a numerical approach to the selection of basis for frame-encoded mri

A numerical approach to the selection of basis for frame-encoded MRI

Zhihua Xu*

Texas Center for Applied Technology, Texas A&M University, College Station, TX 77843-3407, USA

Received 24 November 2002; accepted 20 May 2003

Abstract

A numerical approach was proposed to systematically investigating the suitability of various bases for frame-encoded magneticresonance imaging (MRI). Several basic requirements were established for the ideal encoding frame basis. Three different orders of splinebases were extensively examined with simulation of 2-dimensional MRI. Analyses of imaging time and signal-to-noise ratio showed thatthe compactly supported quadratic spline basis is a strong candidate for high quality MRI with a short imaging time. The simulations alsosuggested that encoding with three levels of resolution is the best choice. © 2004 Elsevier Inc. All rights reserved.

Keywords: Frame encoding; MRI simulation; SNR; Spline basis

1. Introduction

The conventional imaging method for magnetic reso-nance imaging (MRI) is known as Fourier transform encod-ing [1] in which spin-echo signals are measured to representdiscrete samples in the Fourier space (k-space) of the spatialspin density function. A drawback of the Fourier transformencoding is that it requires a long time for data acquisition.Many encoding bases have been proposed to replace theFourier basis in the phase-encoding step. These bases in-clude nonexclusively wavelets [2–5], wavelet packets [6,7],windowed Fourier bases [8,9], the Hadamard basis [10], anddiscrete cosine bases [11]. Noticing that all these bases areframe bases, i.e., they are square integrable in a Hilbertspace, we treated these new encoding techniques from aunified perspective and categorized them as frame encod-ing. For a review of these new encoding techniques anddetails on frame encoding, please refer to [12].

Compared with phase encoding, frame encoding hassome potential advantages such as a much shorter dataacquisition time, suppression of the Gibb’s ringing frompartial volume effects, and dramatic reduction and spatiallocalization of motion artifacts. In addition, it enables theimaging of a moving object against a stationary background[4]. To fully exploit its promised advantages, in our previ-ous papers we solved two most prominent problems presentin the implementation of frame encoding—RF pulse design

[13] and image reconstruction [12]. Another important issueis how to select an encoding basis, because not all bases aregood choices for frame encoding in terms of imaging per-formance, such as imaging time and signal-to-noise ratio(SNR) of the reconstructed image. In this paper, we proposea numerical approach to systematically investigating thesuitability of various bases for frame-encoded MRI. Severalbasic requirements are established for an ideal encodingbasis. As examples, three different orders of spline func-tions and their associated wavelets [14] are extensivelyexamined. The numerical examination procedure consists ofthree steps. In the first step, RF pulses for the spline functionand its associated wavelet function are designed based onour new design method [13]. The Bloch equations with thepresence of an RF pulse are then solved numerically togenerate the excitation profile, which is used in the next stepas an encoding basis function. The second step is simulationof 2D MRI and image reconstruction with the frame algo-rithm [12]. The last step contains some analyses of imagingperformance. Simulation results show that the compactlysupported quadratic spline basis is a strong candidate forhigh quality MRI with a short imaging time. Simulationsalso suggest that encoding with three levels of resolution isbetter than other choices.

2. MRI requirements imposed on the encoding basis

Although all previous work in frame-encoded MRI usedorthogonal or bi-orthogonal bases, bi-orthogonality is not a

* Tel.: �1-(979)-862-3272, fax:�1-(979)-862-3336.E-mail address: [email protected] (Z. Xu).

Magnetic Resonance Imaging 22 (2004) 47–54

0730-725X/04/$ – see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.mri.2003.05.008

necessity for this technique. (Orthogonal bases are actuallyspecial bi-orthogonal bases.) In general, we may use a framefor encoding and its dual for reconstruction [15]. Note thatthe difference between a frame and a biorthogonal basis isthat a frame may be over-complete, i.e., not all its basisfunctions are independent. This alternative greatly increasesthe selection of encoding bases. Another advantage is thatthe over-completeness of frame bases leads to a higher SNR.

Frame encoding uses a shaped RF pulse to excite adistribution of nuclei in the sample. As shown in [13], whenthe Fourier transform of a desired basis function is not finite,one need to truncate it to design an RF pulse with finiteduration. The finite duration of the RF pulse causes theexcitation profile to become wider than the desired basisfunction. The excitation profile is not finite even when thedesired basis function is. Because it is the excitation profileinstead of the desired basis function that is used for frameencoding, the desired basis function does not have to befinite. However, almost all advantages promised by frameencoding are related to the localization property of theencoding basis functions. We introduce effective support tocharacterize the excitation profile that has been shaped likea desired basis function. The effective support, �f, of azero-centered function f(x) is defined as:

for all �x� � �f, �f(x)� � a � max��f(x)��, (1)

where a is a small positive constant. For MRI applications,a may be chosen as 0.01. If two excitation profiles of thesame shape are at least 2�f apart, the interference betweenthem is negligible. Therefore, they can be excited immedi-ately one after another without waiting for one TR (repeti-tion time) to restore the equilibrium state of the sample [1].When encoding basis functions have small effective sup-ports, the number of TRs is small. It is shown in the nextsection how to estimate the number of TRs based on effec-tive supports. For a T2-weighted image, TR must be sub-stantially longer than T1 and a large number of repetitiontimes leads to slow imaging. Consequently, one criterion foran ideal basis function is a small effective support to ensurea short imaging time.

Although many basis functions, e.g., wavelets have smalleffective supports, most of them are not qualified for MRIapplication because of practical considerations regardingthe RF pulse. Usually the spectrum of a basis function istruncated to be used for designing a finite duration RF pulse.This truncation has two major effects. First, it severelydamages the perfect reconstruction relationship between theencoding basis and its dual. Observing that the encodingfunctions, i.e., the excitation profiles, constitute a frame inan L2(R) space, we solve this image reconstruction problemin [12] by applying frame theory. Secondly, as mentionedbefore, the truncation forces the spectrum to be band-limitedand thus causes the excitation profile to be infinitely wide.As a result, the effective support of the excitation profilebecomes larger than that of the desired basis function. An

increase in effective support generally leads to an increasein imaging time. To minimize this impact of truncation, theideal encoding basis function for MRI should have a band-limited spectrum. This implies that a desired basis functionshould at least be smooth and continuous. A parameter, CT,may be defined as a measure of the effect caused by truncation:

CT � 1 �

��T

�T

��̂��2 d�

��

�

��̂��|2 d�

(2)

where �T is the frequency limit for truncation and �̂(�) isthe spectrum of basis function �(x). We remark here that thedenominator term in the expression is the total energy of�(�). Hence, CT may be interpreted as the ratio of energyloss caused by truncation of the spectrum. The smaller CT is,the less is the effect of truncation.

In addition, the basis function should be symmetric orantisymmetric about its center. The x- or y-component of atransverse excitation profile is utilized as an actual basisfunction to encode the spin density function. Because theRF envelope function is real, our pulse design method [13]requires that the desired basis function be symmetric orantisymmetric about its center. This requirement may not benecessary if we have a more general design method for RFpulses. As pointed out in the next section, the flip angle ofRF excitation should be 90° to achieve the theoretical boundof SNR. A generic RF pulse design method meeting thischallenge has not appeared in the literature.

From the above discussion, we may state that the ideal

Fig. 1. A cross-section MRI of human brain used as the original image ofsimulation.

48 Z. Xu / Magnetic Resonance Imaging 22 (2004) 47–54

choice of basis function for MRI as: (1) it has a smalleffective support, and (2) it has a band-limited spectrum, (3)it is symmetric or antisymmetric. Again the last requirementis posted by the lack of a generic RF pulse design method.Certainly, it is desirable that the ideal basis achieves a highSNR. Our previous paper showed the relationship between

the SNR and the encoding basis [12]. It is difficult to find abasis that satisfies all three requirements. Intermediate ordercardinal B-splines, such as quadratic and cubic splines, meettwo of them but their spectra are not exactly band-limited.Fortunately, their spectral magnitudes decay very fast, andthus, are nearly band-limited. In the next section, we sim-

Fig. 2. The excitation profile for linear spline with a nearly 90° flip angle.

Fig. 3. The excitation profile for linear spline wavelet with a nearly 90° flip angle.

49Z. Xu / Magnetic Resonance Imaging 22 (2004) 47–54

ulate 2D MRI with these two splines and their associatedwavelet functions as the encoding basis functions. For com-parison, simulations with the linear spline basis have alsobeen carried out. We take N2, N3, and N4 to denote thesethree spline bases.

3. Simulation and analyses

Our simulation process is divided into two stages: exci-tation and imaging. In the first stage, two RF pulses aredesigned for the scaling function and its associated wavelet,

Fig. 4. The excitation profile for quadratic spline with a nearly 90° flip angle.

Fig. 5. The excitation profile for quadratic spline wavelet with a nearly 90° flip angle.


respectively, and the Bloch equations are then solved nu-merically with the RF pulses to produce two excitationprofiles. Simulation of the Bloch equations is necessarybecause the excitation profiles are used in the followingstages of imaging and analysis. Based on the excitationprofiles, the imaging time is estimated and the SNR is

calculated. In the second stage, frame encoding in the rowdirection and frequency encoding in the column directionare simulated on a gray scale image with 128 128 pixels,which is a cross-section MR image of human brain. Figure1 shows the original image. Because the imaging time andthe SNR are determined by the excitation profiles and our

Fig. 6. The excitation profile for cubic spline with a nearly 90° flip angle.

Fig. 7. The excitation profile for cubic spline wavelet with a 90° flip angle.


image reconstruction algorithm is independent of the objectunder imaging, the choice of original MR image has noeffects on our simulation. Thus the simulation results aregenerally applicable. We assume that an inverse RF pulseafter the excitation selects a plane perpendicular to the thirddirection [3]. After frequency encoding, another inverse RFpulse is applied so that the equilibrium state is recovered forall the spins outside of the excited region.

The method proposed in [13] is adopted to design RFpulses for three different orders of spline functions and theirassociated wavelets. These splines include linear, quadratic,and cubic splines. The desired envelope function Be(t) of anRF pulse can be derived from the following integral equation:

B'e�t� � Be�t� cos ��0

t

�Be��d�� (3)

where � is the gyro-magnetic constant and B'e(t) is theenvelope function derived from the well-known linear re-sponse theory [16]. B'e(t) is proportional to the Fouriertransform of the desired excitation profile. As mentionedabove, the spectrum, i.e., the Fourier transform, of thedesired excitation profile is usually not band-limited, andthus, truncated to obtain a finite-duration RF pulse. Wetruncate the spectrum of the linear spline to keep the mainlobe and a side-lobe. Only the main lobe is kept for thespectra of quadratic and cubic splines. For the spectra oftheir associated wavelets, only one side-lobe on each side ofthe center is kept. The CT of truncation is about 0.05. Thetruncated spectra are then scaled so that the peak flip angleis 90°. An obvious reason for this amplitude scaling is toachieve the SNR bound for a chosen basis. Here SNR isdefined as a square root of the ratio of image energy toimage noise energy. Under the assumption that the noiseenergy is constant, the signal energy, and thus, the SNRincrease by almost 4 times as the flip angle increases from15° to 90°. The SNR reaches its theoretical bound as the flip

angle increases to 90°. After an RF pulse is designed, wesolve the Bloch equations numerically using an FDTD (fi-nite difference in time-domain) method with the RF pulsepresent. The solution is an excitation profile used as anencoding basis function in the next stage of simulation.Notice that we do not use the inverse Fourier transform ofthe RF pulse, which is a very coarse solution provided bythe linear response theory [16]. All six excitation profilesare plotted in Figs. 2–7. These figures show that the differ-ences between the excitation profiles and the desired encod-ing basis functions are small but definitely not negligible.

As done in [12], the imaging process is simulated in twosteps: frame encoding and frequency encoding. Frame en-coding is simulated as a moving average between eachencoding basis function and each row of the original imagethat represents the 2D spin density function. Each row isinterpolated to the same sampling rate of excitation profilesin order to achieve high accuracy. All simulations are donewith six levels of resolution for each of the three splinebases. For details about multiresolution decomposition andreconstruction with wavelets see [14]. The FFT is thencomputed for each column to simulate frequency encodingto produce the k-space data. The k-space data are used toreconstruct the image. Independent normal noise may beadded to the k-space data so that the SNR of reconstructedimages can be studied. After the inverse FFT has beenapplied to each column of the k-space data, images arereconstructed at six levels for each spline basis. An image atthe first level is reconstructed with only projection coeffi-cients of the translated spline functions. Higher level imageshave progressively higher resolutions. The respective dualfunctions constructed by the discrete frame algorithm [12]are used for reconstruction. As examples, we include in Fig.8 the images that are encoded with the quadratic splinebasis. The images encoded with the other two spline basesshow similar quality for each encoding level. As in [12], theNormalized mean square error (NMSE) is calculated tomeasure the overall quality of a reconstructed image. With-out thermal noise, NMSEs for the 6th encoding level are0.0003, 0.0002, and 0.0001, respectively, for linear, qua-dratic, and cubic spline bases. Obviously, they are negligi-ble when compared to the error in a reconstructed imagecaused by thermal noise, which is induced during the im-aging process. For a detailed discussion on the SNR, theNMSE, and thermal noise, see our previous paper [12].

As mentioned above, imaging time is related to the

Table 3Comparison of accumulative repetition time (in unit of TR) for differenttotal levels of encoding with linear, quadratic, and cubic spline bases

Encoding levels N2 N3 N4

6 18 24 294 12 16 193 9 12 142 6 8 9

Table 1Effective supports of the excitation profiles for linear, quadratic, andcubic spline bases

N2 N3 N4

� 1.56 2.07 2.18� 1.38 1.88 2.31

Table 2Comparison of SNR (shown how many times smaller than that ofconventional Fourier imaging) for different total levels of encoding withlinear, quadratic, and cubic spline bases

Encoding levels N2 N3 N4

6 7.9 6.2 7.64 7.9 6.3 7.73 8.0 6.4 7.82 8.4 7.0 8.2


effective supports of encoding basis functions. The effectivesupports of the excitation profiles are listed in Table 1. Inreality, the x- or y-component of transverse magnetization is

used as an encoding function. However, the other compo-nent of transverse magnetization also has interference withthe previous or the next excitation. When measuring the

Fig. 8. Reconstructed images from frame encoding with the quadratic spline basis. Figs. A–F are the images reconstructed at the first to the sixth level.


effective support, it would be more secure to replace theencoding function by the transverse magnetization. There-fore, the interference between two consecutive excitations issurely negligible. As done before, the parameter a is chosento be 0.01. � is the effective support of the spline functionwhile � is that of its associated wavelet.

Let N be the smallest integer that is greater than 2�f ofthe excitation profiles. When exciting the object at a certainlevel, we can organize the excitations into N groups. In eachgroup, any two excitation profiles are at least 2�f away fromeach other, and thus, they can be excited immediately oneafter another. One TR is required for the relaxation torestore the equilibrium state when beginning excitation of adifferent group. This makes (N � 1) TRs for that level.Similarly, when moving excitation to a different level, aperiod of TR is also required. For encoding with six levels,the accumulative TRs are 18, 24, and 29 for encoding withN2, N3, and N4, respectively. In addition to total repetitiontime, the scanning time for each basis function should alsobe included. Let the scanning time for one basis function beTs. The accumulative scanning time in unit of Ts is 139, 150,and 161 for encoding with N2, N3, and N4, respectively.Total imaging time is a sum of the accumulative repetitiontime and the accumulative scanning time. Qualitatively,encoding with N2 is the fastest while encoding with N4 is theslowest. The quantitative difference depends on the valuesof TR and Ts. Typically, TR is about 1 s and Ts is about0.02 s. Using these values, the total imaging time for sixlevels is 19.8 s and 31.2 s, respectively, for encoding withN2 and N4. This means that imaging with N2 is approxi-mately 60% faster than that with N4.

One may notice from Fig. 8, which displays the recon-structed images at different encoding levels, that the firsttwo images are quite blurred, and thus, do not carry muchinformation. Because each encoding level requires a certainnumber of TRs, more encoding levels lead to slower imag-ing. Here we simulate frame imaging with different totalnumbers of encoding levels to determine the best choice.The SNR and the accumulative repetition time are listed inTables 2 and 3, respectively. Difference in accumulativescanning time is not large enough to be considered. Thecomputational formula developed in [12] is utilized to cal-culate the SNR. Table 2 shows how many times smaller theSNR of frame encoding are compared to the SNR of con-ventional Fourier imaging under the assumption that thenoise is independent of imaging methods.

As shown in Table 2, the SNR does not change much asthe total number of encoding levels decreases from 6 to 3.When the number of levels further decreases to 2, the SNRdrops by 5–10%. For the sake of SNR, the total number ofencoding levels should be more than 2. Furthermore, encodingwith two levels provides only two levels of resolution andmay not be applicable to the cases where multiresolutionanalysis is desired. On the other hand, the imaging timeincreases as the total number of encoding levels increases.For example, the imaging time increases by about 30% when

the number of encoding levels increases from 3 to 4. Wesuggest that the best choice is encoding with three levels.

4.Conclusion

Simulation results show that all three splines encodedMRI produce very clear images with the application offrame reconstruction algorithm. When the effect of inde-pendent thermal noise is included, encoding with the qua-dratic spline basis provides the highest SNR, and thus, ismore promising than the linear and cubic spline bases.Imaging time is also estimated. It shows that linear spline isthe fastest of three. Without any degradation of SNR, oursimulation also suggests that the best choice of total encod-ing level is 3 to increase imaging speed as much as possible.It is also worthy to point out that our simulation procedureand analysis are general, and thus, can be readily applied tostudy other types of bases for frame-encoded MRI.

Acknowledgments

This research project is partially supported by the TexasHigher Education Coordinating Board Grant No.ARP999903-168.

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a numerical approach to the selection of basis for frame-encoded mri

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