Improving tissue segmentation of human brain MRI through preprocessing by the Gegenbauer reconstruction method

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1 NeuroIage 20 (2003) Iproving tissue segentation of huan brain MRI through preprocessing by the Gegenbauer reconstruction ethod Rick Archibald, a, * Kewei Chen, b Anne Gelb, c and Roseary Renaut c a The Center for Syste Science and Engineering Research, Arizona State University, Tepe, AZ , USA b Saaritan PET Center, Good Saaritan Regional Medical Center, E. McDowell Road, Phoenix, AZ 85006, USA c Departent of Matheatics and Statistics, Arizona State University, Tepe, AZ , USA Received 5 Septeber 2002; revised 24 March 2003; accepted 24 April 2003 Abstract The Gegenbauer iage reconstruction ethod, previously shown to iprove the quality of agnetic resonance iages, is utilized in this study as a segentation preprocessing step. It is deonstrated that, for all siulated and real agnetic resonance iages used in this study, the Gegenbauer reconstruction ethod iproves the accuracy of segentation. Although it is ore desirable to use the k-space data for the Gegenbauer reconstruction ethod, only inforation acquired fro MR iages is necessary for the reconstruction, aking the procedure copletely self-contained and viable for all huan brain segentation algoriths Elsevier Inc. All rights reserved. Keywords: Segentation; Brain extraction; Edge detection; Gegenbauer reconstruction; Noise. Introduction * Corresponding author. E-ail address: archi@ath.la.asu.edu (R. Archibald). Magnetic resonance iaging (MRI) is a noninvasive procedure that has proven to be an effective tool in the study of the huan brain. The inforation that MRI provides has greatly increased knowledge of noral and diseased anatoy for edical research and is a critical coponent in diagnosis and treatent planning. Iage segentation algoriths for the delineation of 3D anatoical structures, or tissue types, play an iportant role in nuerous research and clinical studies, involving the visualization and quantitative analyzes of anatoical and functional cortical structures (Worth et al., 997; Dale et al., 999; Van Essen et al., 999; Joshi et al., 999). Guided neural surgery using segented agnetic resonance iages has grown in popularity in treating brain-related diseases such as epilepsy and stroke (Khoo et al., 997; Roux et al., 997; Le Bihan, 2000; Taylor, 995). Functional visualization, in which a segented brain can provide an anatoical fraework, has eerged as a proising approach in neuroscience research and in neurosurgical planning owing to the advanceent of brain function studies using functional MRI (fmri) (Kwong et al., 992; Ogawa et al., 992; Liu et al., 2000; Logothetis et al., 200) and positron eission toography (PET) (Chen et al., 200). The segentation of brain iages is also widely used in cortical surface apping, volue easureent, tissue classification and differentiation, functional and orphological adaptation assessent, and characterization of neurological disorders such as ultiple sclerosis, stroke, and Alzheier s disease (Rusinek et al., 99; Suckling et al., 999; Le Goualher, 2000). Moreover, brain segentation is also a required preliinary step for any other iage-processing procedures, such as brain registration, warping (Leieux, 998; Ghanei et al., 2000; Joshi et al., 997), and voxel-based orphoetry (Ashburner and Friston, 2000). The fact that any applications depend on accurate brain segentation has inspired uch work for its iproveent. The focus of this study is to iprove segentation through the use of the Gegenbauer reconstruction ethod as a preprocessing step. The Gegenbauer reconstruction ethod is a high-resolution ethod, introduced by Gottlieb et al. (992), and established as an effective reconstruction /03/$ see front atter 2003 Elsevier Inc. All rights reserved. doi:0.06/s053-89(03)00260-x

2 490 R. Archibald et al. / NeuroIage 20 (2003) ethod for agnetic resonance iaging by Archibald and Gelb (2002a). Using siulated brain phanto data (Brainweb, 2003) and data fro real patients, this study shows that the Gegenbauer reconstruction ethod iproves the quality of the MRI data and the subsequent brain tissue aps generated by SPM (SPM Website, 2003) segentation algorith. 2. Methods 2. Segentation preprocessing using the Gegenbauer reconstruction ethod The Gegenbauer reconstruction ethod, introduced by Gottlieb et al. (992), is a high-resolution iage reconstruction technique which is capable of reconstructing iages with exponential accuracy, including the edges of structures in the iage, without blurring features, hence itigating a coon proble associated with iage filtering. Since the locations of the edges are needed to deterine the regions of soothness in which the iages can be reconstructed (Gottlieb and Shu, 997), a critical first step in all high-resolution reconstruction is edge detection. Edge detection is achieved by a cobination of the edge detection procedure designed by Gelb and Tador (999), (2000a), with the iniization procedure introduced by Archibald and Gelb (2002a). The edge detection and Gegenbauer reconstruction ethods are briefly outlined in the Appendix under sections 6. and 6.2. As described under these sections, the eployent of the FFT algorith for the both edge detection and the Gegenbauer reconstruction ethod ensures that the speed of coputation is of the order of conventional FFT iage reconstruction. 2.2 Windowed reconstructed data Given the Fourier reconstruction of an iage it is possible to deterine the edges via the concentration ethod (Gelb and Tador, 999). Once the edges are known, highresolution iage reconstruction of the entire iage is achieved by using the Gegenbauer reconstruction ethod in each sooth region. Typically the edge detection and Gegenbauer reconstruction procedures require knowledge of the original Fourier coefficients (or k space data), which are ibedded (Liang and Lauterbur, 2000) in RF signal collected during a MR acquisition. However, often what is available to the researchers/physicians are the windowed (or filtered) Fourier reconstruction data, i.e., the conventional MRI data. Although less desirable, this still provides sufficient inforation for both the edge detection and Gegenbauer reconstruction procedures. Proofs of the previous stateent are given in the Appendix in order to deonstrate the broad application of the Gegenbauer reconstruction ethod to MR iaging segentation. This study uses only windowed reconstructed data in order to deonstrate the effectiveness of the edge detection and Gegenbauer reconstruction procedures on conventional MRI data. 2.3 Segentation Segentation was perfored using the well-known Statistical Paraetric Mapping (SPM) progra developed by the ethodology group at the Wellcoe Departent of Cognitive Neurology (SPM Website, 2003). Statistical Paraetric Mapping is publically available software used to test hypotheses about [neuro]iaging data fro single photon eission coputed toography (SPECT), PET, and fmri. Of particular interest to this study is the segentation progra bundled in this software. SPM segents spatially noralized iages into gray atter (GM), white atter (WM), and cerebrospinal fluid (CSF), using a odified ixture odel cluster analysis technique (Ashburner and Friston, 997), and is extended to include a correction for iage intensity nonunifority (Ashburner and Friston, 2000) that arises fro various sources in MR iaging. The version of SPM used in this study was SPM2b. The default settings were used in segentation and brain extraction. This version, copared to previous versions, has iproved segentation and brain extraction utilities, for which details are outlined on the SPM website (SPM Website, 2003). We note that other segentation algoriths (Liang et al., 994; Schroeter et al., 998; Aizenberg et al., 998; Wu et al., 998) when used in conjunction with Gegenbauer reconstruction as a preprocessing step, including SPM99, produce results siilar to those presented in this study (Archibald, 2002). Therefore we conclude that the accuracy of general segentation is iproved by using the Gegenbauer ethod as a presegentation step. 3. Experiental ethods 3. Data acquisition: MNI digital brain phanto The effect of the Gegenbauer reconstruction ethod on the results of brain tissue segentation was evaluated for siulated MRI scans (Brainweb, 2003) of a MNI digital brain phanto (Kwan et al., 996; Cocosco et al., 997; Collins et al., 998). Brainweb is aintained by the Brain Iaging Center at the Montreal Neurological Institute. The siulated MRI data were generated with noise levels of 5, 7, and 9%, with intensity non-unifority (INU) of 20 and 40%. Ten iages were generated for each cobination of noise and intensity non-unifority, for a total of 60 rando iages generated by Brainweb (2003). Each generated iage siulated a T-weighted single channel MRI scan using the SFLASH (spoiled FLASH) pulse sequence collected in the transverse direction. The paraeters for each generated iage were as follows: TR, 8 s; TE, 0 s; flip angle, 30 ; FOV, ; nuber of slices (contig-

3 uous), 256; slice thickness, ; in-slice resolution. 3.2 Data acquisition: real data We used real MRI data fro six consenting noral healthy subjects. Their MR data were acquired using a.5 T Signa syste (General Electric, Milwaukee, WI) and T-weighted, three-diensional pulse sequence (radio frequency-spoiled gradient recall acquisition in the steady state (SPGR), repetition tie, 33 s; echo tie, 5 s;, 30 ; nuber of excitations, ; field of view, 24 c; iaging atrix, ; slice thickness,.5 ; scan tie, 3:36 in). The MR data set consisted of 24 contiguous horizontal slices with an in-plane voxel diension of 0.94 by Validation: MNI digital brain phanto The probability aps for GM and WM for each MNI digital brain phanto that is generated by SPM segentation are copared to the corresponding true tissue probability ap using the average absolute value nor over all voxels, where the nor is defined for an arbitrary iage A N x N y N z as A N x N x N y N z i x N y i y N z a ix,iy,iz. () i z An error easureent is calculated for a given segented tissue probability ap, P S, and the Brainweb truth tissue probability ap P T as EP T, P S P T P S. (2) Note that in addition to tissue probability aps, Brainweb also provides a discrete tissue index ap. Using the tissue index ap as ground truth is ost appropriate in the evaluation of binary segentation. This study analyzes the probability aps generated by SPM, where it is appropriate to use Brainweb s tissue probability aps as ground truth. The index (Dice coefficient) is a paraeter that has frequently been reported in the literature (Collins et al., 999; Van Leeput et al., 999; Shattuck et al., 200) to evaluate the siilarity of two iages. Given two binary iages, S and S 2, the index is calculated as S, S 2 2S S 2 S S 2. (3) Since probability aps are not binary iages, a threshold is applied in order to use the index as a ethod to copare the SPM segented tissue aps to ground truth. For a given segented tissue probability ap, P S, and the Brainweb truth tissue probability ap, P T, two binary iages that represent the body of the tissue are generated by setting each pixel, p, of both probability aps to a new threshold value, s tb,as R. Archibald et al. / NeuroIage 20 (2003) , if p 0.95 s tb 0, else. (4) For these two binary iages the index (3) is calculated, which, in this case, easures how well the segented tissue probability ap approxiates the body of the tissue. Siilarly for a given segented tissue probability ap, P S, and the Brainweb truth tissue probability ap, P T, two different binary iages that represent the boundary or partial volue region are generated by setting each pixel, p, of both probability aps to a new threshold value, s pv,as, if 0.05 p 0.95 s pv 0, else. (5) The index (3) is calculated for these two binary iages and in this case easures how well the segented tissue probability ap approxiates the partial volue region of the tissue. The thresholds defined above allow coparisons of the core and partial value regions of the gray/white tissue atter ore specifically. Therefore, we refer to this ethod as the local easureent. Different thresholds, other than 0.95 and 0.05, have been applied yielding siilar results. 3.4 Validation: real data Since ground truth tissue aps are not available for real MR data, the quality of the SPM segented tissue probability aps via Gegenbauer reconstruction preprocessing was assessed by visually inspecting the quality of the extracted brain cortical surface. The SPM brain extraction utility relies exclusively on segented data to estiate the brain cortical surface. It is assued in this study that visually apparent artifacts in brain extraction, which are reduced through the utilization of the Gegenbauer reconstruction ethod as a presegentation step, deonstrate the ability of this ethod not only to iprove brain extraction but also to iprove segentation. Thus the Gegenbauer reconstruction ethod as a presegentation step is validated for real data in this study by analyzing visually apparent brain extraction artifacts. 4. Results 4. Global error easureents Table displays the average tissue probability ap error easureent (2) for each cobination of noise and intensity non-unifority for both gray atter and white atter tissue. Using the Gegenbauer reconstruction ethod as a segentation preprocessing step reduces the error easureent for each cobination of noise and intensity nonunifority for both tissue types. As the level of noise increases, so does the average aount of error reduction between the original reconstruction and the Gegenbauer

4 492 R. Archibald et al. / NeuroIage 20 (2003) Table Tissue probability ap error easureent (2) statistics for randoly generated MNI digital brain phantos Noise (%) INU (%) Tissue Gray atter White atter Original Gegenbauer Original Gegenbauer Coparison between each segented probability ap for original and Gegenbauer iage reconstruction. All values are ultiplied by 0 2 and are written in the forat of ean standard deviation. reconstruction. The segentation iproveent using Gegenbauer reconstruction is statistically significant for each noise level and inhoogenity level cobination (P 0.05 paired t test). The P values decreased with the level of noise and are of the order 0 5 for the 9% noise level. Depicted in Figs. and 2 are gray atter and white atter segented probability aps for the original and Gegenbauer iage reconstruction of a particular randoly generated MNI digital brain phanto with a 9% level of noise and 40% intensity non-unifority. Visually it can be seen that there is a difference in the accuracy of the probability aps when Gegenbauer reconstruction is used as a segentation preprocessing step. 4.2 Local error easureents Table 2 displays the average tissue probability ap error easureent (2) for each cobination of noise and intensity non-unifority for both gray atter and white atter tissue, where each probability ap is restricted to its partial volue region, which is defined in (5). The results irror Table in the sense that it is evident that by using the Gegenbauer reconstruction ethod as a segentation preprocessing step, the error easureent is reduced for each cobination of noise and intensity non-unifority for both tissue types. Siilarly, it is noticed that as the level of noise increases, the average aount of error reduction between Fig.. Gray atter segented probability aps for the original and Gegenbauer iage reconstruction of a particular randoly generated MNI digital brain phanto with a 9% level of noise and 40% intensity non-unifority.

5 R. Archibald et al. / NeuroIage 20 (2003) Fig. 2. White atter segented probability aps for the original and Gegenbauer iage reconstruction of a particular randoly generated MNI digital brain phanto with a 9% level of noise and 40% intensity non-unifority. the original reconstruction and the Gegenbauer reconstruction for the partial volue region grows. The segentation iproveent using Gegenbauer reconstruction is statistically significant for each noise level and inhoogenity level cobination (P 0.0 paired t test). The P values decreased with the level of noise and are of the order 0 6 for the 9% noise level Siilarity easureent In order to deterine the index (3), binary iages ust be generated fro each segented probability ap. The tissue body region (4) and partial volue region (5) are the generated binary iages used to calculate the siilarity coefficients. Depicted in Figs. 3 and 4 are gray atter and white atter partial volue region binary iages of the segented probability aps for the original and Gegenbauer iage reconstruction of a particular randoly generated MNI digital brain phanto with a 9% level of noise and 40% intensity non-unifority. There is a visual difference in the accuracy of the binary iages of the probability aps when Gegenbauer reconstruction is used as a segentation preprocessing step. Table 3 displays the index for the partial volue region binary iages generated fro both gray atter and white atter segented probability aps. Siilarly, Table 4 displays the index for the tissue body region binary iages Table 2 Tissue probability ap restricted to the partial volue region (5) error easureent (2) statistics for randoly generated MNI digital brain phantos Noise (%) INU (%) Tissue Gray atter White atter Original Gegenbauer Original Gegenbauer Coparison between each segented probability ap restricted to the partial volue region (5) for original and Gegenbauer iage reconstruction. All values are ultiplied by 0 2 and are written in the forat of ean standard deviation.

6 494 R. Archibald et al. / NeuroIage 20 (2003) Fig. 3. Gray atter partial volue region binary iage (5) generated fro the segented probability aps for the original and Gegenbauer iage reconstruction of a particular randoly generated MNI digital brain phanto with a 9% level of noise and 40% intensity non-unifority. Fig. 4. White atter partial volue region binary iage (5) generated fro the segented probability aps for the original and Gegenbauer iage reconstruction of a particular randoly generated MNI digital brain phanto with a 9% level of noise and 40% intensity non-unifority.

7 R. Archibald et al. / NeuroIage 20 (2003) Table 3 The index (3) statistics for tissue partial volue region binary iages (5) generated fro tissue probability aps of MNI digital brain phantos Noise (%) INU (%) Tissue Gray White Original Gegenbauer Original Gegenbauer Coparison between tissue partial volue region binary iages (5) generated fro the segented probability ap for original and Gegenbauer iage reconstruction. All values are written in the forat of ean standard deviation. generated fro both gray atter and white atter segented probability aps. Typically it is considered that 0.7 indicates excellent agreent (Bartko, 99) in binary iages. It is noted that in any instances, particularly for white atter tissue, the use of the Gegenbauer ethod as a preprocessing step iproves the index beyond the critical value of 0.7, establishing that this preprocessing step has the ability to iprove siilarity between the segented tissue probability aps and ground truth so that it is possible to conclude excellent agreent. The index also has erit in the fact that it provides a value that can be used to copare the siilarities between two easureent pairs (Zijdenbos et al., 994), which is the priary reason it is used in this study. It can been seen that there is significant iproveent in the index coefficient for each tissue type and for all cobinations of noise and intensity non-unifority. Furtherore, the iproveent of both siilarity coefficients between the original and Gegenbauer reconstruction increases with the noise level. The iproveent is even ore draatic for the siilarity coefficients than for the error easureents. Using a Wilcoxon signed rank test, the significance level is P 0.0 for every cobination of noise and intensity non-unifority. 4.3 Real data evaluation It has been reported that artifacts in brain extraction include nonbrain tissue such as scalp in the final extracted iage. In our study, we find that four of six extractions presented this artifact, with severity varying fro sub ject to subject. Fig. 5 displays the results of SPM brain extraction of one particular subject, where the view on the right utilized the Gegenbauer reconstruction ethod as a presegentation step. It is deonstrated that all visually apparent artifacts are reduced by using the Gegenbauer ethod. Siilar results were obtained with all subjects in this study with visually apparent artifacts. Additionally for all six subjects, no new visually apparent artifacts were generated. 5. Discussion and conclusion As is deonstrated in section 4, the use of the Gegenbauer reconstruction ethod as a segentation preprocessing step significantly iproves the quality of the probability aps generated fro SPM segentation for both gray and white tissue. This iproveent can be understood in ters of the differences in filtered Fourier reconstruction, which is Table 4 The index (3) statistics for tissue body binary iages (4) generated fro tissue probability aps of MNI digital brain phantos Noise (%) INU (%) Tissue Gray White Original Gegenbauer Original Gegenbauer Coparison between tissue body binary iages (4) generated fro the segented probability ap for original and Gegenbauer iage reconstruction. All values are written in the forat of ean standard deviation.

8 496 R. Archibald et al. / NeuroIage 20 (2003) Fig. 5. SPM brain extraction of one particular subject, where the picture on the right used the Gegenbauer reconstruction ethod as a presegentation step. the traditional ethod of reconstruction for MRI, and the Gegenbauer reconstruction. The filtered Fourier ethod is the iage reconstruction technique used for the real and siulated data in this study. Filters are often introduced to dapen the high-frequency odes which reduce the effects of Gibbs ringing artifact and noise. Filtering is a violation of the data consistency constraint, but is tolerated in order to reduce the Gibbs ringing artifact (Liang and Lauterbur, 2000) and suppress noise. There is iportant inforation that is carried in the high-frequency odes that pertain to finer features of the reconstructed iage. The Gegenbauer reconstruction ethod, introduced by Gottlieb et al. (992), does not violate the data consistency constraint and is capable of reconstructing piecewise sooth functions in sooth intervals with exponential accuracy up to the edges of the interval without blurring the features at the boundaries. This is reflected in section 4.2, in the iproveent of both the tissue partial volue region and the tissue body region. The Gegenbauer reconstruction ethod s ability to reconstruct sharp boundaries is suggestive of the iniization of visual artifacts in brain extraction in section 4.3. Noise prevalent in MRI spectral data is an ipedient to iage reconstruction. This noise is difficult to quantify and cannot be systeatically reoved. A preliinary investigation was conducted by Archibald and Gelb (2002b) to address the effects of noisy spectral data on the ability of the concentration ethod to locate edges and the Gegenbauer ethod to reconstruct iages. As deonstrated throughout section 4, the application of the edge detection and highresolution reconstruction ethods discussed in this paper not only recovers iages with very high accuracy, but is also robust in the presence of noise and additionally reduces the effects of noise. Fig. 6 depicts the ethod for ipleenting Gegenbauer reconstruction as a presegentation step. In order not to violate the data consistency constraint, k-space data should be used for both edge detection and Gegenbauer reconstruction prior to segentation. Often it is the case that only reconstructed data are available; thus this study deonstrates the viability of using reconstructed data for edge detection and Gegenbauer reconstruction to iprove the results of segentation. This research opens new directions in the application of high-resolution techniques in the field of MRI. Acknowledgents Fig. 6. Flow chart of the Gegenbauer reconstruction ethod as a segentation preprocessing step. The authors of this work were supported in part by the Arizona Center for Alzheier s Disease Research (R. Archibald, K. Chen, A. Gelb, and R. Renaut), the Center for Syste Science and Engineering Research at Arizona State University (R. Archibald and A. Gelb), the Departent of

9 R. Archibald et al. / NeuroIage 20 (2003) Matheatics and Statistics at Arizona State University (R. Archibald, K. Chen, A. Gelb, and R. Renaut), the John von Neuann visiting Professorship of the Zentru Matheatik, Technische Universitaet Muenchen (R. Renaut), NSF Grant DMS (R. Renaut), and NSF Grant DMS (A. Gelb). 6. Appendix Section 2. discuses the boundary detection and highresolution reconstruction ethod used in this paper, which are described in greater detail below in sections 6. and 6.2. Both the edge detection and reconstruction ethod require the knowledge of k-space data. Often, as is the case with this study, what is known are the windowed reconstructed data, and therefore it is proven in section 6.3 that using the windowed reconstructed data will only inially affect the results of the Gegenbauer reconstruction ethod. However, we stress that knowledge of the unfiltered k-space data is preferable in order to satisfy the data consistency constraint (Liang and Lauterbur, 2000). 6. Edge detection The concentration edge detection ethod is first described for one diension. Define the jup function [f](x) for a piecewise sooth function as [f](x) f (x) f (x), where f (x) are the right and left side liits of the function at x, f (x) li x3x f(x). Note that [f](x) is 0 away fro a discontinuity and is the value of the jup at a discontinuity. It is shown in Gelb and Tadon (999) that the concentration ethod for detecting jup discontinuities is easily ipleented in the discrete case as N T N fx : i kn sgnk k N 2 f ke ikx 3 fx, as N 3. (6) Here f k are the discrete Fourier coefficients coputed using the fast Fourier transfor (FFT) algorith fro given discrete data f(xj), xj j/n, j 0,..., 2N, as f k 2N 2Nc k j0 f x j e ikxj, c k 2, if k N,, otherwise, and () is deterined in Gelb and Tador (999) as (7) 2 sin, (8) where () is a continuous concentration factor that satisfies C 2 0, and noralized so that 0 d. There are several exaples of adissible concentration factors discussed in Gelb and Tador (999). The exponential concentration factor, where ce, (9) c exp (0) d is particularly effective, as it takes full advantage of the spectral data by rapidly converging away fro the discontinuities. The paraeter is freely chosen, with a typical value 6. We note that when the data given are k-space data, the edge detection ethod (6) can be directly applied and easily calculated with the FFT algorith. In order to deterine the exact intervals of soothness, which is iperative for high-resolution reconstruction, the concentration ethod (6) ust be further enhanced to pinpoint the edges exactly. For this purpose, an edge enhanceent procedure based on a separation of scales has been constructed by Gelb and Tador (2000b), E N T N fx T N fx, if N q/2 T N fx q J crit, 0, if N q/2 T N fx q J crit. () Here J crit is an O() global threshold paraeter signifying the inial aplitude for the jup discontinuity not to be negligible. Since (6) actually locates the neighborhoods of the discontinuities, the exact jup locations are deterined as the corresponding locations of the largest aplitudes E J crit in each neighborhood of adissible jups (i.e., where E J crit ). Note that J crit should be chosen to be consistent with the variation and scaling of the function. Experients show that the paraeter q 2 is adequate for enhanceent. The nonlinear enhanceent procedure (Gelb and Tador, 2000) works well when the discontinuities are located far enough fro each other. (Experientally, the neighborhood of a discontinuity is approxiately 5x.) When the discontinuities are extreely close together, as is the case for MRI scans, where two neighboring pixels ay contain discontinuities, a ore refined procedure near the locations of the discontinuities ust be eployed. We adopt a iniization procedure (Archibald and Gelb, 2002a) given by

10 498 R. Archibald et al. / NeuroIage 20 (2003) where and in M,ai,b i ax x T N hx : in M,ai,b i ax x T N fx M a i 2 T N gx; b i, (2) h x : f x M a i 2 g x; b i (3) i g x; b i x, if x b i, x, if b i x. (4) Since h(x) is a sooth function, T N [h](x) 3 0, and therefore the correct iniization of (2) yields the nuber of discontinuities, M, with the associated positions, b i, and agnitudes, a i, for i,...,m, of the function f. The edge detection ethod and iniization process can be extended to detect the size and position of discontinuities of a ultiple diensional function by holding all but one diension fixed and deterining the edges as a function of the fixed coordinates. This three-diensional procedure is used for all iages processed in this paper. The eployent of the FFT algorith for the boundary detection procedure ensures that the speed of coputation is of the order of the three-diensional FFT. i 6.2 Gegenbauer reconstruction ethod The Gegenbauer reconstruction ethod was developed by Gottlieb et al. (992) and extended in a litany of articles (consult Gottlieb and Shu (997) for references). It is a powerful tool that recovers piecewise sooth functions with spectral accuracy up to the edges in each sooth interval and can therefore be used to copletely eliinate the Gibbs ringing artifact without coproising high resolution at the edges. Let us first introduce the Gegenbauer reconstruction ethod for a one-diensional piecewise sooth function f(x). The Gegenbauer reconstruction ethod is perfored on each sooth interval [a, b] [, ]. Specifically, define a local variable [, ] such that x(), where b a/2 and b a/2. The Gegenbauer reconstruction is then based on the Fourier approxiation of f(x) in[a, b], N f N x f N f ke ik, (5) kn where f k is defined in (7) and is coputed by g x g l C l, (6) l0 where the Gegenbauer polynoial C n (x) is an orthogonal polynoial of order n that satisfies x 2 2C k xc n xdx h n, k n, 0, k n, where (for 0) with h n C n C n 2 n (7), (8) n 2 n!2. (9) The approxiate Gegenbauer coefficients, g l h l are coputed by the FFT algorith as where 2 2f N xc l d, (20) kn g l c,l, k e ik f k, (2) kn if k 0, c,l, k, i l l J l k 2, if k 0. k (22) Here J l (k) is the Bessel function. It is noted here that soe sooth intervals ay consist of too few points to construct an approxiation. The Gegenbauer reconstruction requires at least a theoretical iniu of points to for an approxiation (Gottlieb and Orszag, 977). Therefore, in these intervals, the values at each grid point are assued constant and equivalent to the values deterined at the edges by the edge detection ethod (Archibald and Gelb, 2002a). The paraeters and depend upon the nuber of points, N I, in the subinterval that is reconstructed. A specific requireent is that N I. Recent work deonstrates how the paraeters and can be optiized for a particular subdoain (Gelb, 2003). For siplicity, we choose the paraeters such that with ax, in ax, N I 4, (23) where [N I /4] is the nearest integer to N I /4, and ax 2. The Gegenbauer reconstruction ethod can be directly extended to ultiple diensions by perforing reconstruc-

11 R. Archibald et al. / NeuroIage 20 (2003) tion in sooth regions. The reconstruction will have exponential accuracy up to the edges of each sooth region. Three-diensional Gegenbauer reconstruction is used for all the iages processed in this paper. The coputational cost of the Gegenbauer reconstruction ethod is of the order of the three-diensional FFT. We note that different values can be chosen for and in each diension, which can be optiized by the size of the particular subdoain as deonstrated in Gelb (2003). However, for siplicity we adopt the paraeters for each diension according to (23). Convergence analysis of the Gegenbauer reconstruction ethod for the one-diensional case is provided in the next section. 6.3 Windowed transfored data in edge detection One factor contributing to the Gibbs ringing artifact for piecewise sooth functions is the slow decay rate of the Fourier expansion coefficients (Gottlieb and Orszag, 977). A window (or filter) increases the rate of decay by attenuating the higher order coefficients, which in turn controls the effects of the Gibbs oscillations. The windowed discrete Fourier reconstruction is coputed as N f w N x w k f ke ikx, (24) kn where w k w(k/n) is a window function. Given reconstructed data on equally spaced points the inverse transfor will result in the spectral data f kw w k f k, (25) where w is the widow function used in reconstruction. An edge detection procedure designed by Gelb and Tador (999, 2000a), and briefly outlined in section 6. uses Fourier coefficients to deterine boundaries. In this study, the k-space data are not available but rather the windowed k-space data are known (25). Theore 6. proves that the inforation required for high-resolution reconstruction, naely, the position of edges, is preserved if windowed k-space data are used in the edge detection procedure provided that the window function satisfies two basic requireents. The first requireent is that the window function, w, is w C 2 0,. This is the case with all coonly used widowed functions used to eliinate the Gibbs ringing artifact (Gottlieb and Shu, 997). The second requireent is that 0 w d 0, where is the concentration factor used in the edge detection ethod. The flexibility in the choice of the concentration factor allows adissible window functions. In this paper the window function used by Brainweb is a sinc function (Kwan et al., 999), and we use a exponential concentration factor (9) in edge detection, which together satisfy both requireents. Theore 6. Using spectral data (25) in the concentration ethod (6) preserves the positions of the jup discontinuities, provided that and w C w d 0, (26) :0 w C 2 0,. (27) Proof. Using the spectral data (25) the concentration ethod (6) will becoe N T,w N fx : i kn sgnk k w e ikx Define the function N 2 f k N i sgnkw k kn k N 2 f keikx. (28) w w, (29) C w where C w is defined in (26). Since w C 2 (0, ) and is a concentration factor, and w C w 0 C 2 0,, w d w d. (30) C w C w0 Thus w is a concentration factor. Hence and i C w N kn sgnkw k k N 2 f ke ikx 3 fx (3) T N,w fx 3 C w fx. (32) Thus it is possible to deterine the position of the discon-

12 500 R. Archibald et al. / NeuroIage 20 (2003) tinuities. The size of the jups will be off by a factor of C w, but is acceptable since only the positions of the jups are necessary in Gegenbauer reconstruction. 6.4 Filtered transfored data in the Gegenbauer reconstruction ethod As shown in Gottlieb et al. (992), if f(x) is an arbitrary L 2 function on [, ], then the error in the ax nor ax x f x g l C l x ax x f x l0 fˆl C l x ax x fˆl C l x l0 l0 g l C l x RE, l0 TE,, N (33) is exponentially convergent for N, where (2/27)e. The proof provided in Gottlieb et al. (992) shows that both the regularization error (RE) and the truncation error (TE) are exponentially convergent. If it is assued that what is known is the windowed reconstruction (24), then the error in the ax nor becoes where ax x f x g,w l C l x ax x f x l0 fˆl C l x ax x fˆl C l x l0 l0 g,w l C l x, (34) l0 g,w l h l x 2 2f N w xc l xdx. (35) If the error in (34) reains exponentially convergent, then the Gegenbauer reconstruction ethod utilizing the coefficients (35) generated fro the windowed Fourier reconstruction will still yield high accuracy. The following theore is a slight odification of the theore in Gottlieb et al. (992) that proves the exponential convergence of the truncation error in (34). Specifically all of the original arguents for the exponential convergence proof are the sae. Theore 6.2 If f(x) isanl 2 function on [, ], then there exists a constant A which is independent of,, and N, such that the truncation error defined as TE,, N ax x fˆl g,w l C l x, (36) l0 satisfies the estiate TE,, N A 2!2 2 where w is a window (or filter) function. N, (37) Proof. Before we prove the theore for an arbitrary L 2 function on [, ], consider the special function f(x) e inx with n N. For this special case f N w (x) 0 and we obtain fˆl g,w l C l x C l x h l x 2 2e inx C l xdx. (38) In Batean (953; p. 23) there is an explicit expression for (38) given by h l x 2 2e inx C l xdx 2 n i l l J l n. (39) Here J (x) is the Bessel function. Since J (x) for all x and 0 (Abraowitz and Stegun, 970; p. 362), we have, for 0 l, fˆl g,w lc l x C l n 2 i l l l l 2 l!2 n 2 2!2 n 2, (40) where in the second step we used the forula l 2 C l, 0, (4) l!2 and in the last step we used the fact that (l )(l 2)/l! is a increasing function of. We now return to the general function f(x), which by Gottlieb and Shu (997) satisfies

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