Phantom-Based Evaluation of Isotropic Reconstruction of 4-D MRI Volumes using Super-Resolution Eric Van Reeth∗ , Ivan WK Tham† , Cher Heng Tan‡ and Chueh Loo Poh∗ ∗ School

of Chemical and Bioengineering, Nanyang Technological University, Singapore of Radiation Oncology, National University Cancer Institute , Singapore ‡ Department of Diagnostic Radiology, Tan Tock Seng Hospital, Singapore

† Department

Abstract—This article investigates the feasibility of isotropic super-resolution reconstruction on 4-D (3-D + time) thoracic MRI data. 4-D MRI sequences generally have high temporal resolution to characterize dynamic phenomena but poor spatial resolution, creating highly anisotropic voxels elongated in the slice-select dimension. Isotropic post-acquisition reconstruction can be obtained using super-resolution algorithms. A new MRI compatible phantom design that simulates lung tumour motion is introduced to evaluate the feasibility and performance of the proposed superresolution algorithm in the context of 4-D MRI. Several orthogonal low-resolution acquisitions of the phantom are performed through time using a fast true 3-D gradient echo based sequence. The acquired volumes are then registered and combined using a totalvariation based regularizer super-resolution algorithm to obtain the high-resolution volume. The quality of the reconstruction is evaluated by measuring the mutual information between the reconstructed volume and a direct isotropic 3-D acquisition. Subjective and objective evaluations show the superiority of our approach compared to the averaging method. This article also discusses the influence of various parameters such as the number of low-resolution scans used and the influence of automatic motion estimation versus known displacement. Index Terms—Medical imaging, Super-resolution, MRI, Phantom, Interpolation, Volume reconstruction

I. I NTRODUCTION In many medical applications, high-resolution three-dimensional (3-D) magnetic resonance imaging (MRI) images are required to facilitate early and accurate diagnosis. In practice, the direct acquisition of high-resolution volumes can require long

scanning times which might not be applicable to study dynamic phenomena (foetal imaging, thoracic or abdominal imaging). To prevent motion artifacts, a variety of fast acquisition techniques have been developed in both 2-D and 3-D. For example, fast 2-D acquisitions can be performed with half-Fourier acquisition single-shot spin-echo or steady state free precession gradient recalled echo sequences, and true 3-D acquisitions can be obtained using fast gradient-echo sequences or echo-planar sequences (SSEPI). A sequence of 3-D acquisitions of the same volume over time creates a 4-D dataset that represents the dynamic evolution of the anatomy. High temporal resolution is often sought but is generally reached at the expense of spatial resolution in both through-plane and in-plane dimensions [1] to maintain decent signal-to-noise ratio (SNR). The slice thickness is usually increased which can lead to a poor detail characterization in the throughplane dimension of small but clinically significant tissues. In order to retrieve high-resolution information, super-resolution reconstruction (SRR) methods have been successfully applied to MRI data. SRR is a post-acquisition technique that combines lowresolution independent observations to reconstruct a high-resolution image. Unlike classic interpolation techniques (linear, bi-cubic, zero-paddinginterpolation), SRR is able to add significant frequency content into the reconstructed volume and efficiently remove the aliasing that was initially present in the images due to the low acquisition sampling rate.

A. Previous Work Increasing the in-plane resolution of MR images using SRR has been a subject of controversy when shifted acquisitions of static subjects were combined [2]–[4]. In any case, even if motion ensures that independent information is present in every image and provides a sufficient basis for SRR to succeed, Mayer and Vrscay have shown in [5] that the potential resolution enhancement in the inplane direction is very limited. For these reasons, most studies have focused on reconstructing images with isotropic voxels by matching the through-plane resolution to the in-plane resolution. Isotropic reconstruction has been successfully applied on both static and moving subjects. Static subjects are out of the scope of this paper but a complete review of such applications can be found in [6], [7]. Moving subjects SRR has been a subject of growing interest since first encouraging results were shown in [8]. In this paper, Rousseau et al. have combined three orthogonal 2-D multislice stacks of foetal brain images to reconstruct an isotropic high-resolution 3-D image. Since motion can occur between slice acquisitions, a slice-tovolume registration step was needed to create a coherent volume for each plane. Subsequently, the three orthogonal volumes were registered to each other and combined to produce the isotropic output. A rigid motion model with six degrees of freedom (three rotations, three translations) was used and showed to be able to correctly capture the foetus head motion. A similar approach using scattered data interpolation was proposed later in [9] before Gholipour et al. formalized this approach under a SR framework in [10]. More recently, SRR isotropic reconstruction using orthogonal acquisitions was applied to tongue data in [11]. In this approach, no motion occurred during the acquisition of each volume but the subject moved between each acquisition for swallowing purposes. Consequently, the authors had to perform a local deformable registration to match the three orthogonal volumes. This study shows the applicability of SRR techniques to different types of motion, as long as the registration is able to provide sub-pixel accuracy. Finally, isotropic SRR was applied in the context of diffusion MRI

using single-shot echo-planar imaging (SSEPI) orthogonal volumes in [12]. Unlike previous acquisition sequences, SSEPI allows to acquire a whole slab covering the subject after a single excitation. This acquisition protocol produces an anatomical representative 3-D volume not corrupted by motion artifacts but with low spatial resolution. Subsequent spatial alignment of the acquired volumes was performed using distortion information obtained by the acquisition of a field-map and rigid registration which provided sufficient information for isotropic reconstruction. B. Scope of the article This article investigates the applicability of isotropic SRR on 4-D (3-D + time) MRI datasets to ultimately improve lung tumour characterization. Volumes acquired with true 3-D sequences present different characteristics than 2-D multi-slice scans: i) the whole volume is excited by the RF pulse instead of thin slice selection, ii) an additional phase-encoding gradient is needed to encode the third dimension, iii) a 3-D Fourier inverse transform of the k-space is applied to reconstruct the volume in the physical space. Dinkel et al. have shown the possibility of lung tumour tracking using 4D MRI in [13]. In [14], Yang et al. have successfully monitored the lung anatomy during the breathing cycle using a 4-D MRI TWIST (Timeresolved Angiography With Interleaved Stochastic Trajectories) sequence. This true 3-D fast gradientecho sequence allows to scan the whole thoracic cage in less than a second and offers reasonable SNR and limited distortions. However, the acquired volumes present strong partial volume effect due to the low spatial resolution which currently limits the accurate assessment of anatomical details, accurate registration/segmentation and positioning of lung tumours. Enhancing the resolution of these scans is likely to improve such processing which is the reason why the applicability of SR algorithms is investigated in this context. This paper proposes a new phantom design that simulates typical lung tumour motion during the respiratory cycle. Motion is simulated by successively translating and rotating the phantom at six

pre-determined positions. At each position, orthogonal volumes are acquired in the coronal, axial and sagittal planes using the TWIST sequence and are subsequently combined using SRR to reconstruct an isotropic volume. Additionally, an isotropic high resolution scan of the phantom is performed and used as a reference to evaluate the performance of the isotropic reconstruction. This article aims at investigating the following points: • High-resolution isotropic volume reconstruction can be obtained using a series of orthogonal true 3-D anisotropic scans of a moving subject with rigid motion. • 3-D automatic registration is able to accurately estimate the phantom motion, despite the presence of strong partial volume effect and the different volume coordinate systems. The difference between the automatic registration based reconstruction and the reconstruction based on the exact (known) displacement is analysed. • SRR based on automatic registration leads to a clear resolution enhancement in the throughplane dimension and outperform the classic averaging method. The influence of various parameters such as the number of combined volumes and the accuracy of estimated registration versus known registration are evaluated and discussed. Detailed descriptions of both the phantom and the algorithm used in this study are given respectively in section II and III. Results are presented in section IV and discussed in section V which also includes future studies that can be derived from this work. II. M ATERIAL A. Phantom The phantom is presented in Figure 1. It is composed of three equally spaced plastic grid layers which are fixed into an acrylic box filled with a saline solution (sodium chloride 0.9%). The size of the acrylic box is 11.1 × 6.4 × 8.3 cm3 and the grid is composed of 1.5 × 1.5 cm2 squares. Each grid layer is 0.9 cm thick, and the spacing between each grid is also set to 0.9 cm. The phantom can be fitted into an acrylic support shown in Figure 2.

The support presents six distinct positions at which the phantom is successfully moved between each acquisition to simulate the motion. We will refer at the successive phantom positions by {P1 , . . . , P6 }. The simulated motion is representative of a lung tumour motion during a respiratory cycle. It follows an hysteresis loop pattern as suggested in [15]. 3-D translations are performed, with the main displacement corresponding to the cranio-caudal axis following what has been observed in typical tumour displacement during a breathing cycle in [16]. One additional rotation around the vertical axis was added to challenge the registration process. The maximal amplitude of the translation is 2.5 × 4.5 × 2.0 cm3 which falls into the range of typical lung tumour displacement range presented in [16], and the maximal rotation is 15◦ . B. MRI acquisition Successive orthogonal acquisitions of the phantom were performed at each position in the coronal, axial and sagittal planes using the TWIST true 3-D sequence with a 3T MRI scanner (Trio, Siemens, Erlangen, Germany) and a 6 channel body matrix coil. Each volume was acquired with a voxel size of 1.5 × 1.5 × 8 mm3 and took between 4s and 5s to acquire depending on the plane orientation. This scanning time is considered to be an applicable breath-hold time in practice even for patients with tumours. A total of eighteen volumes were acquired, corresponding to the three orthogonal scans at each of the six positions. The MRI scans are presented in Figure 3 (a)-(c) for each acquisition plane. The outof-plane acquisitions are also shown and linearly interpolated to match the in-plane resolution of the acquisition. The air bubbles that can be seen on the photos in Figures 1 and 2 were not present during the acquisition and did not disrupt the scans. In this study, the in-plane resolution was chosen so that the grid geometry could be correctly visualized. However the slice thickness is purposely set to a high value in order to introduce aliasing in the through-plane dimension. The through-plane sampling rate is indeed too low to capture the alternation of the grids as can be observed in the outof-plane images of Figure 3(a)-(c). Additionally, a

(a) Axial View

(b) Coronal View

(c) Sagittal View

(a) Axial Scan

Fig. 1: Orthogonal views of the phantom used in this study.

(b) Coronal Scan

(a) Position P3

(b) Position P6

Fig. 2: Phantom on its support at two distinct positions, P3 and P6 . Rotation and spatial shifts can be observed comparing the two images.

high-resolution reference volume shown in Figure 3(d) was acquired at 1.5 × 1.5 × 1.5 mm3 and took 24s to acquire. The reference volume is ultimately compared with the isotropic reconstruction obtained with the SR algorithm described in next section.

(c) Sagittal Scan

(d) Isotropic Scan

III. M ETHODS A. Registration algorithm In this study, image registration is used to compute the displacement of the phantom between two positions. Before being registered, the volumes are re-sampled to the same isotropic resolution with a tri-linear interpolation to apply registration on equally sampled volumes. A rigid, six degree-offreedom (3-D translation, 3-D rotations) algorithm is considered using the intensity mean-square error as the metric to be minimized. The challenge consists of performing accurate registration between volumes acquired in different planes. B. Super-resolution algorithm A SR regularized approach is used to estimate the optimal high-resolution solution X, from the classic

(e) Average Reconstruction

(f) SRR Reconstruction

Fig. 3: (a)-(d) Original MRI scans at position P1 . (e) and (f) are respectively the average and SRR reconstructions. The first column corresponds to the axial plane, second and third column correspond to the coronal and sagittal planes.

observation model of equation 1. This model assumes that the low-resolution observations that are acquired are displaced, blurred, down-sampled and noisy versions of the ideal high-resolution image X as illustrated in the following equation: Yk = Dk Bk Gk X + Vk , k = {1, . . . , N }

(1)

In this equation, N represents the total number of low-resolution observations used to estimate X, X is the high-resolution image given at an arbitrary reference position kr , Gk is the geometric transformation from the image Yk to the reference image Ykr , Bk is the space-variant blur operator modelling the PSF of the imaging process, Dk is the downsampling operator and Vk is an additive zero mean Gaussian noise. In this study, we consider a space invariant 3-D PSF so that the operator Bk becomes B. To be specific, the PSF is assumed to have a Gaussian profile and the full width at half maximum in the slice-select direction is set equal to the slice thickness as mentioned in [6]. Finally, the operator Gk encompasses the plane acquisition orientation transformation and the displacement given by the registration step. From the model of equation 1, the optimal highˆ should minimize the distance resolution solution X between the real observations YK and the estimated ˆ so that: observations Yˆk = Dk Bk Gk X "N #

2 X

ˆ X = arg min (2)

Yk − Yk ˆ X

k=1

This term is often referred as the data fidelity term in the literature. Considering only this term ˆ would result in an ill-posed underto estimate X determined problem and regularization procedures are generally adopted to stabilize the solution and improve the convergence rate. Regularizers usually limit the apparition of high-frequency content in the reconstructed image to avoid noisy solution but tend to over-smooth the image. Total variation (TV) regularizers have been successfully applied to SR methods and showed interesting edge preserving properties [17] while limiting the apparition of noise by penalizing the total amount of variation inside the estimated image. The total variation operator is

defined as the L1 norm of the gradient operator: ΓT V (X) = ||∇X||1 Farsiu et al. have introduced in [18] a TV-based regularizer called bilateral-TV that proved to produce high-quality results and to be computationally efficient which is an interesting consideration when considering 3-D data processing. We propose to extend this regularizer in 3-D to fit our purpose and define the tri-lateral regularizer as: ΓT T V (X) =

P X

lmn α|m|+|l|+|n| ||X − Sxyz X||1

l=−P m=−P n=0 l+m+n≥0

lmn = Sxl Sym Szn regroups where the operator Sxyz respectively the 3-D horizontal, vertical and depth shift operators by a distance of l, m and n pixels. Note that with P = 1, the trilateral-TV is proportional to the classic TV operator. The addition of the prior term into the global minimization term gives the following formulation: "N #

2 X

ˆ ˆ X = arg min

Yk − Yk + λΓT T V (X) ˆ X

k=1

2

where λ balances the influence of the prior term versus the data fidelity term. It is common to use high λ values for highly under-determined problems to compensate the lack of observations. In this study, parameters were fixed regardless of the combination of low-resolution inputs in order to compare the results on a similar basis (P = 2, λ = 0.005, α = 0.75). A gradient descent algorithm with fixed step size (β = 2) was used to iteratively converge towards the optimal solution as illustrated in Equation 3. in which T represents the transpose operator, sgn the sign function and Id the identity matrix. A stable solution was considered to be reached when the difference between two successive iterations was below a fixed threshold which typically corresponded to less than ten iterations in practice. IV. R ESULTS The SR reconstruction is compared to the isotropic scan acquisition which is taken as a refer-

ˆ n+1 = X ˆ n +β X

N hX k=1

GTk B T DkT (Yk − Yˆkn )−λ

P X

i −(lmn) lmn ˆ n ˆ n −Sxyz α|m|+|l|+|n| [Id−Sxyz ] sgn(X X ) (3)

l=−P m=−P n=0 l+m+n≥0

ence. Mutual information is used as a comparison metric because PSNR and mean-square error rely on intensity consistence which is not guaranteed between two MRI scans. Three methods are compared. The averaging method (AVE), the SR method based on known motion (TV-SRk) and the SR method based on computed motion (TV-SRc). A. Testing procedure For each method, various combinations of lowresolution inputs are tested and averaged to compute the mean quality improvement of the reconstructed volume. Each test consists of combining a similar number of axial, coronal and sagittal volumes. However, the positions of the volumes for each plane are chosen independently and randomly. Let N be the total number of combined low-resolution scans, N will take successive values of {3, 6, 9, 12, 15, 18}. Figure 4 represents the mutual information results for increasing values of N . For a given N , the graph shows the average mutual information result obtained from 10 distinct volume combinations per position, and in turn averaged for all six positions. Notice that the value at N = 1 corresponds to the average mutual information result between the reference isotropic volume and each of the lowresolution initial scans (axial, coronal and sagittal) at all six positions. B. Interpretation Figures 3(e) and 3(f) show the resulting isotropic reconstructed volume using the AVE and TV-SRk methods with 18 low-resolution inputs. Figure 4 illustrates the mutual information evolution of: i) the average reconstruction of registered low-resolution inputs using known displacement (AVE), ii) the result of the total variation based SR method using computed displacement (TV-SRc), iii) the result of the total variation based SR method using known

displacement (TV-SRk). First of all, the superiority of the proposed TV-SR method is clearly visible on the images shown in Figures 3(e) and 3(f). Edge sharpness and contrast are strongly enhanced by the proposed SR method. High frequencies are introduced and successfully preserved by the TV-based regularizer. Visual inspection of the phantom structure also shows a high level of similarity between the reconstructed volume and the direct isotropic acquisition. This subjective comparison is confirmed by the mutual information evolution shown in Figure 4. Both TV-SRk and TV-SRc reach much higher values than the AVE method. The AVE curve slightly increases until 6 low-resolution volumes are combined and then reaches a steady value. This suggests that averaging up to 6 volumes decreases the noise level that is quite high in the low-resolution scans which in turn increases the mutual information with the isotropic scan which is less noisy. However, combining more than 6 volumes has a very low impact on the mutual information which allows to conclude that no new information is added to the averaged volume as could be expected of this method. On the opposite, both TV-SR results are improved when more lowresolution inputs are considered which demonstrates the resolution improvement ability of such methods. Secondly, it is interesting to compare both TV-SR to study the impact of registration errors. Figure 4 illustrates the superiority of the TV-SRk method over the TV-SRc, emphasizing the importance of accurate registration. Looking at the TV-SRc curve, in addition to the fact that the overall reconstruction is globally inferior to TV-SRk, the relative improvement between two consecutive values is also lower. The addition of small registration errors gradually lowers the quality of the reconstruction. This suggests that in practice, resolution improvement is

1.85

MI Value (bits)

1.8 1.75 1.7 1.65

TV-SRk TV-SRc AVE

1.6 0

2

4

6 8 10 12 14 Number of LR volumes

16

18

Fig. 4: Mutual Information between the highresolution reference and the reconstructed SR volume using different combinations of low-resolution (LR) volumes.

possible using automatic registration based SR but that the overall resolution improvement is limited by the accuracy of the registration. After some point, adding more low-resolution volumes will not significantly improve the reconstruction quality. C. Computational complexity In practice, a stable estimation of the solution was typically obtained in less than 10 iterations of equation 3. It has been noticed that when more low-resolution volumes are used, a lower number of iterations is needed to converge. In other words, introducing more observations into the estimation problem improves the convergence rate. To reduce the computational burden at each iteration, the displacements between positions were computed before the SR process. The resulting computational time is a function of the number of low-resolution volumes used, ranging from 60 to 280 seconds in average when using 3 to 18 low-resolution volumes. Tests were run using Matlab, The MathWorks Inc. version 7.5.0, on an Intel Double Quad Core Xeon CPU with 2.00 GHz. V. C ONCLUSION In this article, the feasibility of SRR on 4-D MRI data was investigated using a phantom designed for this purpose. The phantom offers

the possibility to access a high-resolution ground truth to which the reconstructed results can be compared. Motion is simulated by displacing the phantom to pre-determined positions which allows to compare the reconstruction based on automatically registered motion (TV-SRc) that would be used in a real application, to the one based on ground-truth displacement (TV-SRk). Both subjective and objective evaluations clearly proved the superiority of the proposed TV-SR approach over the averaging method in terms of resolution enhancement. It proves that despite the strong partial volume effect imposed by the short scanning times, registration is still able to estimate the phantom displacement. SRR allows to reach isotropic reconstruction which could not be directly acquired in practice because of subject motion. It also reduces the partial volume effect, improves contrast and SNR values which usually limit the accuracy of image analysis algorithms. In the context of dynamic thoracic imaging, enhancing the resolution of 4D sequences should help to improve the segmentation and characterization (volume, motion path) of anatomical structures of interest including lung tumours. The off-line nature of the algorithm offers the possibility to obtain isotropy from low-resolution volumes with few changes on the acquisition protocol and almost no increase in scanning time. In our case, six low-resolution scans could be acquired within the same scanning time as the direct isotropic acquisition. It can be seen from Figure 4 that using six low-resolution orthogonal volumes already leads to a high quality reconstruction. Finally, the presented phantom will allow future investigations on other aspects of SRR. In this article, the influence of the number of lowresolution scans used for SRR is studied. Future work will include the influence of the number of planes used. It is expected that increasing the number of planes will lead to better isotropic reconstruction. This study could help to know the acquisition protocol that optimizes the acquisition time versus reconstruction quality ratio. Future work will also include improvements on

the phantom design. Instead of being parallel to the acquisition planes, the plastic grids will be orientated so that they intersect all three scanning planes, which is likely to be closer to a real-case scenario. This is expected to give more realistic information about the influence of each plane when combined into the SR process. Moreover, such a phantom is a useful tool to evaluate and compare different SR algorithms and regularizers. This work has shown that registration errors significantly affect the quality of the reconstruction. The proposed phantom could be a good evaluation tool for the development of registration-error robust SRR algorithms. In addition, many SRR approaches have been proposed recently but few comparisons have been performed due to the lack of common evaluation data. Such real-scanned phantom data could be used to compare the performance of various SRR algorithms in the context of 4-D MRI. ACKNOWLEDGEMENTS This work was supported by Singapore National Medical Research Council (NMRC) under Grant number NMRC/NIG/1033/2010. R EFERENCES [1] A. Webb, Introduction to Biomedical Imaging, ser. Metin Akay, N. Y. Wiley-Interscience, Ed. IEEE Press Series in Biomedical Engineering, 2003. [2] Y. Yeshurun and S. Peled, “Superresolution in mri: Application to human white matter fiber tract visualization by diffusion tensor imaging,” Magnet Reson Med, vol. 45, pp. 29–35, 2001. [3] K. Scheffler, “Superresolution in mri?” Magnet Reson Med, vol. 48, no. 2, pp. 408–408, 2002. [Online]. Available: http://dx.doi.org/10.1002/mrm.10203 [4] Q. M. Tieng, G. J. Cowin, D. C. Reutens, G. J. Galloway, and V. Vegh, “Mri demodulation frequency changes provide different information,” Magnet Reson Med, vol. 66, no. 6, pp. 1513–1514, 2011. [5] G. S. Mayer and E. R. Vrscay, “Measuring information gain for frequency-encoded super-resolution mri,” Magn Reson Imaging, vol. 25, no. 7, pp. 1058–1069, 2007. [6] E. Van Reeth, I. W. K. Tham, C. H. Tan, and C. L. Poh, “Super-resolution in magnetic resonance imaging: A review,” Concept Magnetic Res - Part A, vol. 40A, no. 6, pp. 306–325, 2012. [Online]. Available: http://dx.doi.org/10.1002/cmr.a.21249 [7] H. Greenspan, “Super-resolution in medical imaging,” Comput J, vol. 52, pp. 43–63, 2009.

[8] F. Rousseau, O. Glenn, B. Iordanova, C. RodriguezCarranza, D. Vigneron, J. Barkovich, and C. Studholme, “A novel approach to high resolution fetal brain mr imaging,” in Medical Image Computing and Computer-Assisted Intervention - MICCAI, T. Jiang, N. Navab, J. Pluim, and M. Viergever, Eds. Palm Springs, California, USA: Springer Berlin / Heidelberg, October 2005, pp. 548–555. [9] S. Jiang, H. Xue, S. Counsell, M. Anjari, J. Allsop, M. Rutherford, D. Rueckert, and J. V. Hajnal, “Diffusion tensor imaging (dti) of the brain in moving subjects: Application to in-utero fetal and ex-utero studies,” Magnet Reson Med, vol. 62, no. 3, pp. 645–655, 2009. [Online]. Available: http://dx.doi.org/10.1002/mrm.22032 [10] A. Gholipour, J. A. Estroff, and S. K. Warfield, “Robust super-resolution volume reconstruction from slice acquisitions: Application to fetal brain mri,” IEEE T Med Imaging, vol. 29, no. 10, pp. 1739–1758, 2010. [11] J. Woo, Y. Bai, S. Roy, E. Z. Murano, M. Stone, and J. L. Prince, “Super-resolution reconstruction for tongue mr images,” in SPIE Medical Imaging, D. R. Haynor and S. Ourselin, Eds., vol. 8314, no. 1. SPIE, 2012, p. 83140C. [12] B. Scherrer, A. Gholipour, and S. Warfield, “Superresolution reconstruction to increase the spatial resolution of diffusion weighted images from orthogonal anisotropic acquisitions,” Med Image Anal, vol. 16, no. 7, pp. 1465– 1476, 2012. [13] J. Dinkel, C. Hintze, R. Tetzlaff, P. E. Huber, K. Herfarth, J. Debus, H. U. Kauczor, and C. Thieke, “4d-mri analysis of lung tumor motion in patients with hemidiaphragmatic paralysis,” Radiother Oncol, vol. 91, no. 3, pp. 449 – 454, 2009. [Online]. Available: http://www.sciencedirect.com/ science/article/pii/S0167814009001455 [14] Y. Yang, C. H. Tan, and C. L. Poh, “Visualization of lung using 4d magnetic resonance imaging,” in Proc. of the First International Symposium on Bioengineering, 2011. [15] E. Kyriakou and D. R. McKenzie, “Dynamic modeling of lung tumor motion during respiration,” Phys Med Biol, vol. 56, no. 10, pp. 2999–3013, MAY 21 2011. [16] T. Neicu, H. Shirato, Y. Seppenwoolde, and S. B. Jiang, “Synchronized moving aperture radiation therapy (smart): average tumour trajectory for lung patients,” Phys Med Biol, vol. 48, no. 5, p. 587, 2003. [Online]. Available: http://stacks.iop.org/0031-9155/48/i=5/a=303 [17] L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1-4, pp. 259–268, 1992. [18] S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE T Image Process, vol. 13, no. 10, pp. 1327–1344, 2004.