Jason P. Stockmann 1 and R. Todd Constable 1,2 Yale University, Department of Biomedical Engineering 1, Department of Diagnostic Radiology 2, New Haven, CT RECONSTRUCTION FROM ARBITRARY PROJECTIONS Parallel imaging performance is typically optimized by selecting a coil array geometry that provides maximal spatial information along the linear phase encoding direction(s) being undersampled [1]. Recent work has approached this problem in reverse, instead tailoring the encoding gradient to the shape of the available coil profiles. In addition to improving parallel imaging performance, non-linear encoding functions hold the potential to reduce gradient switching times (due to reduced dB/dt excursion) and to allow matching of gradient shapes to anatomy [2]. The introduction of non-linear encoding functions motivates the exploration of image reconstruction algorithms that can handle arbitrary combinations of coil profiles, gradient functions, and – ultimately – gradient pulse shapes. In this work, we demonstrate the suitability of the Kaczmarz iterative projection algorithm [3] – also known as Algebraic Reconstruction Technique (ART) – to the reconstruction of “O-space” images from projections made using combinations of spherical harmonics [Oral #761]. This row-action method has found application in the past to Computed Tomography and cryo-electron microscopy. O-space acquisitions play out much like a conventional radial k-space trajectory, but with the addition of the Z2 spherical harmonic to create projections of the sample along rings concentric about a chosen point in the FOV. The X and Y gradient strengths are varied between echoes to move the location of the ring center placement (CP). ROW-ACTION SOLUTION TO SIGNAL EQUATION Non-linear gradients project the object onto a non-Fourier basis set, so the familiar concept of k-space is of only limited value in O-space imaging. Instead, the signal equation is approached directly: Stacking all acquired echoes into a single column vector, the discrete version of the integral equation may be expressed in matrix form, where A describes the hybrid encoding function. For a typical acquisition, A is large ( 32,768  16,384), making direct computation of the pseudo-inverse memory-intensive and computationally demanding. Instead, we obtain the unknown object  (x,y) via the Kaczmarz iterative projection algorithm. Each row of A is treated as a basis function whose inner product with the object produces one sampled time point in the echo. The difference between this inner product and the echo point is used to weight the amount of the basis function that is added to the next update of the estimator,  n+1, where a m,q denotes the row of matrix A corresponding to the q th coil, m th CP, and t th time point. In the course of one iteration, the algorithm proceeds through each row of A in succession until all rows have been used. Convergence is achieved in a small number of iterations even for naïve choices of  0 [3]. The algorithm is however highly sensitive to inconsistency in the set of equations; in such cases a regularized version of the algorithm is recommended [4]. SIMULATIONS Undersampled datasets are simulated in M ATLAB by applying the hybrid encoding functions to a 128  128 numerical phantom to calculate the echoes using the above formula. Uncorrelated noise with standard deviation equal to 2% of the peak phantom intensity is added to the phantom prior to each echo point calculation. The following cases are considered:  Single uniform coil datasets: O-space vs. undersampled conventional radial k-space data having 32 spokes. Radial k-space reconstruction is performed using the Kaczmarz algorithm.  8-Coil datasets: O-space vs. SENSE reconstructions for 8-coil acquisitions with reduction factors R = 4 and R = 8 using a time-equivalent amount of O-space data.  O-space datasets employing 128, 256, or 512 points in the readout direction, along with correspondingly increased gradient strengths. The number of projection “rings” within the object grows with the square of the gradient strength, potentially improving resolution. Extra noise is added to reflect the increase in sampling BW. REFERENCES 1.Sodickson DK et al. MRM 1997;38:591– Hennig J et al. MAGMA 2008;21: Herman GT et al. Comput. Biol. Med. 1976;6: Censor Y. Row-action Methods for huge and sparse systems and their applications. SIAM Review, 1981;23(4): RESULTS Uniform coil recons from 32 echoes Hybrid encoding/basis function Echo at time t produced by q th coil and m th center placement Object source at (x,y) = (0,0) source at (x,y) = (85,0) KACZMARZ ITERATIVE RECONSTRUCTION FOR ARBITRARY HYBRID ENCODING FUNCTIONS Off-center Quadratic Encoding Function + + B0 offset = + Encode using first and second-order spherical harmonincs Correspondence: This poster is available at: mri.med.yale.edu/individual/stockmann Simulations were performed to find an efficient encoding scheme consisting of CPs distributed throughout the FOV. The following encoding scheme was found to perform well for a variety of noise levels and surface coil geometries [see Abstract #4556] : 128  128 Numerical phantom Conventional Radial O-space O-space reconstruction shows slightly higher noise level but reduced artifacts. R = 4 (32 echoes) 8-Coil Reconstructions R = 8 (16 echoes) O-space reconstructions are robust even for a high degree of undersampling. With R = 8, the reduction factor equals the number of coils and the SENSE reconstruction are ill-conditioned, leading to severe noise amplification. For O-space reconstructions, substantial improvements in resolution are observed as the gradient strength is increased and more readout points are taken, albeit at the cost of a reduction in SNR. This suggests that gradient strength may be adjusted as a way to trade SNR for resolution in O-space imaging. The point spread function of the 16-echo encoding scheme was obtained by reconstructing echoes formed using point sources with a uniform coil. As expected, blurring occurs along a ring concentric about the center location chosen for each of the 16 projections: 8-Coil Array Number of readout points CONCLUSION The Kaczmarz algorithm shows great promise for reconstructing datasets generated using arbitrary hybrid encoding functions. By operating on a single row of the encoding matrix at a time, the algorithm makes large matrix problems tractable without placing extreme memory demands on the workstation. ACKNOWLEDGEMENTS The authors are grateful to Hemant Tagare for his insight into iterative methods for solving large matrix equations. SENSE YALE UNIVERSITY Faculty of Engineering Faculty of Engineering Department of BIOMEDICAL ENGINEERING