spaced data for all flips – G is the SPIRiT kernel – F -1 the inverse Fourier transform – Represents data consistency, self-consistency, and smoothness
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Cartesian-based Acceleration Methods Parallel imaging – SENSE – poses the reconstruction problem in the image domain With coherent aliases, the problem can essentially become one of bookkeeping: keeping tracking of which pixels were folded onto a point then solving for the original pixels knowing the coil sensitivities Optimal if coil sensitivities known Limited to uniform undersampling – GRAPPA – frequency domain, over each coil Uses a calibration region to learn how to interpolate samples with various configurations of surrounding collected data points Limited to uniform undersampling
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Cartesian-based Acceleration Methods Parallel imaging – SPIRiT – optimization problem in the frequency domain over each coil Adopts the idea of a calibration region from GRAPPA but only a single kernel interpolating from all surrounding points and coils Key insight: applying the SPIRiT kernel, i.e. interpolating, on the reconstructed data should give back the same image: the result must be self-consistent Enforce data and self-consistency for each coil image Handles any sampling pattern (including non-cartesian) Compressed Sensing – multiple domain solution – Exploits sparsity in natural images, typically in the Wavelet domain – Enforces data consistency and sparsity – Must have incoherent random undersampling to distinguish large Wavelet coefficients from background sampling artifacts
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Solution The solution to the alternate problem can be formulated as a Projection Over Convex Sets algorithm Enforce each part of the problem in turn and iterate until convergence Slow but simple to implement
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Result
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Aggressive 5x random undersampling Velikina used overall R=3.95 (R=3 for first and last 2 angles, R=5 else) Slight signal gain in the center of the brain but no significant improvement Computation time was about 1hr for one slice with 8 cores Considered a compressed sensing variant as another approach Result
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Compressed Sensing Problem F is Fourier transform Ψ is Wavelet transform λ 1 based on knowledge that image is about 85% sparse λ 2 set so that 2 nd deriv. is about 25% sparse
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Result
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Aggressive 5x random undersampling Effectively no improvement at all with regularization Solution converges within 5 minutes
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Another Compressed Sensing Problem Should we instead be forming a hybrid space and jointly enforcing sparsity in the Wavelet and 2 nd derivative domains? The sparsifying transform is now the Wavelet transform of the 2 nd derivative images This fails to converge!
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Wavelet Coefficients
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ΨΔ 2 Coefficients
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Conclusions The Wavelet transform of the 2 nd derivative images is not as sparse as the Wavelet transform alone – It is a poor sparsifying transform, explains why solution did not converge Unable to reproduce the findings of Velikina, not sure 2 nd deriv. is the correct thing to minimize – Only small for large angles well past the Ernst angle, which don’t need to be collected anyway but not sure what subset of angles they ultimately used
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Ideas Collect more angles? Pfile numbering problem Linearize the signal curve first by dividing by the flip angle since sinα ≈ α in this range – If perfectly linear, the 2 nd deriv. would be 0 everywhere, there would only be content in the initial “position” and “velocity” frames – Led to strange behavior with negative values in the reconstruction View sharing + SPIRiT?
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