Spatial Smoothing in fmri Using Prolate Spheroidal Wave Functions

Transcription

1 Human ain Mapping 29: (2008) Spatial Smoothing in fmri Using Polate Spheoidal Wave Functions Matin A. Lindquist 1 * and To D. Wage 2 1 Depatment of Statistics, Columbia Univesity, New Yok, New Yok 2 Depatment of Psychology, Columbia Univesity, New Yok, New Yok Abstact: The acquisition of functional magnetic esonance imaging (fmri) data in a finite subset of k- space poduces ing-atifacts and side lobes that distot the image. In this aticle, we exploe the consequences of this poblem fo functional imaging studies, which can be consideable, and popose a solution. The tuncation of k-space is mathematically equivalent to convolving the undelying tue image with a sinc function whose width is invesely elated to the amount of tuncation. Spatial smoothing with a lage enough kenel can eliminate these atifacts, but at a cost in image esolution. Howeve, too little spatial smoothing leaves the inging atifacts and side lobes caused by k-space tuncation intact, leading to a potential decease in signal-to-noise atio and statistical powe. Thus, to make use of the high-esolution affoded by MRI without intoducing atifacts, new smoothing filtes ae needed that ae optimized to coect k-space tuncation-elated atifacts. We develop a polate spheoidal wave function (PSWF) filte designed to eliminate tuncation atifacts and compae its pefomance to the standad Gaussian filte in simulations and analysis of fmri data on a visual-moto task. The PSWF filte effectively coected tuncation atifacts and esulted in moe sensitive detection of visual-moto activity in expected bain egions, demonstating its efficacy. Hum ain Mapp 29: , VC 2007 Wiley-Liss, Inc. Key wods: fmri; spatial smoothing; polate spheoidal wave function; Gaussian smoothing; pepocessing; spatial filteing INTRODUCTION In functional magnetic esonance imaging (fmri) studies it is common pactice to spatially smooth the acquied data pio to pefoming statistical analysis. Spatial smoothing involves bluing the functional MRI images by convolving the image data with a filte kenel, most fequently a Gaussian, though othe types of kenels (e.g., *Coespondence to: Matin A. Lindquist, 1255 Amstedam Ave, 10th Floo, MC 4409, New Yok, NY matin@stat.columbia.edu Received fo publication 5 Januay 2007; Revised 10 July 2007; Accepted 11 July 2007 DOI: /hbm Published online 2 Novembe 2007 in Wiley InteScience (www. intescience.wiley.com). sinc kenel) may also be used. Gaussian smoothing is implemented in majo softwae packages such as SPM (Statistical Paametic Mapping, Wellcome Institute of Cognitive Neuology, Univesity College London), AFNI (Analysis of Functional Imaging Data), and FSL (FMRI softwae libay, Oxfod). It is used pimaily to minimize the eos in goup analysis intoduced by the spatial nomalization of bains into a common space and to make data confom to the assumptions of Gaussian andom field theoy if it is used fo coection fo multiple compaisons (Wosley and Fiston, 1995). In addition, if the spatial extent of a egion of inteest (ROI) is lage than the spatial esolution, smoothing may educe andom noise in individual voxels and incease the signal-to-noise atio (SNR) within the ROI (Rosenfeld and Kak, 1982; Smith, 2003). Although it is advantageous to smooth data fo these easons, thee ae also obvious costs in spatial VC 2007 Wiley-Liss, Inc.

2 Spatial Smoothing Using PSWF Figue 1. (A) A cicula egion of adius 22.5 mm is placed in an image with FOV 240 mm and theoetical k-space measuements ae calculated coesponding to a egion of k-space. () Image econstuction with no smoothing. The inging is due to the fact that k-space is tuncated. (C) The image in () is smoothed using a Gaussian filte with 4 mm FWHM. The inging atifacts ae educed but not completely eliminated. (D) The image in () is smoothed using a Gaussian filte with 12 mm FWHM. The inging atifacts ae eliminated, at the cost of educed image esolution. The images ae plotted on the log-scale to bette illuminate the inging atifacts. [Colo figue can be viewed in the online issue, which is available at esolution. With lage sample sizes, highe field stengths, and othe advances in imaging technology, many goups may wish to take advantage of the high potential spatial esolution of fmri data and minimize the amount of smoothing. Howeve, thee is a cost to not smoothing images that is pehaps unde-appeciated. The cost comes fom the fact that data ae acquied in a finite potion of k-space, which is often quite limited in functional (T2*) acquisition schemes. To obtain a pefect econstuction of an image, an infinite numbe of k-space measuements need to be made. The estiction of sampling to a finite space intoduces bluing in the image: this tuncation is mathematically equivalent to convolving the image with a sinc function whose width is invesely elated to the amount of k-space that is sampled. The esult is inging atifacts and side lobes - aliased potions of the image that extend beyond the actual location of the imaged object - that distot the images (Fig. 1). Such atifacts, if significantly lage, may at best educe signal to noise atio, and at wost esult in mis-localization of functional activation. Spatial smoothing of images (e.g., with a Gaussian filte) can amelioate these poblems. The pocess of spatial smoothing is equivalent to applying a low-pass filte to the sampled k-space data. It has the effect of alteing the spatial esolution in the image, as it educes the intensity of high-fequency points in k-space, as well as changing the image s esulting point-spead function. If the fequencies sampled in the images ae highe than the filte cutoff, the high fequency infomation (i.e., fine spatial esolution) will be lost, though the side lobes will be eliminated (Fig. 1D). Altenatively, if the fequencies sampled ae lowe than those in the filte, diect application of the filte will not be able to completely eliminate the side lobes. In this case, k-space is insufficiently sampled to suppot the applied filte and significant tuncation atifacts will emain (Fig. 1C). The effective smoothing applied to the images will theefoe be wide than intended. Thus, when image data ae athe coasely sampled (e.g mm voxels in a 240 mm field of view) special cae needs to be taken to minimize tuncation atifacts. In this aticle we show that in cetain situations a Gaussian filte will not allow fo contol fo these seious effects and should not be used when applying a naow filte (e.g., with FWHM <8 mm) to a low esolution (e.g ) image. We show in a seies of simulations that failue to take pope cae of these issues will lead to a decease in SNR and to deceased powe in the esulting statistical tests. In pincipal, one can make a distinction between two sepaate, but elated, pepocessing steps. The fist is the need to coect atifacts because of the econstuction of finite k-space data, and the second is smoothing as a means of inceasing signal-to-noise and validating cetain statistical techniques. While, most smoothing kenels ae designed pimaily to deal with the latte issue, they can if popely designed also be used to handle the fome. One method that is well-suited to contolling tuncation atifacts is the 0th ode polate spheoidal wave function (PSWF) (Landau and Pollak, 1961, 1962; Slepian and Pollak, 1961). The PSWF is the function, with compact suppot on a fixed subset of k-space, which maximizes the signal ove a cetain pedefined subset of image-space. Computationally, this function is the lagest eigenfunction of the finite, o tuncated, Fouie tansfom. It should be noted that discete polate spheoidal sequences (Slepian and Pollak, 1961) have peviously appeaed in the fmri liteatue as pat of the multitape technique fo tempoal smoothing (Mita and Pesaan, 1999; Mita et al., 1997). The attactive featues of the PSWF spatial smoothing filte ae discussed in this aticle. Simulations show that the powe to detect activation is significantly inceased when using the PSWF filte instead of a Gaussian filte if the FWHM is below 8 mm. Futhe data fom a visual task is used to illustate its efficiency when applied to fmri data. In this wok we will focus on compaing the PSWF filte with its Gaussian countepat. Howeve, it is impotant to note that a numbe of othe studies have been concened with finding altenatives to smoothing with a fixed Gaussian filte. Fo example, statistical analysis famewoks using 1277

3 Lindquist and Wage Gaussians of vaying width (Poline and Mazoye, 1994; Wosley et al., 1996) and otations (Shafie et al., 2003) have been poposed. As an altenative to using a Gaussian kenel, a numbe of aticles have suggested the use of wavelets (see Van De Ville et al., 2006 fo an excellent oveview of the liteatue). In addition, data-dependent smoothing methods have also been suggested. One pomising example is the use of anisotopic diffusion (Kim et al., 2005). It should be noted that while the appoach we ae suggesting is data-independent, as ae both the Gaussian and wavelet techniques, it does depend on the acquisition paametes. METHODS Theoy Spatial smoothing Spatial smoothing involves bluing the functional MRI images by applying a moving aveage filte to the images. When smoothing an image, each voxel is effectively tansfomed into the weighted sum ove a ROI, which consists of the voxels lying unde the kenel of the filte. The size of the kenel is detemined by the full width at half maximum (FWHM), which measues the width of the kenel at 50% of its peak value. Hence, voxels that lie within the ange defined by the FWHM ae weighted highe than voxels lying outside of this ange. To cay out spatial smoothing in MRI, a smoothing matix, epesenting the filte, is constucted which has the same size as the image. The image matix is convolved with the smoothing matix by fist Fouie tansfoming both the image and the smoothing matix, and then calculating the invese Fouie tansfom of the poduct of these two matices. The pocess of convolving the image using a smoothing kenel can be witten as efðxþ ¼ FðuÞKðx uþ du ¼ ^FðkÞ^KðkÞ e ixk dk; ð1þ X X whee K(x) is the smoothing matix, Kˆ (k) its Fouie tansfom, Fˆ(k) is the expeimental sampling function in k- space and F(x) is its coesponding invese Fouie tansfom (the image). Hee F (x) epesents the value of the smoothed data at the coodinate point, x, of the image. Typically when pefoming spatial smoothing in fmri one uses a Gaussian kenel, though othe choices ae also possible. Howeve, because of the populaity of the Gaussian we will concentate on studying its popeties in this paticula aticle. It can be expessed on the fom: K U ðxþ /exp x2 2 2 : ð2þ Convolving the image using the Gaussian kenel is equivalent to multiplying the k-space data by the Fouie tansfom of the kenel function, which in tun can be witten as: ^K U ðkþ /exp 1 2 ð2pþ2 k 2 : ð3þ Studying the Fouie tansfom of the Gaussian in Eq. (3), it is clea that it also follows a Gaussian distibution, in this case with a standad deviation equal to (2p) 21. The width of the smoothing kenel is detemined by the amount of spatial smoothing equied and is witten in tems of the FWHM. It is impotant to note that the elationship between the FWHM and the standad deviation of the Gaussian can be witten as ¼ FWHM p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4þ 2 lnð2þ Figue 2A (fist and second ow-ight column) shows an example of K F (x) and Kˆ F(k) in one-dimension. y studying the shape of Kˆ F(k) it is clea that application of this filte devalues points in the oute egions of k-space and is theefoe equivalent to low-pass filteing. A well known fact about the Gaussian distibution states that 99.7% of its mass is contained in a egion lying within Figue 2. (A) A boxca function (top left) is multiplied with a Gaussian kenel (top ight). The line unde the Gaussian shows the fequency extent of the boxca compaed to the Gaussian. Afte pefoming the invese Fouie tansfom on the poduct, the esult is equivalent by the Fouie Convolution Theoem, to convolving the point spead function (cente left) with a Gaussian kenel (cente ight). The esults of this convolution ae shown in the bottom ow. () A boxca function (top left) is multiplied with a kenel (top ight) designed to have the same fequency extent as the boxca. Afte pefoming the invese Fouie tansfom on the poduct, the esult is equivalent to convolving the point spead function (cente left) with the invese Fouie tansfom of the kenel (cente ight). The esults of this convolution ae shown in the bottom ow. The filte effectively contols fo the inging atifacts. 1278

4 Spatial Smoothing Using PSWF thee standad deviations of its mean. ecause of the symmetic natue of the Gaussian, this egion will make up a cicula shape in two-dimensions (spheical in theedimensions). Hence, 99.7% of the mass of Kˆ F(k) will lie within a cicula egion with adius 3(2p) 21. Using this popety, we can define an altenative measue fo the extent of a filte. Let the width of the kenel in k-space be defined as W, if 99.7% of the kenels mass lies in the ange [20.5W,0.5W]. Though the suppot of the Gaussian is infinite, the width of Kˆ F(k) is elated to the FWHM by Eq. (4) and can be expessed as W ¼ 6 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lnð2þ 1 p FWHM : Similaly, we can define the width of the filte, K F (x), in image-space as T, if 99.7% of the kenels mass in imagespace lies in the ange [20.5T,0.5T]. Hence, the width can be expessed using Eq. (4) as Effective smoothing ð5þ T ¼ 3FWHM p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð6þ 2 lnð2þ As afoementioned, thee is an additional souce of bluing inheent in the econstuction of MR images. This bluing is due to the fact that sampling is pefomed ove a finite subset of k-space. To obtain a pefect econstuction of the undelying image one would need to sample an infinitely lage egion of k-space. Unfotunately, this is not possible and k-space is by necessity tuncated. This tuncation esults in a inging atifact in the image which is due to the unde-sampling of high spatial fequency components of k-space (Fig. 1). y tuncating k-space we ae pefoming an opeation that is mathematically equivalent to convolving the undelying tue image with a sinc function, whose width is invesely elated to the amount of tuncation. It should be noted that thoughout this aticle we ae assuming that we ae dealing with images that have been econstucted using the invese discete Fouie tansfom (IDFT). Othe econstuction algoithms (e.g., SENSE and GRAPPA based econstuction) give ise to othe types of atifacts than those descibed ealie. Howeve, since the bandwidth issues ae compaable fo all econstuction techniques, the inging atifacts ae expected to be simila. Fo this eason, and the fact that most pactitiones still wok diectly with images econstucted using the IDFT, we concentate on atifacts elated to the IDFT in this wok. If one wee to apply a filte with 4 mm FWHM diectly to a image, the esulting smoothed image would not possess the exact amount of smoothness that would be expected, as the applied filte is too naow to completely eliminate the side-lobes of the sinc function that aise because of tuncation (Figs. 1C and 2A). It is impotant to ealize that diectly applying the filte to the image-space data is equivalent, by the Fouie convolution theoem, to applying a tuncated (in k-space) vesion of the filte to the k-space data. In the continuation we will efe to effective smoothing as the bluing in the image because of the combination of the applied filte and the tuncation effects. The effective smoothing kenel will thus be the convolution of the applied filte and the sinc function that aises because of finite sampling (Fig. 2-last ow). We measued the effects of inging when applying a Gaussian filte, with vaying FWHM (3 12 mm), diectly to an image. If no tuncation effects wee pesent we would expect 99.7% of the mass to lie within 63 (o FWHM) of the cente of the effective smoothing kenel, that is, within the width of the kenel. It is clea fom Figue 3A that the popotion of the effective smoothing kenel that lies in this ange is geatly affected fo filtes with FWHM less than 8 mm. Futhe, accoding (A) The popotion of the effective Gaussian smoothing kenel that lies futhe than 63 (o FWHM) fom the cente of the filte when N 5 64 and the FOV is equal to 200 and 240 mm (dashed and bold line espectively). () The actual and Figue 3. expected FWHM plotted fo a Gaussian filte fo the case when N 5 64 and the FOV is equal to 200 and 240 mm (dashed and bold line espectively). Fo compaison puposes the dotted line shows whee the actual and expected FWHM coincide. 1279

5 Lindquist and Wage to Figue 3, the FWHM of the effective smoothing kenel deviates fom the applied FWHM. It is clea that fo naow filtes (FWHM <8 mm) this deviation becomes athe significant. Fo example, when the FOV is equal to 240 mm, a Gaussian filte with FWHM of 4 mm applied diectly in image-space will have an effective FWHM of 5.35 mm when the effects of finite sampling ae included. In addition ove 25% of its mass will lie outside of its intended width. It would clealy be beneficial to design a filte that is able to educe the effects of tuncation atifacts in the image. In the next section we discuss a filte that povides optimal contol ove tuncation atifacts. The polate spheoidal wave function filte The PSWF filte is the function, with compact suppot on a fixed set of k-space, which maximizes the signal ove a cetain pedefined subset of image-space. To find this function, we begin by setting up the poblem fomulation. As k- space consists of a sequence of discete measuements duing MRI data acquisition, it should theefoe be chosen to be a discete space. On the othe hand, image-space, denoted by X in this aticle, can be chosen to be eithe a discete space consisting of a collection of the image voxels (e.g., the N 3 N voxels in an image) o as a continuous space. We discuss the latte below. Next, conside a convex ROI, denoted, in image-space and the k-space sampling egion, A (e.g., the collection of measued points on a gid see Fig. 4A,). The objective is to find the filte function g(k), that satisfies the following two citeia: 1. It takes the value 0 fo points outside of A. 2. Its invese Fouie tansfom, G(x), has maximal signal concentation in, that is, the atio R jgðxþj 2 dx k ¼ R jgðxþj 2 dx X is maximized ove all possible functions fo which the fist citeion holds. These conditions ensue that g(k) is the function with compact suppot on A, whose invese Fouie tansfom has the least amount of signal leakage outside of the egion. Thus, it is the function with optimal contol ove inging atifacts outside of. Note that k takes values between 0 and 1 and can be used to calculate the amount of signal leakage that the filte gives ise to outside of the pedetemined ROI. As an additional constaint, the denominato of the atio in Eq. (7) is set equal to one. This simplifies the poblem to finding the function, g(k), with nom equal to one, whose invese Fouie tansfom maximizes, k ¼ jgðxþj 2 dx: ð8þ ð7þ Figue 4. A cicula egion in image-space (top left) and a k-space subset A (top ight). The PSWF filte g(k) (bottom ight) and its invese Fouie tansfom G(x) (bottom left) coesponding to the choice of A and. Note the minimal inging pesent in the image-space filte. Using Paseval s identity, the poblem can be witten in matix fom as unde the constaint k ¼ maxfg þ ^K gg ð9þ g þ g ¼ X k j 2A gðk j Þgðk j Þ¼1: ð10þ Let us define Kˆ to be the Fouie tansfom of the indicato function of the egion, which can be witten: ^K ðkþ ¼ expf 2piðx; kþg dx ð11þ Then it can be shown (Lindquist, 2003; Lindquist et al., 2006; Yang et al., 2002; Shepp and hang, 2000) that the kenel, Kˆ, is given by, ^K ðk; k 0 Þ¼^K ðk k 0 Þ ð12þ fo k,k 0 [ A. A moe detailed deivation of these esults can be found in the Appendix. Fo simple egions (e.g., cicles, squaes, and ellipses) thee exist analytical expessions fo Kˆ, which allow fo easy computation of the kenel defined in Eq. (12). The kenel Kˆ is positive definite which implies that all of its eigenvalues k k, k 5 0,...a 2 1, ae non-negative, 1280

6 Spatial Smoothing Using PSWF whee a denotes the numbe of points sampled in k-space (e.g., a 5 N 2 ). In fact, 1 k 0 k 1... k a 1 0: ð13þ The eigenfunction, g 0, associated with the lagest eigenvalue, k 0, is temed the 0-ode solution, and povides the solution to the poblem stated in Eq. (7). In addition, k 0 is equal to the faction of the total signal intensity in calculated accoding to Eq. (7) and povides a quantitative measue of the signal leakage that the eigenvecto gives ise to. A value of k 0 close to 1 indicates little signal leakage, while the leakage inceases as k 0 deceases. The set of eigenfunctions g 0, g 1,... g a21 have some inteesting popeties. As we will always choose A to be symmetical about the cente of k-space (k 5 0), the set of eigenvectos can always be taken to be eal and othogonal on. In addition, the index i not only anks the eigenvalues, but also specifies the numbe of zeos the function g i has within the egion (Pecival and Walden, 1993). The function g 0 will theefoe be positive ove the whole egion, a popety it shaes with the Gaussian kenel. y applying the filte g(k) diectly to the k-space data and theeafte taking the invese Fouie tansfom of the filteed k-space data, the esulting image will epesent the tue image convolved with the function G(x). Theefoe the PSWF can be used as a filte to spatially smooth fmri data. The amount of smoothing applied will depend upon the size of the egion. In ou discussion of the PSWF filte it will often make moe sense to talk about the filtes width instead of the taditional FWHM used fo the Gaussian. As long as k 0.997, the width of the PSWF is bounded by the diamete of the egion, as this condition ensues that less than 0.3% of the signal lies outside of. The PSWF filte has the impotant popety that any othe choice of filte will give ise to a geate amount of signal leakage outside of the spatial coveage egion. Hence, the PSWF filte is chosen to minimize the amount of signal leakage, and theeby inging, outside of the egion. This popety leads us to believe that it has excellent potential as a spatial smoothing filte in fmri. In pactice, one can constuct a smoothing kenel by taking the IDFT of g(k) and econstucting it onto a matix with the same dimensions as the image. Smoothing can theeafte be pefomed in an analogous manne as fo Gaussian, which is descibed algebaically in Eq. (1). In ou implementation, A is chosen to coespond to the sampled egion of k-space (Fig. 4) and is chosen to be a cicula egion with diamete FWHM (equal to 6) to ensue a kenel with simila width as a Gaussian filte of a given FWHM. Figue 4C,D show examples of G(x) and g(k) coesponding to the egions A and. The PSWF filte shaes many of the same qualities as the Gaussian filte. Howeve, it is completely concentated on a subegion, A, of k-space, while the Fouie tansfom of the Gaussian kenel has infinite suppot. If one wishes to obtain a filte that moe exactly mimics the popeties of the Gaussian, A can be chosen as a cicula egion with diamete equal to the minimum of N and W, whee N denotes the image dimensions and W the width of the Gaussian one seeks to mimic. This citeia ensues that the PSWF filte will have equivalent width in the fequency domain as the Gaussian wheneve possible (W > N fo filtes with FWHM < 8 mm), and will theefoe give ise to filtes with simila popeties in image-space. It should be noted that the idea of matching the equivalent FWHM of a function is not new, and has been done befoe fo diffeent wavelet functions (Fadili and ullmoe, 2004; Van De Ville et al., 2003). Figue 5 shows the effective FWHM and leak- (A) The popotion of the PSWF kenel (bold) and the Gaussian kenel (dashed) that lies futhe than 63 (o (1.274 FWHM) fom the cente of the filte when N 5 64 and the FOV is equal to 200 mm. () The actual and expected FWHM plotted fo the Figue 5. PSWF and Gaussian filte fo the case when N 5 64 and the FOV is equal to 200 mm (bold and dashed line espectively). Fo compaison puposes the dotted line shows whee the actual and expected FWHM coincide. 1281

7 Lindquist and Wage Figue 6. The effective smoothing kenel obtained using the PSWF (bold) and the Gaussian (dashed) with 4 mm (left) and 12 mm (ight) FWHM (N 5 64, FOV mm). While the filtes ae oughly equivalent fo the wide FWHM, the naow PSWF filte shows bette contol of inging atifacts and slightly widened FWHM compaed to its Gaussian countepat. age fo a PSWF constucted in this manne, when N 5 64 and FOV mm. Also included ae simila plots fo the Gaussian filte, which is equivalent to those shown in Figue 3. It is clea that the PSWF filte gives ise to filtes that have equivalent popeties to the Gaussian fo FWHM > 8 mm. In addition, fo naow filtes, it is clea that the PSWF filte gives impoved contol ove the amount of leakage outside of the width of the filte. The pice one pays fo this is a slight incease in the effective FWHM of the filte. Fo compaison puposes we show in Figue 6 two examples of the combined effects of tuncation and smoothing when using PSWF and Gaussian kenels. The left panel shows the esults fo a naow filte (4 mm FWHM), while the ight panel shows the esults fo a wide filte (12 mm FWHM). While the effective smoothing kenels ae oughly equivalent fo the wide filte, the naow PSWF filte shows bette contol of inging atifacts and slightly widened FWHM compaed to its Gaussian countepat. This is consistent with the esults shown in Figue 5. Simulation 1 Simulation Studies To moe closely study the issues involved with tuncation atifacts we pefomed a simulation study, in which a cicula egion of adius 22.5 mm was placed in the cente of a blank image with FOV 240 mm (Fig. 1A). Using the simple geomety of the image, theoetical k-space measuements wee calculated coesponding to a egion of k-space. The image was then econstucted using the invese fast Fouie tansfom (IFFT) (Fig. 1) and Gaussian smoothing with 4 and 12 mm FWHM was applied to the esulting images. In addition, a PSWF filte with the same width as the Gaussian with 4 mm FWHM was applied to the data. Simulation 2 (Pat A) The image descibed in the pevious simulation (Fig. 1A) was eceated 160 times. In 80 of the images the intensity fo voxels within the cicle was set to 1.0. These images wee denoted as OFF images. In the emaining 80 images the intensity was set to These ae denoted as ON images. The images wee theeafte odeed accoding to the paadigm of 4 epetitions of 20 OFF/20 ON images and Gaussian noise was added to the data with a standad deviation coesponding to a Cohen s d of 0.25 (Cohen, 1988). Theoetical k-space measuements wee calculated coesponding to a egion of k-space and the images wee econstucted at this esolution using the IFFT. The esulting data was smoothed using both PSWF and Gaussian filtes with FWHM equal to 4, 8, and 12 mm. The esulting data was analyzed using the standad GLM appoach. This pocedue was epeated 1,000 times and the statistical powe to detect activation was calculated fo each filte type and width. (Pat ) The pocedue outlined in pat A was epeated using a moe ealistic activation patten (Fig. 8). Using a gay matte mask a egion of activity was ceated. The esulting data was intepolated to 10 times its size ( ) and the fast Fouie tansfom was used to calculate k-space data. The cental egion was used to mimic the effects of finite k-space sampling. Using this data set the same expeimental paadigm as descibed in (A), with 4 epetitions of 20 OFF/20 ON images, was ceated. The data was analyzed in an analogous manne as descibed in pat A. Expeiment Nine students at the Univesity of Michigan wee ecuited and paid $50 fo paticipation in the study. All human paticipant pocedues wee conducted in accodance with Institutional Review oad guidelines. The ex- 1282

8 Spatial Smoothing Using PSWF peimental data consisted of a visual paadigm conducted on the nine subjects. It consisted of a blocked altenation of 11 s of full-field contast-evesing checkeboads (16 Hz) with 30 s of open-eye fixation baseline. locks of unilateal contast-evesing checkeboads wee pesented on an in-scanne LCD sceen (IFIS, Psychology Softwae Tools). Spial-out gadient echo images wee collected on a GE 3T fmri scanne (Noll et al., 1995). Seven oblique slices wee collected though visual and moto cotex, mm voxels, TR s, TE 5 25 ms, flip angle 5 90, FOV 5 20 cm, 410 images. Data fom all images wee coected fo slice-acquisition timing diffeences using 4- point sinc intepolation (Oppenheim et al. 1999) and coected fo head movement using six-paamete affine egistation (Woods et al., 1998) pio to analysis. Fo each subject Gaussian filtes with FWHM of 4, 8, and 12 mm wee applied to the slice of the data, which contained the lagest signal ove the visual cotex (slice No. 3). Theeafte, thee PSWF filtes, with equivalent width wee applied to the same data set, giving a total of six seies of smoothed images fo each subject. Next a standad GLM analysis was pefomed on each of the 54 data sets (six seies of smoothed images fo each of the nine subjects). The data was thesholded and the numbe of active voxels in the visual cotex was counted fo each seies and a matched-pais sign-test was pefomed fo each level of smoothness. The test was designed to detemine whethe thee is a significant diffeence in the numbe of active voxels between the two filte types acoss subjects. Simulation 1 RESULTS Simulation Studies Figue 7. The image in Figue 1 is smoothed using a Gaussian filte with FWHM equal to (A) 4 mm and () 12 mm, as well as with a PSWF with width equivalent to a 4 mm Gaussian (C). The images (A C) ae plotted on the log-scale to bette illuminate the inging atifacts. (D) A coss-section though the cente of each image (A C, along with Fig. 1A,) illustates the inging effects appaent in each case. [Colo figue can be viewed in the online issue, which is available at The simulated k-space data was econstucted using the IFFT (Fig. 1) and Gaussian smoothing with 4 and 12 mm FWHM was applied to the esulting images (Fig. 7A,). It is appaent that significant inging atifacts exist in the econstucted nonsmoothed image. It is also clea that the filte with 4 mm FWHM was able to minimize, but not completely eliminate, these atifacts. The filte with 12 mm FWHM successfully eliminated these atifacts, at the pice of deceased spatial esolution. Next, we applied a PSWF filte with the same width as the Gaussian with 4 mm FWHM and applied this to the same data set (Fig. 7C). It is clea that the PSWF was able to effectively contol the inging atifacts, while still maintaining a simila spatial esolution as the 4 mm Gaussian. Figue 7D shows a cosssection though the cente of each image to bette illustate the inging effects appaent in the vaious images. In paticula note the smooth behavio of the PSWF compaed to the 4 mm Gaussian. Simulation 2 The functional simulation studies descibed in the Methods section wee epeated 1,000 times fo each filte type and width. The esulting data was analyzed using a stand- TALE I. The total numbe of significant voxels (P-value <0.005) found in the visual cotex fo data smoothed using both PSWF and Gaussian kenels fo thee diffeent levels of smoothing (4, 8, and 12 mm) Subject 4 mm 8 mm 12 mm PSWF Gauss PSWF Gauss PSWF Gauss P-value The esults ae shown fo each of the nine subjects included in the expeiment. The last ow show P-values fom a sign test fo matched pais, which tests whethe thee is a significant diffeence in the numbe of active voxels between the two smoothing methods. The null hypothesis of no diffeence is ejected fo both 4 and 8 mm smoothing at the 5% level of significance. 1283

9 Lindquist and Wage Figue 8. (A) The simulated activation patten and ROC cuves fo Simulation 2A obtained using Gaussian (dashed) and PSWF (bold) filtes with 3 diffeent FWHM (4, 8, and 12 mm). Fo naow filtes (e.g., 4 mm) using a Gaussian filte will lead to a significant decease in statistical powe compaed to the PSWF. () The same esults fo Simulation 2. Analogous esults hold in this situation. Figue 9. Statistical paametic maps, obtained fom 5 andomly chosen subjects, showing voxels with significant activation (P-value <0.005) fo each of the thee diffeent degees of smoothing. Red epesents voxels that ae significant fo both data that was smoothed using the PSWF and the Gaussian filte. lue indicates voxels that wee significant fo the PSWF data only and yellow voxels that wee significant fo the Gaussian data only. 1284

10 Spatial Smoothing Using PSWF the FWHM is 4 mm, thee is a significant incease in the numbe of active voxels when using the PSWF filte, which is consistent with the incease in powe shown in Figue 8. DISCUSSION Figue 10. The diffeence in the numbe of significant voxels obtained using the PSWF and Gaussian kenel fo a vaiety of thesholds (t* 5 2,... 5 coesponding to P-values anging fom to 0) using filtes with FWHM 4, 8, and 12 mm. It is clea that thee is a significant incease in the numbe of active voxels in the 4 mm case when using the PSWF filte that is attibutable to the incease in powe shown in Figue 8. ad GLM appoach and false positive ates and powe wee assessed. Figue 8 shows eceive opeating chaacteistic (ROC) cuves fo the Gaussian (dashed) and PSWF (bold) filtes coesponding to each of the thee diffeent FWHM (4, 8, and 12 mm) and both simulation types (A and ). Fo both simulations thee is minimal diffeence in powe between the two filte types fo wide filtes (e.g., 8 and 12 mm), while fo naow filtes (e.g., 4 mm) using a Gaussian filte will lead to a significant decease in statistical powe compaed to the PSWF. Expeimental Results Table I shows the numbe of significant voxels (P-value <0.005, uncoected) in the visual cotex fo each subject, obtained using both filte types fo each of the thee diffeent degees of smoothing. The P-values in the last ow show the esults of a sign test fo matched pais. The null hypothesis of no diffeence in the numbe of active voxels is ejected (P-value <0.05) fo both 4 and 8 mm smoothing. The incease in the numbe of active voxels fo the data smoothed using the PSWF voxel is due to the inceased SNR in the data, as a esult of minimizing the signal leakage, and is consistent with the theoetical esults. Figue 9 show the esults fo five andomly chosen subjects fo each of the 3 degees of smoothing. It is clea that thee is an inceased numbe of active voxels in the PSWFsmoothed data fo naow (8 mm) kenels. Finally, Figue 10 shows the diffeence in the numbe of significant voxels between the two smoothing methods as a function of the chosen theshold. It is clea that fo the case when In this aticle we have discussed two potential souces of bluing in fmri data. The fist is due to the algoithm used to econstuct the image fom finite k-space data and is geneally consideed an unwanted atifact of image econstuction. The second is due to the voluntay application of a smoothing kenel - typically used to incease SNR. The sinc function shown in Figue 2 is the smoothing (spatial esponse function) that aises when econstuction is pefomed using the IDFT. It is impotant to note that othe econstuction algoithms will ultimately give ise to othe spatial esponse functions and may theefoe be moe o less effective in dealing with issues such as Gibbs inging. Fo example, data obtained using paallel imaging techniques, do not econstuct the image using the invese DFT. In these situations the spatial esponse function is much moe complicated than the sinc assumed in this aticle. While the esults of this aticle can not be eadily genealized to all econstuction techniques, we have ultimately decided to focus on the atifacts that aise fom using the DFT to econstuct the images, as this is the type of data most pactitiones will likely have to wok with. In addition, as each econstuction technique is faced with the same bandwidth poblems the inging atifacts should ultimately take a simila fom. Howeve, it should be noted that the PSWF methodology outlined in this aticle have been genealized to k-space data that is sampled on noncatesian k-space (Lindquist et al., 2006). Hence, this would in pincipal allow fo econstuction and smoothing of data obtained in any manne using the PSWF methodology. In this wok we have illustated that when applying a naow Gaussian filte (e.g. 4 mm FWHM) to low-esolution data (e.g., ), an insufficient numbe of measuements ae made in k-space to suppot the applied filte. Hence the smoothed images will contain moe bluing than oiginally intended and the effective smoothing will be geate than the nominally applied smoothing. The tuncation atifacts can be effectively contolled by applying a tapeed filte function diectly to the k-space data. Thee ae a numbe of ways of tapeing the filte function. Fo example one can apply a Hamming filte to the tuncated Gaussian, theeby eliminating its steep edges. Howeve, in this wok we intoduce anothe tapeed filte function, based on the theoy of the PSWF, which can be used to contol the amount of inging in the esulting smoothed images. The PSWF has simila popeties to the Gaussian, with the added benefit that it povides optimal contol ove tuncation atifacts. The PSWF filte has an additional benefit that has not peviously been discussed. Thoughout this aticle we have defined the spatial coveage egion of 1285

11 Lindquist and Wage ou filte, given by, to be a cicula egion in ode to obtain an isotopic filte compaable to a Gaussian. Howeve, it is impotant to note that thee is nothing in the theoy of the PSWF that limits us to constucting isotopic filtes, as canbechosentotakeotheshapesaswell.fo example, we may want to choose to be an oval o some othe blob-like shape and a filte with spatial coveage coesponding to this egion can be constucted using the PSWF methodology (Lindquist et al., 2006; Yang et al., 2002). In pincipal, the amount of smoothing applied is detemined by the expected extent of activation. y the matched filte theoem, optimal filteing is povided by using a filte with the same spatial extent as the activation. Of couse this egion is neve known exactly, making it difficult to detemine the optimal amount of smoothing in pactice. Howeve, it is easonable to assume that the activation patten takes place in a coheent blob in the bain and it would be difficult to ague that the patten would be ippled in the manne that would be pesent afte the application of a naow Gaussian filte. We theefoe feel that the PSWF is moe in the spiit of the matched filte theoem than the Gaussian. Howeve, with the intoduction of high-field scannes and state-of-the-at MR imaging techniques, the poblematic spatial esolutions unde discussion in this wok (e.g., matix 3.75 mm esolution) may eventually become obsolete. In these situations the advantages of the PSWF filte ove the Gaussian ae less clea. On the othe hand, in these situations the ational fo smoothing at all is questionable. It should be noted that the Convolution theoem (illustated in Fig. 2) only holds when the filte is applied diectly to the complex image-space data and most eseaches instead wok with magnitude images. Howeve, since the PSWF filte is positive and eal-valued in ou implementation (like the Gaussian), it does not matte whethe we apply the filte to the complex image-space data o to the positive magnitude images. The esults will be equivalent egadless and the Convolution theoem is still valid. The implementation of the filte is staightfowad and Matlab code is available fom the authos ( Thee is a one-time cost in calculating the filte coesponding to a specific width, FOV and matix size. This calculation involves calculating the lagest eigenfunction of an a 3 a matix. Howeve, if we want an n-dimensional smoothing kenel we can appoximate them as the sepaable poduct of n one-dimensional 0-ode PSWFs. This significantly simplifies calculation with minimal loss of efficiency. Simulation studies showed an incease in powe compaed to Gaussian smoothing with a small FWHM (<8 mm). In addition, the expeimental data pesented in this aticle clealy showed the benefits of using a PSWF filte. In geneal, we feel that the incease in powe shown in both simulations, in conjunction with the inceased numbe of activated voxels in the expeimental data, togethe make a stong case fo the PSWF methods supeioity when naow kenels (<8 mm) ae equied fo smoothing fmri data. These esults ae consistent with the theoy, which states that fo naow filtes the PSWF has supeio concentation popeties compaed to the Gaussian. y applying the PSWF filte we ae able to incease the signal-to-noise by maintaining contol ove tuncation effects. While SNR is one impotant issue elating to spatial smoothing, also impotant ae the statistical issues of intesubject vaiability and statistical validity. When using a tapeed filte such as the PSWF these issues can only be positively impacted, as a eduction of the inging effects can only be good fo the subsequent statistical analysis, as it allows fo constuction of a kenel that bette matches the extent of activation. Finally, we feel that it is easonable to assume that the PSWF filte is intechangeable to the Gaussian in the sense that both can be used to povide the appopiate amount of smoothness needed to make the assumptions of the andom field theoy valid. In detemining the appopiate filte width equied fo valid infeence, we suggest using the same ule-of-thumb typically used fo a Gaussian kenel (e.g., 2 3 voxels). CONCLUSION In this wok a new spatial smoothing filte fo fmri is intoduced. The filte is based on the use of PSWFs and povides optimal contol of tuncation atifacts pesent in MR images. The PSWF filte has the impotant popety that any othe choice of filte will give ise to a geate amount of signal leakage outside of the width of the filte. This popety leads us to believe that it has excellent potential as a spatial smoothing filte in fmri. Simulation studies showed a significant incease in powe compaed to the taditionally used Gaussian filte when smoothing with naow filtes (e.g., FWHM < 8 mm). Expeimental data fom a visual paadigm showed when smoothing with naow filtes the PSWF filte gave ise to a significant ise in the numbe of active voxels in the visual cotex compaed to data smoothed withacompaablegaussianfilte.thisiseisattibutable to an incease in SNR. REFERENCES Cohen J (1988): Statistical Powe Analysis fo the ehavioal Sciences, 2nd ed., Hillsdale, NJ: Elbaum. Fadili MJ, ullmoe ET (2004): A compaative evaluation of wavelet-based methods fo hypothesis testing of bain activation maps. NeuoImage 23: Kim HY, Giacomantone J, Cho H (2005): Robust anisotopic diffusion to poduce enhanced statistical paametic map fom noisy fmri. Comput Vis Image Undest 99: Landau HJ, Pollak HO (1961): Polate spheoidal wave functions, fouie analysis and uncetainty, II. ell Syst Tech J 40: Landau HJ, Pollak HO (1962): Polate spheoidal wave functions, fouie analysis and uncetainty, III. ell SystTech J, 41: Lindquist MA (2003): Optimal data acquisition in fmri using polate spheoidal wave functions. Int J Imag Syst Technol 13:

12 Spatial Smoothing Using PSWF Lindquist MA, hang C-H, Glove G, Shepp L, Yang QA (2006): Genealization of the two dimensional polate spheoidal wave function method fo non-ectilinea MRI data acquisition methods. IEEE Tans Image Pocess 15: Mita PP, Pesaan (1999): Analysis of dynamic bain imaging data. iophys J 76: Mita PP, Ogawa S, Hu X, Ugubil K (1997): The natue of spatiotempoal changes in ceebal hemodynamics as manifested in functional magnetic esonance imaging. Magn Reson Med 37: Noll DC, Cohen JD, Meye CH, Schneide W (1995): Spial K- space MR imaging of cotical activation. J Magn Reson Imaging 5: Oppenheim AV, Schafe RW, uck JR (1999). Discete-Time Signal Pocessing, 2nd ed. Uppe Saddle Rive, NJ: Pentice Hall. Pecival D, Walden A (1993): Spectal Analysis fo Physical Applications. Cambidge, UK: Cambidge Univesity Pess. Poline J, Mazoye (1994): Analysis of individual bain activation maps using hieachical desciption and multiscale detection. IEEE Tans Med Imag 4: Rosenfeld A, Kak AC (1982):Digital Pictue Pocessing, Vol. 2., Olando, Floida: Academic Pess. Shafie K, Sigal, Siegmund D, Wosley K (2003): Rotation space andom fields with an application to fmri data. Ann Stat 31: Shepp L, hang CH (2000): Fast functional magnetic esonance imaging via polate wavelets. Appl Comput Hamonic Anal 9: Slepian D, Pollak HO (1961): Polate spheoidal wave functions, fouie analysis and uncetainty, I. ell Syst Tech J 40: Smith SM (2003):Pepaing fmri data fo statistical analysis. In:Jezzad P, Matthews PM,Smith SM, editos. Functional MRI: An Intoduction to Methods. Oxfod, UK: Oxfod Univesity Pess. Van De Ville D, lu T, Unse M (2003):Wavelets Vesus Resels in the Context of fmri: Establishing the Link with SPM. In the Poceedings of the SPIE Confeence on Mathematical Imaging: Wavelet Applications in Signal and Image Pocessing X, San Diego, CA, 5207: Van De Ville D, lu T, Unse M (2006): Sufing the bain-an oveview of wavelet-based techniques fo fmri data analysis. IEEE Eng Med iol Mag 25: Woods RP, Gafton ST, Holmes CJ, Chey SR, Mazziotta JC (1998): Automated image egistation. I. Geneal methods and intasubject, intamodality validation. J Comput Assist Tomog 22: Wosley K, Maett S, Neelin P, Evans A (1996): Seaching scale space fo activation in PET images. Hum ain Mapp 4: Wosley KJ, Fiston KJ (1995): Analysis of fmri time-seies evisited Again. Neuoimage 2: Yang QX, Lindquist MA, Shepp L, hang CH, Wang J, Smith M (2002): Two dimensional polate spheoidal wave functions fo MRI. J Magn Reson 158: APPENDIX Fo a given A and, the polate spheoidal wave function, g(k), is obtained by finding the function whose invese Fouie tansfom maximizes the atio in Eq. (7). With the pope nomalization of G(x) we can assume that jgðxþj 2 dx ¼ 1: ða1þ 1 The poblem then becomes finding the solution to the following equation, subject to the above constaint on G(x): k ¼ max jgðxþj 2 dx: ða2þ We can ewite the integal on the ight hand side of Eq. (A2) as follows: jgðxþj 2 dx ¼ GðxÞGðxÞ dx whee X X ¼ >: gðkþ e i2pðx;kþ >; >: gðlþ e i2pðx;lþ >; dx k2a l2a 8 9 ¼ X X gðkþgðlþ e i2pðx;k lþ dx >: >; k2a l2a ¼ X X gðkþgðlþ^k ðk; lþ k2a l2a ^K ðk; lþ ¼ e i2pðx;k lþ dx ða3þ ða4þ Combining Eqs. (A2) and (A3), the poblem now becomes solving the equation ( ) X k ¼ max gðkþgðlþ^k ðk; lþ ða5þ o, altenatively, X k2a l2a n o k ¼ max g þ ^K g ða6þ whee Kˆ is an a 3 a matix with elements given by Kˆ (k,l) fo k,l [ A. It is well known that the solution to this poblem is the lagest eigenfunction of the matix Kˆ. This gives us the 0-ode polate spheoidal wave function filte. One can constuct a smoothing kenel by taking the IDFT of g(k) and econstucting it onto a matix with the same dimensions as the image. Smoothing can theeafte be pefomed in an analogous manne as fo Gaussian, which is descibed algebaically in Eq. (1). 1287