2792 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

A Generalization of the Two-DimensionalProlate Spheroidal Wave Function Method forNonrectilinear MRI Data Acquisition Methods

Martin A. Lindquist, Cun-Hui Zhang, Gary Glover, Lawrence Shepp, and Qing X. Yang

Abstract—The two-dimensional (2-D) prolate spheroidal wavefunction (2-D PSWF) method was previously introduced as anefficient method for trading off between spatial and temporalresolution in magnetic resonance imaging (MRI), with minimalpenalty due to truncation and partial volume effects. In the 2-DPSWF method, the k-space sampling area and a matching 2-DPSWF filter, with optimal signal concentration and minimal trun-cation artifacts, are determined by the shape and size of a givenconvex region of interest (ROI). The spatial information in thereduced k-space data is used to calculate the total image intensityover a nonsquare ROI instead of producing a low-resolutionimage. This method can be used for tracking dynamic signalsfrom non-square ROIs using a reduced k-space sampling area,while achieving minimal signal leakage. However, the previoustheory is limited to the case of rectilinear sampling. In order tomake the 2-D PSWF method more suitable for dynamic studies,this paper presents a generalized version of the 2-D PSWF theorythat can be applied to nonrectilinear data acquisition methods.The method is applied to an fMRI study using a spiral trajectory,which illustrates the methods efficiency at tracking hemodynamicsignals with high temporal resolution.

Index Terms—Magnetic resonance imaging, nonuniformk-space sampling, prolate spheroidal wave function (PSWF),rapid data acquisition, signal processing.

I. INTRODUCTION

MAGNETIC resonance imaging (MRI) signals (raw data)are acquired in the frequency-domain (k-space), which

is typically sampled on a rectangular Cartesian grid, and thenFourier transformed into the spatial domain (image space). Atradeoff between spatial and temporal resolution is often used toincrease the image sampling speed required by many dynamicapplications [1]–[6]. However, reduction of spatial resolutionin an image leads to a stronger partial-volume effect, whichdecreases the sensitivity of dynamic signals. These problemsare frequently encountered in rapid imaging and chemical shift

Manuscript received January 7, 2005; revised November 28, 2005. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Attila Kuba.

M. A. Lindquist is with the Department of Statistics, Columbia University,New York, NY 10027 USA (e-mail: martin@stat.columbia.edu).

C.-H. Zhang and L. Shepp are with the Department of Statistics, RutgersUniversity, New Brunswick, NJ 08854 USA (e-mail: cunhui@stat.rutgers.edu;shepp@stat.rutgers.edu).

G. Glover is with the Department of Radiology, Lucas MR Center, StanfordUniversity, Stanford, CA 94305 USA (e-mail: gary.glover@stanford.edu).

Q. X. Yang is with the Department of Radiology and Bioengineering (Centerfor NMR Research), College of Medicine, Pennsylvania State University, Uni-versity Park, PA 16802 USA, and also with The Milton S. Hershey MedicalCenter, Hershey, PA 17033 USA (e-mail: qyang@psu.edu).

Digital Object Identifier 10.1109/TIP.2006.877314

imaging. The origin of the problem is the disparity between thesize and irregular geometry of the region of interest (ROI), andthe rectangular voxel produced by the fast Fourier transform [7].

In order to solve this problem, the two-dimensional prolatespheroidal wave function (2-D PSWF) method was developedto address the issues inherent in the fast Fourier transform[8]–[12]. To efficiently reduce the number of acquired datapoints, the 2-D PSWF method tailors the k-space sampling areaaccording to the size and shape of a predetermined convex ROIand creates a matching 2-D PSWF filter to optimally reducetruncation effects. In this method, the spatial information in thereduced k-space data is used to calculate the total signal inten-sity over a nonsquare ROI instead of producing a low-resolutionimage. This method is ideally suited for localized spectroscopyand tracking dynamic signals from nonsquare ROIs using areduced k-space sampling area, while achieving minimal signalleakage.

Because hardware restrictions (gradient strength andslew rate) and the threshold of neuro-stimulation by rapidlyswitching gradients set a physical limit for image acquisitionrates, nonrectilinear data acquisition methods such as radialand spiral imaging are often employed. For these samplingschemes, k-space is sampled nonuniformly and the samplingpoints do not lie on a rectangular grid. The original 2-D PSWFtheory was developed only for the rectilinear sampling case. Tofurther develop the 2-D PSWF method to increase its suitabilityfor dynamic MRI and CSI studies, this paper presents a gener-alized 2-D PSWF theory that can be applied to nonrectilineardata acquisition methods. An example of its implementationfor dynamic MRI is presented using a spiral k-space samplingmethod.

II. THEORY

A. Fast fMRI Using the 2-D PSWF

Let us begin with a brief review of the basic theory of the2-D PSWF in the context of its application to fMRI before pro-ceeding to develop the general case. As k-space consists of asequence of discrete measurements during MRI data acquisi-tion, it should, therefore, be chosen to be discrete. On the otherhand, image space, denoted by in this paper, can be chosen tobe either discrete or continuous. In previous applications of the2-D PSWF theory to fMRI [8]–[12], image space was definedas a discrete space consisting of a collection of image elements(voxels). For an image, the space consists of ele-ments. However, it may be more advantageous to instead treat

1057-7149/$20.00 © 2006 IEEE

LINDQUIST et al.: GENERALIZATION OF THE TWO-DIMENSIONAL PROLATE SPHEROIDAL WAVE FUNCTION METHOD 2793

Fig. 1. (a) Discrete/discrete case—k-space consists of discrete measurements made inside of the region A. Image space consists of the N voxel elements con-tained in the N �N image. B consists of a set of voxels. (b) Discrete/continuous case; image space has support .

image space as continuous, as the underlying object being im-aged should be considered continuous. Fig. 1 gives an illustra-tion of both approaches towards setting up the problem formu-lation. In this section, we focus on the discrete/continuous case,as the discrete/discrete case has already been studied extensivelyin previous work [8]–[12].

Consider a convex ROI, , in image space. The objective ofthe 2-D PSWF method is to determine the optimal sampling re-gion in k-space, , of a predetermined size , which maximizesthe total signal over the region in image space [Fig. 1(b)].This is achieved by designing a k-space sampling region basedon the size and shape of given , and combining this with amatched 2-D filter that maximizes the energy concentration in

. The key to the method is finding the matched filter function, that satisfies the following two criteria.

1) It takes the value 0 for points outside of .2) Its inverse Fourier transform, , has maximal signal

concentration in , i.e., the ratio

(1)

is maximized over all possible functions for which cri-terion 1) holds.

As an additional constraint, the denominator of the ratio in (1)is set equal to one. This simplifies the problem to finding thefunction with norm equal to one, whose inverse Fouriertransform maximizes

(2)

The problem of finding is a 2-D generalization ofthe theory of PSWF [13]–[18]. Using Parseval’s identity, theproblem can be written in matrix form as

(3)

under the constraint

(4)

Suppose is the Fourier transform of the indicator functionof the region , denoted , which can be written

(5)

Then, it can be shown that the kernel, , is given by

(6)

for . For simple regions (e.g., circles, squares, el-lipses) there exist analytical expressions for , which allow foreasy computation of the kernel defined in (6). The solution tothe problem stated in (3) is the largest eigenvector of the matrix

. The corresponding largest eigenvalue provides a quan-titative measure of the signal leakage that the eigenvector givesrise to. A value of close to 1 indicates little signal leakage,while the leakage increases as decreases.

It can be shown in an analogous manner that, for the casewhen image space is instead considered discrete [Fig. 1(a)], the

2794 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

Fig. 2. Two largest eigenvalues for the case when B is a small circular region with fixed radius (radius = 7:5 mm, FOV = 200 mm) and A is a circular regionwith variable radius, containing a equally spaced k-space measurements. As a increases, both eigenvalues approach 1. The appropriate size ofA is when the largesteigenvalue is equal to one and the second largest is significantly below 1.

problem becomes finding the largest eigenfunction of thematrix with elements given by

(7)

for .Once we calculate the optimal filter corresponding to some

fixed , for a given , we let vary among all possible k-spaceregions of a given size . For each possible region, we ob-tain a corresponding vector for which the stated cri-teria hold. Finally, we choose the region whose corresponding

makes the maximal concentration ratio as large aspossible. The problem of finding the optimal sampling region

is a difficult mathematical problem, as it involves an expo-nential growth in computer searches for an sizeimage. To avoid such an extremely computer-intensive search,a heuristic sampling scheme is introduced in [10], which com-puter searches show to be near optimal. In addition, it can beshown that the optimal choice of , when is a circle, is a cir-cular region in k-space. This choice coincides with the heuristicscheme, which for the convex regions typically used in thismethod winds up giving rescaled and rotated versions of cen-tered in the middle of k-space.

It should be noted that, in one dimension, one has an as-sociated differential operator with the same eigenfunctions[13]–[16], which can be used to deduce conditions for when themultiplicity of the largest eigenvalue is equal to one. However,in higher dimensions, such conditions do not exist, and ifwe are not careful, we may not have a unique eigenfunction,and, hence, have more than one answer to the maximization

problem stated in (3). If the multiplicity of the largest eigen-value is greater than one, it is beneficial to shrink the size of thesampling region until the relationship holds.Our experiment shows that we are able to consistently obtain adominant eigenvalue that has multiplicity one, for appropriateregions, using this approach. Fig. 2 shows a plot of the twolargest eigenvalues for the case when is a small circularregion with fixed radius (radius mm, mm)and is a circular region with variable radius, containingequally spaced k-space measurements. As increases, it isclear from Fig. 2 that the value of the two largest eigenvaluesboth approach 1. However, as the largest eigenvalue increasesfaster, there exists a range of values for , such that the largesteigenvalue is equal to one and the second largest is significantlybelow 1. In this particular example, this corresponds to a regionconsisting of approximately 800 points in k-space. This valueindicates an appropriate size for the sampling region .

Once we have calculated the 2-D PSWF filter , the nextstep is to apply it to the raw data. Let us assume that isan experimental sampling function in k-space and is itscorresponding Fourier transform (the image). It can be shown[8], [9] that we can calculate the integral of the signal intensityover , using the formula

(8)

as long as on , which holds for the convex regionsthat we have studied. Equation (8) shows that the integral of the

LINDQUIST et al.: GENERALIZATION OF THE TWO-DIMENSIONAL PROLATE SPHEROIDAL WAVE FUNCTION METHOD 2795

image intensity over can be approximated using a reducedarea of k-space, , since outside of . Thus, ifwe are interested in obtaining the total signal intensity over agiven ROI in an image, the sampling of k-space can be reducedto a region .

A 2-D PSWF filter calculated for a specific at can be usedfor all translationally shifted ROIs in image space. An alterna-tive ROI at can be obtained by shifting toin image space, where is a displacement vector. The signal in-tensity over the new ROI can be calculated as follows:

(9)

Thus, and the associated function can be used for cal-culation of any ROI with the same shape and size. Therefore,having prior knowledge of the exact location of an ROI is ofless importance, as one can simply shift the ROI if necessaryduring postprocessing.

B. Nonuniform Sampling Case

As the 2-D PSWF method often requires one to sample non-square regions in k-space, it is often more efficient to use a tra-jectory that samples k-space nonuniformly. However, it can beshown that the PSWF operator shown in (6) no longer yields thecorrect solution for the case when the k-space region is sam-pled nonuniformly. Note that throughout we assume that imagespace is continuous and without loss of generality that is aunit square.

When calculating the PSWF function, the denominator of theratio in (1) is set equal to one. In the case when the data is sam-pled on a Cartesian grid, this implies that

(10)

which is an essential constraint for the derivation of (6) to hold.In the case when k-space is sampled nonuniformly, (10) nolonger holds. To see why, let us take a closer look at the de-nominator of the ratio in (1)

(11)

where

(12)

When the k-space points and lay on the Cartesian grid, wehave

ifif .

(13)

Hence

(14)

and, thus

(15)

which implies that (10) will hold. However, when the coordi-nates and do not lie on the Cartesian grid, this relationshipwill not hold. Hence, when dealing with the nonuniform casewe will have to take the kernel, , into consideration while de-riving the PSWF filter.

C. Generalized Prolate Spheriodal Wave Function forNonuniform Sampling

Assume that we nonuniformly sample a finite set of pointsin k-space , which we will refer to as . In MRI,the measured signal at point , , is related to the imagefunction according to the relationship

for (16)

where is a constant and denotes image space. This equationshows that is a multiple of the Fourier transform of theimage function. For notational simplicity, we will assume theconstant is equal to 1 and will subsequently refer to as

.Now, consider a function which is zero outside of

and whose inverse Fourier transform has its energy maximallyconcentrated on a region in image space. That is, is zerofor any point lying outside of and its inverse Fourier transformmaximizes the ratio given in (1). Suppose we apply this functionto the sampled data in the same manner described in the previoussection. Following (9), we can calculate the signal intensity overan ROI , centered at point , as follows:

(17)

2796 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

where

(18)

Hence, the integral of signal intensity over will be ap-proximately equal to the image convolved with a point spreadfunction which depends on the function . By controlling theshape of , we can control the point-spread function and min-imize signal leakage outside of the ROI. Ideally, we would like

to be completely concentrated on a region of the same sizeand shape as B. When k-space is sampled on a Cartesian grid,

, which is the filter covered in previous papers andsummarized in Section II-A. In the nonuniform case, willtake a different form and the calculation of the PSWF functionneeds to be altered accordingly.

D. Determining the Generalized Prolate Spheriodal WaveFunction

In this section, we derive the optimal filter for the case whenk-space is sampled nonuniformly and image space is chosen tobe discrete [Fig. 1(a)]. The continuous image space case is dis-cussed briefly at the end of the section. In the PSWF theory, weare interested in finding the function that solves the followingoptimization problem:

(19)

where is defined in (18).Let denote the set of points in an image space, and letconsist of adjacent voxels that cover the chosen shape of the

ROI. To present the problem in matrix notation, let representthe vector form of , for . Similarly, let and bevectors of and , respectively, for . We will alsomake use of the inner product notation, where

(20)

Further, let us define two operators

(21)

and

(22)

Here, it should be noted that is an and anmatrix. It is important to note two useful facts about the opera-tors and . First

(23)

and, similarly

(24)

Additionally, the matrix is defined as the rows of thematrix that correspond with elements in the ROI and,similarly, will consist of the columns of the matrix thatcorrespond with elements in . When multiplying these twomatrices we will simply write , instead of .

Turning our attention back to (19), for any fixed , wecan rewrite the numerator, in discrete form, as follows:

(25)

Similarly, we can rewrite the denominator as

(26)

Let us define . Since , it holds that.

Let . Now, (25) can be written as

(27)

and the problem defined in (19) can be reformulated as follows:

subject to (28)

The solution to this problem is the largest eigenvalueand the corresponding eigenvector of the matrix

. In the case when k-space is

LINDQUIST et al.: GENERALIZATION OF THE TWO-DIMENSIONAL PROLATE SPHEROIDAL WAVE FUNCTION METHOD 2797

sampled uniformly, , where is theidentity matrix, and can be written as

(29)

Hence, for each

(30)

which is the kernel shown in (7). However, when k-space is sam-pled nonuniformly we have to take the matrix into consider-ation in our continued calculations.

Since is the largest eigenvalue of and its corre-sponding eigenvector, it must hold that

(31)

In general, , and, hence, to simplify the calculation,can be expressed in image space to reduce the matrix size. Let

. Multiplying both sides of (31) by the matrixyields

(32)

This shows that is an eigenvector of the matrix , whichis a matrix that has the same largest eigenvalue, , as

. Once is obtained, we can use it to calculate the optimalfilter . Begin by multiplying both sides of the equation

with the matrix . Together with (31), thisgives the following relationship between and :

(33)

This, in turn, implies that

(34)

Using this relationship, and the fact that , we can ex-press as

(35)

The inverse Fourier transform of is the function for which theratio in (19) is maximized and, therefore, maximizes the energyover the predefined ROI . Equation (35) represents the optimal

PSWF filter for the ROI and representsthe corresponding point spread function.

Once is obtained, the total signal intensity over the ROIcan be evaluated directly from the reduced k-space area usingthe formula

(36)

as shown by (17).In this section, image space was chosen to be discrete. It

should be noted that the problem can be set up in an analogousmanner for continuous image space. However, we ultimately de-termined that treating image space as discrete allowed for a sim-pler expression of the dual kernel in image space (32), which,in turn, greatly simplifies numerical calculation.

E. Calculation of the Generalized PSWF

The main difficulty in applying this method to MRI data, isinverting the matrix . In certain situations, this can be difficultdue to the size of the matrix and to the fact that the smallesteigenvalues can take values close to 0, making the matrix ill-conditioned. When this scenario arises, there are two potentialproblems in calculating . The first difficulty is the calculationof the kernel and the corresponding largest eigenfunction

. The second difficulty is the calculation of . The firstproblem can be solved by substituting withthe pseudoinverse of .

When calculating the kernel , we can avoid the problemof inverting by defining a mapping. Let ,where and . The vectorsform an orthogonal basis of . We can write and in termsof the unitary matrix as follows: and .

Since and , the matrixwill be the projection onto the range of . We can

rewrite the generalized prolate kernel, , in terms of thematrices and

(37)

Using this mapping, we can calculate the kernel withouthaving to invert the matrix . Instead we need to find a basiswhich spans . This can be done in a variety of fashions, forexample, by calculating the singular value decomposition of

. Once we have determined , it is easy to calculateand obtain the largest eigenfunction in the usual manner.

As the matrix works as a weighing function that com-pensates for the nonuniform sampling, an alternative approxi-mate solution that is significantly less computer intensive is tosubstitute the matrix with a matrix with diagonal elementsequal to the values of some density compensation function. Thiswill give a reasonable approximation at a fraction of the costfor most sampling trajectories that one would potentially use inMRI.

2798 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

Fig. 3. Computer phantom images. (a) Image without activation. (b) Image with activation in B. (c) Image with activation in B and 5% white noise.

Fig. 4. Spiral sampling trajectories consisting of (a) 454 points, (b) 910 points, and (c) 3664 points.

III. METHODS

The generalized 2-D PSWF method was implemented inMatlab (Mathworks, Inc.) on a personal computer. To demon-strate its utility for dynamic studies, a high temporal resolutionfMRI experiment was designed to simultaneously track thehemodynamic signals in both visual and motor cortices whilethe subject undergoes a visual-motor activation paradigm. Fol-lowing the steps of the generalized 2-D PSWF method, circularROIs with a radius of 8 mm were chosen for the primary visualcortex and the primary motor cortex (Fig. 8). Since the ROIswere chosen to be circles, the optimal sampling in k-space wasalso circular [10]. This predetermined circular subregion ofk-space was sampled with 3628 points using a spiral trajectory[19] that allowed a minimum repetition time of 60 ms (Fig. 9).After calculating the optimal filter corresponding to thischoice of and , the signal intensity over the circular ROI,

, was calculated with the acquired k-space data using (36).The activation paradigm consisted of six cycles of 30-s inter-

vals. At the beginning of each interval a 100-ms light flash waspresented. The subject was instructed to press a button with theirright thumb immediately after sensing the flash, thereby leadingto activation of the motor cortex. During the 30-s interval, 500

images were acquired rapidly every 60 ms. The sequence was re-peated six times, each time producing a dynamic data set of 500temporal points. The images were obtained in an oblique slicecontaining both primary visual and motor cortices using a spiraltrajectory. The dynamic signals from the predetermined ROIswere acquired with an effective TE 30-ms, flip angle 15 , field ofview 240 240 mm , slice thickness 10.3 mm, matrix 24 24and bandwidth 12.5 kHz. The experiment was performed on a3.0 T whole body scanner (GE magnet, General Electric Med-ical Systems, Milwaukee, WI). A healthy male volunteer (age28) participated in the study after giving written informed con-sent. The activated ROIs in the slice were determined by cor-relating their time-course signal intensity with the experimentalparadigm. For comparison purposes, the acquired k-space datawere also used to reconstruct low-resolution images using con-ventional regridding methods [20] for the dynamic analysis.

IV. RESULTS

In this section, a computer experiment is described toillustrate the concepts and mathematical procedure of the gen-eralized 2-D PSWF method for nonuniformly sampled data.

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Fig. 5. Optimal filter functionp corresponding to each of the three sampling schemes shown in Fig. 4. The x axis represents each point in the order it was sampled.

Fig. 6. Function P (x) and the corresponding signal leakage given by each of the three filters shown in Fig. 5.

An application of the method to fMRI is demonstrated usingexperimental data.

A. Computer Phantom Study

As shown in Fig. 3, we constructed a 64 64 phantom imagecontaining an ellipse representing a human brain that consists of1209 points. The image intensities are assigned values of 1 or0 for the points inside or outside the ellipse, respectively. Twosmaller ellipses having an intensity of 1.1, each consisting of75 points, are placed inside the larger ellipse to simulate ROIswith static contrast to the large ellipse. To simulate a dynamicimage series, this base image is recreated 64 times accordingto a boxcar paradigm consisting of eight cycles of four “ON”and four “OFF” periods. An image is considered in the “ON”state, if there is an “activated” circular ROI with intensity 1.05in the corner of the large ellipse. Conversely, an image is con-sidered in the “OFF” state, if no such ROI exists in the image.The “activated” ROI consists of 21 points. A 5% white noisefloor is added to all 64 images. These dynamic images are sub-sequently transformed into k-space for application of the 2-DPSWF method to measure the dynamic signal change in a givenROI, .

B. Choosing A and B

In this simulation, our ROI is chosen to be the same size andshape as the “activated” circular region consisting of 21 points.

The optimal k-space sampling area, , corresponding to thisROI will also be circular [10]. The spiral sampling trajectoryis the natural choice for sampling a circular k-space region .In order to demonstrate the influence of the size of , threesampling sizes shown in Fig. 4 are given. Our first choice of

[Fig. 4(a)] consists of 454 points and covers approximately11% of the total area of k-space needed for a full image recon-struction of a 64 64 image. We also choose two other sam-pling regions for comparison, one that consists of 910 pointscovering 17% of k-space [Fig. 4(b)] and another that consists of3664 points covering 78.5% of k-space [Fig. 4(c)].

C. Eigenvalues and Eigenvectors

Using the generalized prolate kernel in (32), the optimal filter,, with respect to and , can be calculated. Fig. 5 shows

the optimal filter functions corresponding to the three differentsampling sizes. The signal leakage for each of the three filters isshown in Fig. 6. It is clear that as the sampling size increases,so does the concentration of the signal within the ROI.

In the rectilinear case [8], [9], a lower limit for choosing thenumber of sampling points was given by the relationship

(38)

2800 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

Fig. 7. Time-course signal intensities overB using the 2-D PSWF method andonly 17% of k-space. The signal intensity with full k-space sampling is plottedin dotted lines for comparison.

where is the size of the ROI and is the size of the image.This formula cannot be directly applied to nonuniform data, asthe area of k-space effectively covered is more important thanthe actual number of points sampled. However, by calculatingthe lower limit for in the rectilinear case according to (38),we can determine the appropriate area of the k-space region thatneeds to be covered. For equal to 21 and equal to 64, we findthat we would need to sample 585 points in the rectilinear case.This corresponds to a circular region of k-space with radius 14unit points. It is interesting to note that the first sampling regionis smaller than this heuristic threshold, while the latter two lieabove it.

D. Calculating the Signal Intensity Over B

Once the filter corresponding to the given k-space subsetis obtained, the total signal intensity over can be calculated

directly from the reduced k-space area using (36). In this study,the 2-D PSWF method was applied to data sampled using thesampling trajectory with 910 points in Fig. 4(b). Fig. 7 showsa plot (solid line) of as a function of the imagenumber, , obtained using a reduced k-space sample and the 2-DPSWF method. Also contained in the same figure is a plot ofdynamic signal change over (dotted line), obtained by takingthe sum over directly from the phantom images. It is clear thatwith the 2-D PSWF method the signal intensity over B closelyfollows the true signal change even though only 17% of k-spacewas used in the analysis.

E. Application to Experimental Data

To validate the theoretical analysis and computer modelingresults, the utility of the generalized 2-D PSWF method for dy-namic studies was tested experimentally using a visual-motorstimulation fMRI paradigm. With the generalized 2-D PSWFmethod, the dynamic signal change over each of the two ROIsduring the execution of the stimulation paradigm was obtained.Fig. 10 shows a plot of the average signal intensity in the visual

Fig. 8. Two ROIs (21 voxels) in the (upper) motor and (lower) visual corticessuperimposed on a low-resolution image of the acquired slice.

Fig. 9. Optimal sampling region corresponding to the ROIs given in Fig. 6.The trajectory used in the experiment is superimposed. Compare this with fullk-space (64� 64) which is marked by the black box.

cortex (bold line) and the motor cortex (light line) over the sixruns. A clear delay in motor activation due to reaction time canbe seen. With the increased temporal resolution and SNR, wecan more precisely determine the delay time in motor activationto be approximately 300 ms. For both cortices, the “negativedip” in the time-course plot can be seen clearly. Fig. 11 shows asimilar plot obtained from the same ROIs using images recon-structed with conventional regridding methods. The motor ac-tivation is significantly attenuated as a result of partial volumeeffects.

The generalized 2-D PSWF methodology was able to detectthe signal change in both activated regions of the brain. By trans-lating the filter, the signal intensity over other circular ROIs cen-tered at different areas of the brain could be calculated. No othersignificant activation was consistently detected in other regionsin the chosen slice.

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Fig. 10. Dynamic signal change over the (bold) visual and (light) motor cortex for the first 18 s following visual stimuli. The delay in time-to-peak between thetwo curves is approximately 300 ms.

Fig. 11. Dynamic signal changes over the (bold) visual and (light) motor cortex for the first 18 s following visual stimuli, obtained by integrating the signal overthe ROIs in images reconstructed using regridding methods.

V. DISCUSSION

This paper presents a generalization of the 2-D PSWF methodto nonuniformly sampled k-space data. This generalization isimportant, as k-space is typically sampled in a nonrectilinearfashion for dynamic MRI studies due to hardware limitations.The example presented here samples k-space with a smoothspiral trajectory that eliminates abrupt switching of spatial

encoding gradients. This approach is also advantageous forimplementation of the 2-D PSWF method because it is morestraightforward to sample the tailored nonsquare k-space re-gions using a spiral trajectory rather than using conventionalrectilinear k-space sampling methods.

From a practical point of view, the major difficulty in applyingthe 2-D PSWF method to MR data is the computational diffi-culties that are involved in inverting the matrix . For typical

2802 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

Fig. 12. Fourier transform of the dynamic signal over the motor cortex shown in Fig. 9. A peak at ! = 16 indicates the frequency of the periodic fluctuationapparent in the signal.

MRI sampling trajectories, the matrix is often ill-conditionedbecause the smallest eigenvalues take values close to 0. In thesesituations, one can substitute the inverse with the pseudo-inversethat is a generalization of matrix inversion that can be com-puted using singular value decomposition. An additional diffi-culty is that when is large, the calculation of the inverse orpseudo-inverse is computationally intensive. An alternative ap-proximate solution that is significantly less computationally in-tensive is to substitute a matrix having diagonal elements equalto some density compensation weight for the matrix .This will give a reasonable approximation at a fraction of thecost for most sampling trajectories used in MRI.

In the 2-D PSWF method, the reduction of k-space samplingis realized by trading off spatial resolution for temporal reso-lution while maintaining high SNR. Tailoring the sampling re-gion and matching the 2-D PSWF filter to the shape and size ofa given ROI allow us to optimize the reduction of k-space sam-pling with minimal truncation penalty. The nonsquare shapedROI reduces the signal leakage outside of the ROI, which im-proves the contrast-to-noise ratio (CNR). As demonstrated inFig. 10, the improved temporal resolution provides more de-tailed information about hemodynamic signals of visual andmotor cortices. A “negative dip” that peaks 2s after the onsetof visual stimulation, is clearly visible in the hemodynamic sig-nals from both visual and motor cortices. Since the amplitude ofthe negative dip is much smaller than the positive BOLD effect( at 3T), similar results reported previously generally re-quired signal averaging across multiple runs from multiple sub-jects [21]. With the generalized PSWF method, the time coursesignal from the visual cortex allows the data to reliability char-acterize the hemodynamic response from a single subject. This

is important for the clinical applications where diagnoses of ab-normalities are normally based on a single subject study. Thecorresponding negative dip for the motor cortex is apparent butless pronounced in the plot. This may be due to averaging ef-fect of the temporal signals over all the runs because the timeof subject responding to the visual stimulation can be differentwith each run. With the conventional gridding method, both pos-itive and negative activations in the time-course plot for motorcortex are diminished.

The other advantage of the high temporal resolution providedby the PSWF method is the increased frequency-domain band-width of the dynamic signal. This allows for better identificationand subsequent filtering of undesirable physiologic noises in thehemodynamic signal. As shown in Fig. 11, the characteristicsfor the signal over the motor cortex were interfered by a periodicmodulation in the time course signal. This can be easily iden-tified in the Fourier transform of the temporal signals shown inFig. 12, and subsequently removed by a specific bandpass filter.

Typically, under the conditions needed to increase the tem-poral resolution, tracking the hemodynamic responses is diffi-cult because the image SNR is significantly attenuated by thedrastically reduced TR. However, as seen in Fig. 10, the 2-DPSWF method provides an adequate SNR for high temporal res-olution dynamic studies. Thus, the 2-D PSWF method can be avaluable tool for the studies of dynamics of the brain functionsuch as functional neuron-neuron interaction, synchronizationand connectivity [22]–[24].

VI. CONCLUSION

The 2-D PSWF method addresses issues inherent in the fastFourier transform, such as partial volume effects due to the

LINDQUIST et al.: GENERALIZATION OF THE TWO-DIMENSIONAL PROLATE SPHEROIDAL WAVE FUNCTION METHOD 2803

square-shaped voxel and the inverse relationship between imageresolution and k-space sampling area (temporal resolution).These problems are frequently encountered in rapid imagingand chemical shift imaging. This method uses prior knowledgeof a given ROI and the temporal resolution requirements todesign a reduced sampling area of k-space with a matched 2-DPSWF filter, such that optimal SNR and minimal truncationartifacts are achieved.

This paper introduces a generalized version of the 2-D PSWFmethod that allows its application to nonuniformly sampledk-space data with arbitrary trajectory. This extension is impor-tant; as the k-space subregions normally take a nonsquare shapewhen applying the 2-D PSWF method and sampling on theCartesian grid for a nonsquare k-space is inefficient. To validateour method, the generalized 2-D PSWF method was applied toa high temporal resolution fMRI study using a spiral samplingtrajectory and its effectiveness in tracking the dynamic signalchange over the visual and motor cortices was illustrated.

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Martin A. Lindquist received the Ph.D. degree instatistics from Rutgers University, New Brunswick,NJ, in 2001.

He is currently an Assistant Professor with theDepartment of Statistics, Columbia University, NewYork. His research focuses on developing statisticalmethods for the analysis of functional magneticresonance imaging (fMRI) data.

Cun-Hui Zhang received the Ph.D. degree in statis-tics from Columbia University, New York, in 1984.

He is currently a Professor with the Department ofStatistics, Rutgers University, New Brunswick, NJ.His research interests include empirical Bayes, non-parametric and semiparametric methods, functionalMRI, biased and incomplete data, networks, multi-variate data, and biometrics.

Gary Glover received the Ph.D. degree in electricalengineering from the University of Minnesota,Minneapolis, in 1969.

He has studied biomedical-computed ultrasound,X-ray tomography, and NMR imaging since 1974.He joined GE’s Corporate Research Lab, Schenec-tady, NY, where he initially studied hot electron phe-nomena in semiconductors. He joined GE’s MedicalSystems Labs in 1976 to continue the developmentof CT reconstruction algorithms. He has been a pio-neer in magnetic resonance imaging since 1980. His

present academic interests encompass the physics and mathematics of MRI,with a special focus on rapid scanning methods using spiral and other non-Carte-sian k-space trajectories for dynamic imaging of brain function. Using thesetechniques, fMRI pulse sequences and processing methods are under develop-ment for mapping cortical brain function, with applications in the basic neuro-sciences, as well as for clinical diagnosis and therapy.

Dr. Glover was President of the ISMRM and has received numerous awards,including the JSMRM’s Gold Medal.

2804 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 9, SEPTEMBER 2006

Lawrence Shepp received the Ph.D. degree fromPrinceton University, Princeton, NJ, in 1961.

He was a member of Bell Laboratories from 1962to 1996. Since 1997, he has been a Professor ofStatistics at Rutgers University, New Brunswick,NJ. His major research interests are in probabilistic,combinatorial, and statistical analysis of modelsfor problems arising in physics, engineering, andcommunications; computed tomography; automaticpattern recognition; probabilistic models for phasetransitions; connectedness of random graphs; math-

ematics of finance; and genetics.Dr. Shepp has received numerous awards and distinctions. He received the

Paul Levy Prize in 1966 and the IEEE Distinguished Scientist Award in 1979for work in computed tomography. He is a Member of the National Academyof Sciences (1989), the National Academy of Medicine (Institute of Medicine,1992), the American Academy of Arts and Sciences (1993), and a Fellow of theInstitute of Mathematical Statistics.

Qing X. Yang received the Ph.D. degree in physicsfrom School of Physics, Georgia Institute of Tech-nology, Atlanta, in 1991.

He is currently an Associate Professor with theDepartment of Radiology and Bioengineering,College of Medicine, Pennsylvania State University,University Park, PA. His research interests includerapid MRI data acquisition method development,high-field MRI physics, and clinical applications ofquantitative MRI and fMRI.