Communication Acceleration of multi-dimensional propagator measurements with compressed sensing Jeffrey L. Paulsen a,⇑⇑ , HyungJoon Cho b , Gyunggoo Cho c , Yi-Qiao Song a,⇑ a Schlumberger-Doll Research, One Hampshire St., Cambridge MA, United States b School of Nano-Bioscience and Chemical Engineering, UNIST, Ulsan, South Korea c Korea Basic Science Institute, Ochang, South Korea article info Article history: Received 21 June 2011 Revised 10 August 2011 Available online 23 August 2011 Keywords: NMR Diffusion Compressed sensing Propagator 2D and 3D diffusion propagator abstract NMR can probe the microstructures of anisotropic materials such as liquid crystals, stretched polymers and biological tissues through measurement of the diffusion propagator, where internal structures are indicated by restricted diffusion. Multi-dimensional measurements can probe the microscopic anisot- ropy, but full sampling can then quickly become prohibitively time consuming. However, for incom- pletely sampled data, compressed sensing is an effective reconstruction technique to enable accelerated acquisition. We demonstrate that with a compressed sensing scheme, one can greatly reduce the sampling and the experimental time with minimal effect on the reconstruction of the diffusion prop- agator with an example of anisotropic diffusion. We compare full sampling down to 64 sub-sampling for the 2D propagator measurement and reduce the acquisition time for the 3D experiment by a factor of 32 from 80 days to 2.5 days. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Measuring the directional dependence of diffusion with NMR is a useful tool to characterize the complicated internal microstruc- ture for many anisotropic materials, such as liquid crystals [1–3], ordered polymer structures [4], stretched polymer electrolytes [5], nanostructured materials [6,7], and biological tissues [8,9] and may help further resolve the NMR spectra of such mixtures with DOSY [10,11] as well. Diffusion tensor measurements only re- quire few independent encodings, a minimum of six along different directions, and provide only a simple model (diffusion tensor) for anisotropic diffusion. However, it is well known [12] that this mod- el may lack the angular resolution necessary to identify more com- plex structures, like fiber crossings in biological networks, due to the underlying Gaussian assumption. For diffusive motion, the Gaussian approximation breaks down as the diffusion length ap- proaches the size on any confining structures [12]. For non-biolog- ical porous media applications, such as characterizing the pore structure of rocks, this can often be the case as pore sizes can span orders of magnitude. Variations from non-Gaussian diffusive behavior has motivated a variety of different NMR encoding and reconstruction schemes such as low q scattering experiments [13], q-ball [14] and kurtosis imaging [15]. Common to these approaches, is that they are model- dependent, relying on the highly regular structure of the diffusion propagator, fitting to a targeted basis such as spherical harmonics or performing something along the lines of a cumulant analysis [16]. However, these are not always extendable to the wider range of propagator measurements, for instance in the presence of flow [17] or in propagator–propagator exchange experiments [18]. Fur- thermore, in certain rare pathological systems, such as well or- dered bead packs [19], resonance effects can still be observed corresponding to pore exchange that some of these newer se- quences aim to produce for a wider range of materials. Alternatively, inverting for the model, free diffusion propagator [20] yields the probability distribution P(Dx, Dt) of displacement Dx of the fluid’s individual molecules for one or multiple travel times Dt. Deviations from free diffusion and the shape of the prop- agator yields information about the microscopic structure as these restrict diffusive motion without any additional assumptions placed on the diffusion process. Multidimensional versions of these measurements, looking at the directionality of the diffusion prop- agator [21] are necessary to yield details of the microscopic anisot- ropy, and the generality of the full propagator inversion allows it to be applied to a wider variety of these experiments. While multidimensional measurements are necessary to fully probe the anisotropy of the microstructure from the propagator, they greatly increase the sampling requirements for full recon- struction and hence experiment time. For example, a fully sampled 3D, high-resolution (e.g. 128  128  128, for 128 samples along each dimension) propagator measurement with a 3s repetition 1090-7807/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmr.2011.08.025 ⇑ Corresponding author. ⇑⇑ Principle corresponding author. E-mail addresses: jpaulsen2@slb.com (J.L. Paulsen), hjcho@unist.ac.kr (H. Cho), gyunggoo@kbsi.re.kr (G. Cho), ysong@slb.com (Y.-Q. Song). Journal of Magnetic Resonance 213 (2011) 166–170 Contents lists available at SciVerse ScienceDirect Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr