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Spheroidal Wave Functions

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Spatial Smoothing in fMRI Using ProlateSpheroidal Wave FunctionsMartin A. Lindquist1* and Tor D. Wager21Department of Statistics, Columbia University, New York, New York2Department of Psychology, Columbia University, New York, New YorkAbstract: The acquisition of functional magnetic resonance imaging (fMRI) data in a finite subset of k-space produces ring-artifacts and ‘side lobes’ that distort the image. In this article, we explore the con-sequences of this problem for functional imaging studies, which can be considerable, and propose a so-lution. The truncation of k-space is mathematically equivalent to convolving the underlying ‘‘true’’image with a sinc function whose width is inversely related to the amount of truncation. Spatialsmoothing with a large enough kernel can eliminate these artifacts, but at a cost in image resolution.However, too little spatial smoothing leaves the ringing artifacts and side lobes caused by k-space trun-cation intact, leading to a potential decrease in signal-to-noise ratio and statistical power. Thus, tomake use of the high-resolution afforded by MRI without introducing artifacts, new smoothing filtersare needed that are optimized to correct k-space truncation-related artifacts. We develop a prolatespheroidal wave function (PSWF) filter designed to eliminate truncation artifacts and compare its per-formance to the standard Gaussian filter in simulations and analysis of fMRI data on a visual-motortask. The PSWF filter effectively corrected truncation artifacts and resulted in more sensitive detectionof visual-motor activity in expected brain regions, demonstrating its efficacy. Hum Brain Mapp 29:1276–1287, 2008.VVC2007 Wiley-Liss, Inc.Key words: fMRI; spatial smoothing; prolate spheroidal wave function; Gaussian smoothing; prepro-cessing; spatial filteringINTRODUCTIONIn functional magnetic resonance imaging (fMRI) s tud-ies it is comm on practice to spatia lly smooth the acquireddata prior to performing statistical analysis. Spatialsmoothing invol ves blurring the functional MRI imagesby convolving the image data with a filter kernel, mostfrequently a Gaussian, though other types of kernels (e.g.,sinc kernel) may also be used. Gaussian smoothing isimplemented in major software packages such as SPM(Statistical Parametric Mapping, Wellcome Institute ofCognitive Neurology, University College London), AFNI(Analysis of Functional Imaging Data), and FSL (FM RIBsoftware library, Oxford). It is used primarily to minimizethe errors in group analysis introduced by the spatial nor-maliza tion of brains into a common space and to makedata conform t o the assumptions o f Gaussian randomfield theory if it is used for correction for multiple com-parisons (Worsley and Friston, 1995). In addition, if thespatial extent of a region of interest ( ROI) is larger thanthe spatial resolution, smoothing may r educe randomnoise in individual voxels a nd increase the signal-to-noiseratio (SNR) within the ROI (R osenfeld and Kak, 1982;Smith, 2003). Although it is advantageous to smooth datafor these reasons, there are also obvious costs in spatial*Correspondence to: Martin A. Lindquist, 1255 Amsterdam Ave,10th Floor, MC 4409, New York, NY 10027.E-mail: [email protected] for publication 5 January 2007; Revised 10 July 2007;Accepted 11 July 2007DOI: 10.1002/hbm.20475Published online 2 November 2007 in Wiley InterScience (www.interscience.wiley.com).VVC2007 Wiley-Liss, Inc.rHuman Brain Mapping 29:1276–1287 (2008)rresolution. With larger sample sizes, higher fieldstrengths, and other advances in imaging tec hnology,many groups may wish t o take advanta ge of the highpotential spatial resolut ion of fMRI data and minimizethe amount of smoothing.However, there is a cost to not smoothing images that isperhaps under-appreciated. The cost comes from the factthat data are acquired in a finite portion of k-space, whichis often quite limited in functional (T2*) acquisitionschemes. To obtain a ‘‘perfect’’ reconstruction of an image,an infinite number of k-space measurements need to bemade. The restriction of sampling to a finite space introdu-ces blurring in the image: this truncation is mathematicallyequivalent to convolving the image with a sinc functionwhose width is inversely related to the amount of k-spacethat is sampled. The result is ringing artifacts and sidelobes - aliased portions of the image that extend beyondthe actual location of the imaged object - that distort theimages (Fig. 1B). Such artifacts, if significantly large, mayat best reduce signal to noise ratio, and at worst result inmis-localization of functional activation.Spatial smoothing of images (e.g., with a Gaussian filter)can ameliorate these problems. The process of spatialsmoothing is equivalent to applying a low-pass filter to thesampled k-space data. It has the effect of altering the spa-tial resolution in the image, as it reduces the intensity ofhigh-frequency points in k-space, as well as changing theimage’s resulting point-spread function. If the frequenciessampled in the images are higher than the filter cutoff, thehigh frequency information (i.e., fine spatial resolution)will be lost, though the side lobes will be eliminated (Fig.1D). Alternatively, if the frequencies sampled are lowerthan those in the filter, direct application of the filter willnot be able to completely eliminate the side lobes. In thiscase, k-space is insufficiently sampled to support theapplied filter and significant truncation artifacts willremain (Fig. 1C). The effective smoothing applied to theimages will therefore be wider than intended. Thus, whenimage data are rather coarsely sampled (e.g. 64 3 64 3.75mm voxels in a 240 mm field of view) special care needsto be taken to minimize truncation artifacts. In this articlewe show that in certain situations a Gaussian filter willnot allow for control for these serious effects and shouldnot be used when applying a narrow filter (e.g., withFWHM <8 mm) to a low resolution (e.g. 64 3 64) image.We show in a series of simulations that failure to takeproper care of these issues will lead to a decrease in SNRand to decreased power in the resulting statistical tests.In principal, one can make a distinction between twoseparate, but related, preprocessing steps. The first is theneed to correct artifacts because of the reconstruction of fi-nite k-space data, and the second is smoothing as a meansof increasing signal-to-noise and validating certain


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