Adaptive keyhole methods for dynamic magnetic resonance image reconstruction

Computerized Medical Imaging and Graphics 31 (2007) 458–468

Adaptive keyhole methods for dynamic magnetic resonance image reconstruction Zhaolin Chen, Jingxin Zhang ∗ , Khee K. Pang Department of Electrical and Computer Systems Engineering, Monash University, Clayton 3800, Vic., Australia Received 21 December 2005; accepted 24 April 2007

Abstract Dynamic magnetic resonance imaging (MRI) acquires a sequence of images for the visualization of the temporal variation of tissue or organs. Keyhole methods such as Fourier keyhole (FK) and keyhole SVD (KSVD) are the most popular methods for image reconstruction in dynamic MRI. This paper provides a class of adaptive keyhole methods, called adaptive FK (AFK) and adaptive KSVD (AKSVD), for dynamic MRI reconstruction. The proposed methods are based on the conventional Fourier encoding and SVD encoding schemes. Instead of the conventional keyhole methods’ duplication of un-acquired data from the reference images, the proposed methods use a temporal model to depict the inter-frame dynamic changes and to estimate the un-acquired data in each successive frame. Because the model is online identified from the acquired data, the proposed methods do not require the pre-imaging process, the navigator signals, and any prior knowledge of the imaged objects. Furthermore, the new methods use the conventional keyhole encoding schemes without the bias to any particular object characters, and the temporal model for updating information is in the general form of AR process without the preference to any particular motion types. Hence, the proposed methods are designed as a generic approach to dynamic MRI, other than for any specific class of objects. Studies on dynamic MRI data set show that the new methods can produce images with much lower reconstruction error than the conventional FK and KSVD. © 2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic MRI; FK; KSVD; Adaptive FK and KSVD; Temporal modeling

1. Introduction Magnetic resonance imaging (MRI) has significantly improved the clinical diagnosis because of its unique ability to acquire high resolution images of soft tissues and organs. Over several decades, research into MRI has therefore been very active and broad, ranging from fundamental physics to post-image processing. The books [1,2] provide an excellent introduction to MRI process from a signal processing perspective, and an overview of the current technical development of MRI can be found in [3]. Dynamic MRI has been introduced to capture image sequences for the examination of temporal variations of soft tissues or organs in cancer studies [4], cardiac imaging [5], and various clinical applications of dynamic contrast-enhanced imaging [6–8]. Precise analysis of dynamic

∗ Corresponding author at: 126, Building 72, Department of Electrical and Computer Systems Engineering, Monash University, Clayton 3800, Vic., Australia. Tel.: +61 3 9905 9613; fax: +61 3 9905 3454. E-mail address: [email protected] (J. Zhang).

0895-6111/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compmedimag.2007.04.005

changes can only be possible if a fast temporal sampling rate is ensured (often to satisfy the Nyquist sampling rate). This means the time interval allocated to the acquisition of each frame of the sequence is always limited. High spatial resolution, on the other hand, requires long acquisition period. Due to the fundamental limitation of MRI data acquisition speed, a trade-off between temporal and spatial resolutions has to be made in practice. This is a long standing problem in dynamic MRI studies. To address this problem, some fast imaging techniques have been developed. One of such techniques is parallel imaging [9,10]. This technique uses multiple receiver coils operating in parallel to pick up image data, with each receiver coil acquiring only a fraction of the data. Consequently, the data acquisition process can be speeded up several times which are bounded by the number of coils used. The ultra-fast imaging ability of parallel imaging is indeed very appealing, but their demanding hardware expenses are not negligible at present, and the implementation of parallel imaging technique is still in its infancy. Parallel to the aforementioned fast imaging methods, alternative methods have also been developed to achieve fast imaging within the current MR scanner structures and without expensive

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

hardware modifications. These alternative methods are based on optimal reduction of k-space sampling and optimal recovery of un-acquired data/information, where ‘optimal’ is defined with respect to the assumptions on the imaging process. One of such classes of methods is based on the well known keyhole idea [11]. Keyhole assumes that the changes among images are mostly encoded in the low spatial frequency components in k-space, and there is no significant changes at high spatial frequencies among the sequential k-spaces. Therefore, only low frequencies are acquired during scanning. Since Fourier encoding method was used for generating k-spaces in the seminal work [11], this method is called Fourier keyhole (FK). Similar to FK, keyhole singular value decomposition (KSVD) encoding methods [12–15] assume that most dynamic information is only embedded in the eigenimages associated with the large singular values of images (for simplicity, large eigenimages here after), and acquire only these large eigenimages during scanning. Since low frequency components or large eigenimages always contain the most energy of images, keyhole is a generic approach requiring no prior knowledge about the objects being imaged. Despite keyhole provides an optimal sampling scheme in the sense of maximal energy preservation, the loss of high frequencies or small eigenimages significantly degrade its performance in some dynamic MRI applications where the high frequency components change significantly, e.g. the border of objects changes over time. For details of the prohibitive limitation of FK and KSVD, please see [16]. To overcome this difficulty, the methods based on prior information of the object being imaged have also been investigated. Modeling motion harmonics in time with the aid of navigating signals is investigated in [17], and dynamic acquisition of cardiac images associated with aperiodic and respiratory motion is investigated in [18,19]. Even though these model based methods have shown good results, they are restricted to the applications where the model derives. Moreover, prior knowledge based methods assign the objects being imaged to specific models before collecting any data, hence, they often require pre-imaging process for model parameter estimation. This paper provides a class of adaptive keyhole methods that have been partially reported in our previous work [20,21] for dynamic MRI reconstruction. The proposed methods use a temporal model to estimate the un-acquired data. Different from the above mentioned prior knowledge based methods, the temporal model in this paper is identified directly from acquired data and is used to depict the inter-frame dynamic changes. The rest of this paper is organized as follows. In Section 2, the conventional keyhole methods are reviewed. In Section 3, the adaptive methods are introduced. The derivation and verification of the proposed model are provided. In Section 4, the new methods are tested on MRI data sets, and the analysis is made based on the results. Section 5 presents conclusion. 2. Conventional keyhole methods In order to understand the proposed methods, a brief review of FK and KSVD reduced encoding methods is given below. Consider a temporally varying 2D object O(r, t), where r denotes the spatial position of the object, and t denotes the discrete time

459

(temporal frames). In conventional keyhole methods, O(r, t) is decomposed into two parts as follows, O(r, t) = Omaj (r, t) + Omin (r, t),

(1)

where Omaj (r, t) and Omin (r, t) represent the major and minor information parts of the t th image frame, respectively. In the case of FK encoding, the spatially encoded data set is S(ω, t) = FT2(O(r, t)), where ω denotes the spatially sampled frequency, and FT2(·) denotes the 2D discrete Fourier transform with respect to r. The S(ω, t) is the so called k-space of the t th frame. With the Fourier transform, Eq. (1) can hence be described in frequency domain as ⎡ h1 ⎤ S (ω, t) ⎢ ⎥ FT2(O(r, t))=S(ω, t)= ⎣ Sl (ω, t) ⎦ : =Sl (ω, t) ⊕ Sh (ω, t), Sh2 (ω, t) (2) where Sl (ω, t) and Sh (ω, t) denote the low frequency and high frequency components of S(ω, t), corresponding, respectively, to Omaj (r, t) and Omin (r, t), and ⊕ denotes the direct sum of Sl (ω, t) and Sh (ω, t) along column direction. In order to reduce the spatial sampling time, conventional FK method acquires a full k-space only at the beginning of the sequence (t = 0) and uses it as the reference frame S(ω, 0). In following frames (t > 0), FK acquires only Sl (ω, t) and substitutes Sh (ω, t) with Sh (ω, 0). The image sequence is obtained by Fourier transform of Sl (ω, t) ⊕ Sh (ω, 0) for each t. In the case of KSVD encoding, the SVD of O(r, t) gives O(r, t) = U H (t)Λ(t)V H (t) where (•)H is the complex conjugate transpose, and U(t) and V (t) are, respectively, the left and right singular vector spaces of O(r, t). The encoding vectors U H (t)/U(t) and V H (t)/V (t) are normally preselected and fixed [22,23]. Hence, the SVD encoded data set is  Ul Λ(t) = U(t)O(r, t)V (t) ≈ UO(r, t)V = O(r, t)[ V l V s ] Us  U l O(r, t)V l U l O(r, t)V s = U s O(r, t)V l U s O(r, t)V s  Λl (t) 0 = := Λl (t) ⊕ Λs (t). (3) 0 Λs (t) In the above equation, Λl (t) is the submatix of large (most significant) singular values, Λs (t) the submatix of small (least significant) singular values, and U l (U s ) and V l (V s ) are, respectively, the left and right large (small) singular vector subspaces. U l Λl (t)V l is the large eigenimages corresponding to Omaj (r, t), and U s Λs (t)V s is the small eigenimages corresponding to Omin (r, t). Similar to FK method, Λ(t) is acquired only once at t = 0 and is used as the reference frame. For all t > 0, only Λl (t) is acquired, in the form of a whole image, by using the SVD encoding along vertical and horizontal directions; whereas, Λs (t) is replaced with Λs (0) from the reference frame. Final H H image sequence is obtained using O(r, t) = (U l ) Λl (t)(V l ) + H s H s s (U ) Λ (0)(V ) . If only one encoding direction, e.g. left

460

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

encoding vectors, is used, the encoding formula (3) can then be written as  U l O(r, t) := Λl (t) ⊕ Λs (t), Λ(t) ≈ U s O(r, t) and the image sequence can be obtained using O(r, t) = H (U l ) Λl (t) + (U s )H Λs (0) which is a simplified version of SVD. As can be seen from the above, in both FK and KSVD, the dynamic information of Omin (r, t) (Sh (ω, t) or Λs (t)) is lost. This may degrade, to some extend, the quality of reconstructed images. In order to partially recover this information, the proposed adaptive methods use an AR process to model the dynamic changes, and to estimate the lost information. 3. Adaptive keyhole method

tically too complicated to obtain the parameters of this general form model since the computation involved is prohibitive. To overcome this difficulty as well as maintain the model’s ability to capture temporal information, it is assumed that for ω in Sh (ω, t) ⊕ Sm (ω, t), the spatial prediction order P = 0. This assumption actually turns off the spatial domain prediction and gives a direct separation of ω and t. It is certain that separating ω and t of Omin (r, t) is not always valid, but the experimental studies in the following section have justified it by producing images with reduced artifacts. Similar to the assumption made in this work, the spatial domain and time domain separation of the whole objects has been previously used by Liang et al. [17] and by Aggarwal et al. [18], yielding promising results. Taking P = 0 in Eq. (6) gives the following expression, S(ω, t) = −

To develop the proposed adaptive keyhole methods, the object O(r, t) is first decomposed into O(r, t) = Omaj (r, t) + Omed (r, t) + Omin (r, t),

(4)

where Omaj (r, t), Omed (r, t) and Omin (r, t) are, respectively, the major, medium and minor information parts of the t th image frame. Performing the Fourier transform of Eq. (4) with respect to r gives S(ω, t) = S (ω, t) ⊕ S (ω, t) ⊕ S (ω, t), m

h

(5)

where Sl (ω, t), Sm (ω, t) and Sh (ω, t) are the k-space representation of Omaj (r, t), Omed (r, t) and Omin (r, t), respectively. Similar to conventional FK method, the Sm (ω, t) and Sl (ω, t) are acquired at each frame. To be comparable with FK, the number of data points acquired in Sm (ω, t) and Sl (ω, t) for the proposed adaptive FK method equals those in Sl (ω, t) for FK. Since Sl (ω, t) and Sm (ω, t) in (5) are acquired at each t, only h S (ω, t) needs to be estimated for each t th frame, t ≥ 1. In general, S(ω, t) as a function of ω and t can be regarded as the impulse response of a 2D IIR filter shown in the expression below S(ω, t) =

δ(ω, t) , L

P

am q1−m bn q2−n

L, P ≥ 0, a0 b0 = 1,

(6)

m=0 n=0

q1−m

am S(ω, t − m)

m=1

3.1. Adaptive Fourier keyhole (AFK) method

l

L

q2−n

where δ(ω, t) is a 2D impulse function, and are shift operators in temporal domain and spatial frequency domain, respectively. Note that ω indicates 2D spatial frequency positions, and therefore, ideally there would be two spatial shift operators along both row and column directions in a Cartesian sampled k-space. However, for simplicity only a single direction of regression along either column or row is considered here. A column-wise ARMA modeling method for a single image reconstruction has previously been used by Smith et al. [24,25]. The expression (6) models well the dynamic relationships of the spatial frequency ω and the temporal frames t. But it is prac-

+ δ(ω, t), ω ∈ {ω|Sh (ω, t) ⊕ Sm (ω, t)}.

(7)

Apparently from (7), if the model parameters ai are known, then any components of Sh (ω, t) ⊕ Sm (ω, t), t > 0, can be calculated sequentially using their corresponding components in S(ω, 0). Thus, Eq. (7) can be used to fill in those subblocks of S(ω, t), t > 0, which are deliberately un-acquired in FK method. Note that Eq. (7) holds for all components of Sh (ω, t) and m S (ω, t), which implies S(ωm , t) = −

L

am S(ωm , t − m) + δ(ω, t),

(8)

m=1

where S(ωm , t) denotes a particular frequency component in Sm (ω, t) that is acquired at each time t in FK imaging scheme. The estimation of the parameters of (8) is an AR-model parameter estimation problem. The forward prediction errors of Eq. (8) is given by Ef =

N L

1

|S(ωm , t) + am S(ωm , t − m)|2 . N t=L+1

(9)

m=1

The post-window least square algorithm [26] can be used to minimize the above cost functions and to obtain the estimated parameters aˆ m . With the estimated aˆ m , an estimate of S(ωm , t) can be calculated as h Sˆ (ω, t) = −

L

h

aˆ m Sˆ (ω, t − m).

(10)

m=1 h Replacing Sh (ω, t) in (5) with Sˆ (ω, t) gives the updated t th frame of k-space h ˆ S(ω, t) = Sˆ (ω, t) ⊕ Sm (ω, t) ⊕ Sl (ω, t).

(11)

Taking in Eq. (11) h Sˆ (ω, 0)=Sh (ω, 0),

h Sˆ (ω, t − m) = 0,

for t − m < 0, (12)

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468 h ˆ t) can be sequentially calculated starting Sˆ (ω, t) and hence S(ω, ˆ from t = 1. The reconstructed image O(t) can be obtained by ˆ the 2D Fourier transform of S(ω, t) with respect to ω. The procedure above is summarized in the following steps:

461

t th frame shown below, δ(t)

O(r, t)= 2π

L

P

,

L, P ≥ 0, a0 b0 =1.

am q1−m bn e−jnωr

m=0 n=0

(1) Acquire the fully sampled reference k-space S(ω, 0) and the truncated following k-space S(ω, t), t = 1, 2, . . . , N, as in conventional FK. (2) Calculate S(ωm , t) as described above. (3) Choose the order L with L < N for the AR model (7), use S(ωm , t) and the least square algorithm to obtain the estimated AR model parameter aˆ i . h (4) Use Eqs. (10) and (12) to calculate sequentially Sˆ (ω, t), starting from t = 1 up to t = N. (5) Use Eq. (11) to form the estimated full dimension k-space ˆ sequence S(ω, t), t = 1, 2, . . . , N, perform 2D Fourier ˆ transform on each frame of S(ω, t) to obtain the reconˆ structed images O(t), t = 1, 2, . . . , N. 3.1.1. Comparison of AFK and FK ˆ AFK and FK all use the estimated k-space S(ω, t) for followh ˆ ˆ ing frames. In AFK, S(ω, t) is generated by S(ω, t) = Sˆ (ω, t) ⊕ m l S (ω, t) ⊕ S (ω, t), where Sˆ h (ω, t) is adaptively updated for ˆ each (frame) time t using (10). While in FK, S(ω, t) is genˆ erated by S(ω, t) = Sh (ω, 0) ⊕ Sl (ω, t), where Sh (ω, 0) is from the reference frame that is fixed for each (frame) time t. Hence some high frequency dynamic information is lost in FK.

The development of proposed AKSVD is very similar to that of AFK. Consider again the decomposition in (4). Instead of performing Fourier encoding, one may apply the singular value decomposition encoding on the object and obtain the following SVD encoded data Ul

⎤

⎢ ⎥ Λ(t) ≈ ⎣ U m ⎦ O(r, t)[ V l Us

Vm

= Λl (t) ⊕ Λm (t) ⊕ Λs (t).

The numerator of the above equation is derived by the Fourier transform of δ(ω, t) with respect to ω since δ(ω, t) ≡ δ(ω)δ(t). In order to estimate the temporal changes, the similar assumption to that in AFK is made: for Omed (r, t) + Omin (r, t), P = 0. Based on this assumption, we obtain the following simplified expression, Omed (r, t) + Omin (r, t) =−

L

am (Omed (r, t − m) + Omin (r, t − m)) +

m=1

δ(t) . 2π

(15)

From the SVD encoding formula (13), we know that m )H + (U s )H Λs (t) Omed (r, t) + Omin (r, t) = (U m )H Λm (t)(V  m U (V s )H . Therefore, if we left multiply and right multiply Us [ V m V l ] on (15), we then obtain (Λm (t) ⊕ Λs (t))= −

L

am (Λm (t−m) ⊕ Λs (t − m)) +

m=1

δ(t) . 2π (16)

Similar to the parameter estimation in AFK, the singular vales σm (t) in Λm (t) acquired at each frame are used as the AR output. The post-window least square algorithm [26] is used for the ˆ s (t) is generated by estimation of am , and the final estimated Λ

3.2. Adaptive keyhole SVD (AKSVD) method

⎡

(14)

ˆ s (t) = − Λ

L

ˆ s (t − m), aˆ m Λ

m=1

ˆ s (t) = Λs (0), Λ

ˆ s (t − m) = 0, Λ

for t − m < 0.

(17)

ˆ s (t), a low rank estimate of Replacing Λs (t) in (13) with Λ O(r, t) for each t (frame) can be calculated by

Vs ]

ˆ t) = (U l )H Λl (t)(V l )H + (U m )H Λm (t)(V m )H O(r, (13)

In the above equation, Λl (t) is the submatix of large (most significant) singular values, Λs (t) the submatix of small (least significant) singular values, and Λm (t) is the singular values between Λl (t) and Λs (t). The decomposition of (13) maps the expression (4) to the SVD encoded space. According to keyhole, the encoded data set Λl (t) and Λm (t) are acquired at each frame, and therefore only Λs (t) needs to be estimated. To develop the proposed AKSVD, we adopt the frequency and time domain model (6). The Fourier transform of S(ω, t) with respect to ω gives the corresponding image of each

ˆ s (t)(V s )H . + (U s )H Λ

(18)

4. Examples The grapefruit data sequence used in this experiment was designed and acquired by Dr. W.S. Hoge at Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA. It was obtained on a 1.5 Tesla GE Signa LX scanner using a standard 2D Fast Spin Echo sequence (TR: 50 ms, TE: 7 ms). The data was acquired using 128 phase encodes along the horizontal direction (R/L) and 256 data points along the vertical

462

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

readout direction (A/P) for each of the total 30 frames. This frame sequence intents to capture the scenarios of a chop-stick “needle” inserting into a grapefruit. For each image the grapefruit was pulled out from the scanner and the chop-stick inserted slightly farther into the fruit, and then the fruit was repositioned in the scanner. The images reconstructed from this fully sampled data sequence (128 × 256 × 30) are considered as the experimentally true images (To form the 256 × 256 image size, the zero-padding was done before the Fourier transform of the corresponding k-space data). For Fourier encoding methods, we assumed that only the central 64 phase encodes were allowed for each subsequent frame, which offers a 2-fold accelerated scanning. The frequency points at row 32 and column 128 on each k-space map of the sequence were selected as S(ωm , t). However, in this paper the difference values, S(ωm , t) := S(ωm , t) − S(ωm , t − 1) instead of S(ωm , t) were used to form a sixth order AR process, and to estimate the parameters. Finally, the estimated sequence was obtained by (10) and (11). For SVD encoding methods, the 16 largest left vectors were used to encode the fully sampled images as the low rank estimation of each subsequent frame, which gives an acceleration factor of 8. The σm (t) := σm (t) − σm (t − 1) was used to obtain a fourth order AR model. The final images are obtained by (17) and (18). The comparisons of the overall performance of different methods are detailed in Table 1. The visual results are displayed in Figs. 1–9 10 and 11 the relative reconstruc. Figs. 256 compare 256 256 tion error, 256 | |/ i=1 j=1 i,j i=1 j=1 |ai,j |, where |i,j | is the

Table 1 Reconstruction methods and its reconstruction error Figures

Methods

Reconstruction errors (averaged over frames, %)

Fig. 1 Figs. 2 and 6 Figs. 3 and 7 Figs. 4 and 8 Figs. 5 and 9

Experimental true images FK AFK KSVD AKSVD

0 16.59 13.43 38.31 29.05

absolute value of the difference at the (i, j) th pixel of the t th frame, and |ai,j | is the absolute value at the (i, j) th pixel of the t th frame. From the visual observations in Figs. 2–5, the sequences of images estimated by the proposed methods show less blurry than their counterpart methods. Further, as can be observed from the error images in Figs. 6 and 8, both FK and KSVD methods produced images with greater incorrect information especially on the boundaries of the grapefruit. This is because the shape of the grapefruit and its position change with further inserting the needle into it at each time, and the inherited shape information from reference image is re-imaged on the successive images and hence results in blurry images. This clearly indicates that the conventional methods are inadequate for the estimation of such significant interimage changes. The conventional methods’ information duplication from reference is actually an inaccurate prediction and consequently brings incorrect information into the reconstructed images. In contrast, AFK in Fig. 7 and AKSVD

Fig. 1. Images reconstructed from fully sampled k-space as the experimentally true images. Each image is 256 × 256 pixels, and only displayed the central 182 × 182 pixels.

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

463

Fig. 2. FK images. Each displayed image is 182 × 182 pixels in size.

in Fig. 9 incorporate the temporal model that is identified across frames and used to estimate the un-acquired image information Omin (r, t), and hence, better reconstructions are obtained. From relative reconstruction error comparisons in Figs. 10 and 11, it

can be seen that both the proposed method produced a much lower level of errors than those of conventional methods for all frames. This confirms that the new methods have an enhanced ability to track the dynamical changes of the object.

Fig. 3. AFK images. Each displayed image is 182 × 182 pixels in size.

464

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

Fig. 4. KSVD images. Each displayed image is 182 × 182 pixels in size.

Fig. 5. AKSVD images. Each displayed image is 182 × 182 pixels in size.

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

Fig. 6. FK error images. Each displayed image is 182 × 182 pixels in size.

Fig. 7. AFK error images. Each displayed image is 182 × 182 pixels in size.

465

466

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

Fig. 8. KSVD error images. Each displayed image is 182 × 182 pixels in size.

Fig. 9. AKSVD error images. Each displayed image is 182 × 182 pixels in size.

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

467

are popularly performed to obtain the prior knowledge of the imaged objects; (4) unlike the prior knowledge based imaging methods, they can generally be used for any type of objects due to the texture-independent information compression ability of Fourier/SVD encoding and the online identified temporal model from data. Even though the initial aim of this paper is to give a class of generic methods for dynamic MRI reconstruction, the presented methods can also be potentially useful for 3D multislice imaging by replacing the temporal dimension to another spatial dimension. Acknowledgements Fig. 10. Relative error comparison of FK and AFK methods for all frames.

The authors gratefully acknowledge Dr. W.S. Hoge at Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA. for providing MRI data for this work. References

Fig. 11. Relative error comparison of KSVD and AKSVD methods for all frames.

Furthermore, by using Fourier and SVD encoding which give the object-independent optimal energy compression ability, the proposed methods can be expected as more generic methods than other prior knowledge based methods for solving such reduced encoding problem in dynamic MRI. However, one of the major practical limits of AFK and AKSVD is the requirement of relatively long image sequence for the pursuit of precisely estimated model parameters. 5. Conclusion A class of adaptive keyhole methods for dynamic MRI sequence estimation have been presented in this paper. The experimental studies have demonstrated its superiority over the conventional FK and KSVD methods. The major advantages of the new methods lie in (1) they produce the reduced reconstruction error by which images are more accurately recovered; (2) the proposed methods are based on the Fourier and SVD imaging schemes and therefore can be applied onto machines without any special rf pulse sequence; (3) they do not require any pre-imaging process [19] or navigating process [17] which

[1] Liang Z-P, Lauterbur PC. Principles of magnetic resonance imaging. IEEE Press Series in Biomedical Engineering; 2000. [2] Vlaardingerbroek M, den Boer J. Magnetic resonance imaging theory and pratice. 3rd ed Berlin/Heidelberg/New York: Springer-Verlag; 2003. [3] Riederer S. Current technical development of magnetic resonance imaging. IEEE Eng Med Biol Mag 2000;19:34. [4] Wallis F, Gilbert F. Magnetic resonance imaging in oncology: an overview. JR Coll Surg Edinb 1999;44:117. [5] Yang P, Kerr A, Liu A, Liang D, Hardy C, Meyer C, et al. New realtime ineractive cardiac magnetic resonance imaging system complements echocardiography. J Am Coll Cariol 1998;32:2049. [6] Hayes C, Padhani AR, Leach MO. Assessing changes in tumour vascular function using dynamic contrast-enhanced magnetic resonance imaging. NMR Biomed 2002;15:154. [7] Benjaminsen IC, Graff BA, Brurberg KG, Rofstad EK. Assessment of tumor blood perfusion by high-resolution dynamic contrast-enhanced MRI: a preclinical study of human melanoma xenografts. Magn Reson Med 2004;52:269. [8] Noworolski S, Henry R, Vigneron D, Kurhanewicz J. Dynamic contrastenhanced MRI in normal and abnormal prostate tissues as defined by biopsy, MRI, and 3D MRS. Magn Reson Med 2005;53:249. [9] Sodickson D, Manning W. Simultaneous acquisition of spatial harmonics (SMASH): ultra-fast imaging with radiofrequency coil arrays. Magn Reson Med 1997;38:591. [10] Pruessmann KP, Weiger M, Scheidegger MB, Peter B. SENSE: sensitivity encoding for fast MRI. Magn Reson Med 1999;42:952. [11] van Vaals J, Brummer M, Dixon W, Tuithof H, Engels H, Nelson R, et al. Keyhole method for accelerating imaging of conrast agent uptake. Magn Reson Med 1993;3:671. [12] Zientara G, Panych L, Kuethe D. Dynamically adaptive MRI method using the SVD. In: First Meeting of the Society of Magnetic Resonance, vol. 4. 1994. [13] Zientara G, Panych L, Jolesz F. Dynamically adaptive MRI with encoding by singular value decomposition. Magn Reson Med 1994;32:268. [14] Mitsouras D, Edelman A, Panych L, Jolesz F, Zientara G. Fast mutipleexcitation multiple-echo spin-echo pules sequence for non-Fourier spatial encoding. In: Proc. ISMRM 9th Scientific Meeting. 2001. [15] Panych L. Partial non-Fourier encoded MRI. In: Proc. IEEE International Symposium on Biomedical Imaging. 2002. [16] Hoge W, Miller E, Lev-Ari H, Brooks D, Panych L. A doubly adaptive approach to dynamic MRI sequence estimation. IEEE Trans Image Process 2002;11:1168. [17] Liang Z-P, Jiang H, Hess CP, Lauterbur PC. Dynamic imaging by model estimation. Int J Imaging Syst Technol 1997;8:551.

468

Z. Chen et al. / Computerized Medical Imaging and Graphics 31 (2007) 458–468

[18] Aggarwal N, Zhao Q, Bresler Y. Spatio-temporal modeling and minimum redundancy adaptive acquisition in dynamic MRI. In: Biomedical Imaging, 2002. Proceedings, IEEE International Symposium. 2002. [19] Aggarwal N, Bandyopadhyay S, Bresler Y. Spatio-temporal modeling and adaptive acquisition for cardiac MRI. In: Second IEEE International Symposium on Biomedical Imaging: Macro to Nano, vol. 1. 2004. p. 628. [20] Chen Z, Zhang J, Pang KK. Adaptive k-space updating methods for dynamic MRI sequence estimation. In: Proc. of 27th Anual International Conference of the IEEE EMBS. 2005. [21] Chen Z, Zhang J, Pang KK. Adaptive keyhole SVD method for dynamic MRI reconstruction. In: The Eighth International Symposium on Signal Processing and Its Applications. 2005. [22] Zientara G, Panych L, Jolesz F. Keyhole SVD encoded MRI. In: Proc. SMR. Second Annual Meeting. 1994. [23] Panych L, Oesterle C, Zientara G, Henning J. Implementation of a fast gradient-echo SVD encoding technique for dynamic imaging. Magn Reson Med 1996;35:554. [24] Smith MR, Nichols ST, Henkelman RM, Wood ML. Application of autoregressive moving average parametric modeling in magnetic resonance image reconstruction. IEEE Trans Med Imaging 1986;5MI:132. [25] Yang J, Smith MR. Constrained and adaptive ARMA modeling as an alternative to the DFT-with application to MRI. Control Dyn Syst 1996; 77:225. [26] Stoica P, Moses R. Introduction to spectral analysis.Upper Saddle River, New Jersey: Prentice-Hall Inc.; 1997. p. 07458. Zhaolin Chen received his bachelor in electrical engineering from Harbin Institute of Technology, China, in 2002. He is now working towards to his PhD degree

in the Department of Electrical and Computer System Engineering, Monash University, Australia. His research interests are in MR image reconstruction, system identification, spectrum estimation and filter banks applications. Jingxin Zhang received MEng and Phd degrees in electrical engineering, in 1983 and 1988, from Northeastern University, China. Between 1988 and 1992, he was with Northeastern University, China, as associate professor. Since 1989, he has held research positions in University of Florence, Italy, University of Melbourne, and Cooperative Research Center for Sensor Signal and Information Processing, Australia, and senior lecturer position in University of South Australia and Deakin University, Australia. He is currently with the Department of Electrical and Computer Systems Engineering, Monash University, Australia. He is the author and coauthor of over 130 research papers in diverse research areas such as adaptive and predictive control, time varying systems, robust filtering, multirate signal processing and medical imaging. His current research interests are in control and signal processing and their applications to medical and industrial systems. Dr. Zhang is recipient of 1989 Fok Ying Tong Educational Foundation (Hong Kong) for the outstanding Young Faculty Members in China and 1992 China National Education Committee Award for the Advancement of Science and Technology. Khee K. Pang is Honorary Research Associate at Monash University. He was formerly the deputy Director of Centre for Telecommunications and Information Engineering (CTIE). He has over thirty years of teaching and research experience in Australia, USA, and Japan. As Principal Investigator of three consecutive large research projects, he was involved in research in a number of areas, which include video and multi-media communications, source and channel coding, wireless communications and digital signal processing.