Fourier Reconstruction from Non-Uniform Spectral Data Anne Gelb - PowerPoint PPT Presentation

Fourier Reconstruction from Non-Uniform Spectral Data Anne Gelb School of Mathematical and Statistical Sciences Arizona State University anne.gelb @ asu.edu February Fourier Talks February 17 2011 Research supported in part by National Science Foundation grants DMS 0510813 and DMS 0652833 (FRG).

Introduction Non-Uniform Data Current Methods Alternate Approaches 1 Introduction Magnetic Resonance Imaging Sampling Patterns in MR Imaging Challenges in Cartesian Reconstruction Spectral Reprojection 2 The Non-Uniform Data Problem Problem Formulation The Non-harmonic Kernel Reconstruction Results using the Non-harmonic Kernel 3 Current Methods Reconstruction Methods Error Characteristics 4 Alternate Approaches Spectral reprojection Incorporating Edge Information Spectral reprojection for Fourier Frames Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction MR Imaging Process An acquired MR signal can be written as ( 2 D slice) Z Z ρ ( x, y ) e − i ( k x x + k y y ) dx dy S ( k x , k y ) = ρ ( x, y ) is a measure of the concentration of MR relevant nuclei (spins) and Z t Z t k x = γ k y = γ G x ( τ ) dτ, G y ( τ ) dτ 2 π 2 π 0 0 G x and G y are the applied gradient fields We denote the signal acquisition space k = ( k x , k y ) as “ k -space” Fourier Coefficients 10 8 1 6 0.8 4 f ( ω k x , ω k y ) | 0.6 2 0 0.4 k y | ˆ −2 0.2 −4 0 −6 100 120 100 −8 80 50 60 40 20 −10 0 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 ω ky ω kx k x (a) Fourier transform (b) k -space sampling trajectory Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Sampling Patterns Spiral scan trajectory 10 0.5 8 0.4 6 0.3 4 0.2 2 0.1 0 0 k y k y −2 −0.1 −4 −0.2 −6 −0.3 −8 −0.4 −10 −0.5 −10 −8 −6 −4 −2 0 2 4 6 8 10 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 k x k x (c) Cartesian Sampling (d) Non-Cartesian Sampling – Spiral Imaging Figure: MR Imaging Sampling Patterns 1 1 Spiral sampling pattern courtesy Dr. Jim Pipe, Barrow Neurological Institute, Phoenix, Arizona Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Advantages and Disadvantages of each Pattern Cartesian Imaging Advantages Disadvantages Well understood and often used in Susceptible to undersampling – causes equipment aliased images Simple reconstruction procedure Gradient field waveforms can mean sharp transitions in collection Computationally efficient through use of the standard FFT Non-Cartesian Imaging Advantages Disadvantages Less susceptible to aliasing artifacts – Complex reconstruction procedure (if aliased images are of “diagnostic possible!) quality” Computationally demanding – FFT Easier to generate gradient field algorithm no longer applies waveforms Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Advantages and Disadvantages of each Pattern Cartesian Imaging Advantages Disadvantages Well understood and often used in Susceptible to undersampling – causes equipment aliased images Simple reconstruction procedure Gradient field waveforms can mean sharp transitions in collection Computationally efficient through use of the standard FFT Non-Cartesian Imaging Advantages Disadvantages Less susceptible to aliasing artifacts – Complex reconstruction procedure (if aliased images are of “diagnostic possible!) quality” Computationally demanding – FFT Easier to generate gradient field algorithm no longer applies waveforms Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Challenges in Cartesian Reconstruction The Gibbs Phenomenon affects the reconstruction of piecewise-smooth functions. occurs when data is sampled uniformly or non-uniformly. Two important consequences: Gibbs ringing artifact – presence of non-physical oscillations in the vicinity of discontinuities. Reduced rate of convergence to first order even in smooth regions of the reconstruction. Why is this important? Oscillations cause post-processing problems in tasks like segmentation, edge detection etc. Filtering oscillations may cause loss of important information. The reduced order of convergence means more Fourier coefficients are needed to obtain a good reconstruction. Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction The Gibbs Phenomenon – An Example Z π N f ( k ) = 1 X f ( k ) e ikx , ˆ ˆ f ( x ) e − ikx dx S N f ( x ) = 2 π − π k = − N Fourier Reconstruction Log Error in Reconstruction 0 True 2 Fourier −1 1.5 −2 1 −3 Log 10 |e| S N f(x) 0.5 −4 0 −5 −0.5 −6 −1 −7 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x x (a) Reconstruction (b) Reconstruction error Figure: Gibbs Phenomenon, N = 32 Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction The Gibbs Phenomenon – An Example Z π N f ( k ) = 1 X f ( k ) e ikx , ˆ ˆ f ( x ) e − ikx dx S N f ( x ) = 2 π − π k = − N Fourier Reconstruction Log Error in Reconstruction 0 True N=32 2 Fourier N=64 N=256 −1 1.5 −2 1 −3 Log 10 |e| S N f(x) 0.5 −4 0 −5 −0.5 −6 −1 −7 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x x (a) Reconstruction (b) Error does not go away by increasing N Figure: Gibbs Phenomenon Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Shepp Logan MRI Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Reconstruction from Fourier Data by Spectral Reprojection (Gottlieb, Shu et. al.) Assume that Fourier coefficients { ˆ f k } N k = − N are given for a piecewise analytic function f ( x ) in [ − 1 , 1] Use edge detection to determine sub-intervals of smoothness [ a, b ] Compute the Fourier partial sum in [ a, b ] : S N f ( x ) = P N k = − N ˆ f k e ikπx Reproject S N f ( x ) onto a new basis for x ∈ [ a, b ] : ` ´ P M S N f ( x ) → f ( x ) Many other algorithms available (Banerjee & Geer, Driscoll & Fornberg, Eckhoff, Jung & Shizgal, Solomonoff, Tadmor & Tanner, more ...) Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Spectral Reprojection (Gottlieb & Shu) Goal: approximate a function that is smooth but not periodic for ξ ∈ [ − 1 , 1] (after linear transformation from x ∈ [ a, b ] ) We could use an orthogonal polynomial expansion, M a l = 1 X P M f ( x ( ξ )) = a l ψ l ( ξ ) , γ l < f, ψ l > ω l =0 but we have the Fourier expansion N X ˆ f k e ikπx ( ξ ) S N f ( x ( ξ )) = k = − N Can we use S N f to approximate a l ? What should P M look like? Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data

Introduction Non-Uniform Data Current Methods Alternate Approaches Magnetic Resonance Imaging Sampling Patterns Challenges in Cartesian Reconstruction Spectral Reprojection (Gottlieb & Shu) Determine appropriate orthogonal polynomial basis { ψ l ( ξ ) } M l =0 on [ − 1 , 1] Build Fourier approximation inside smooth region [ a, b ] : N X f k e ikπx ( ξ ) , ˆ S N f ( x ( ξ )) = x ( ξ ) = ǫξ + δ k = − N Expansion coefficients for ψ l ( ξ ) are approximated by 1 b l = γ l < S n f, ψ l > ω ≈ a l Reprojection in smooth region [ a, b ] : m X P m ( S N f ( x ( ξ ))) = b l ψ l ( ξ ) l =0 Anne Gelb Fourier Reconstruction from Non-Uniform Spectral Data