2D and 3D Shepp-Logan Phantom in the Fourier and Image Domains

Fig 1. The three-dimensional rendering of the 3D Shepp-Logan phantom. [Feel free to download the Mathematica file to generate a similar 3D graphics yourself.]

Fig 2. The animated gif of the reconstructed image domain data computed from the analytical 3D Shepp-Logan phantom in the Fourier domain. [Click on the image to run the animation]

Fig 3. The 2D Shepp-Logan phantom.

Fig 4. The reconstructed 2D image from the analytical 2D Shepp-Logan phantom in the Fourier domain.

The same month after the publication of our work, Ref. [1], in Magnetic Resonance in Medicine, we released our software to two different sites, Matlab Central and Wolfram Library Archive. Matlab Central and Wolfram Library Archive are two very wonderful sites for sharing software. Since software development is a on-going process, it is not convenient to share ideas, comments and to update/upload the software on these sites. Hence, only major releases will be made on these sites and I shall use this google site for sharing ideas and comments. Of course, your comments or questions are welcome.

The archive file, STBBFolder, is basically the same as the one on the Matlab Central. Thanks to Christine, a student from Germany, whose questions dated Aug 11, 2009 have led me to add a few lines of MATLAB codes to illustrate the steps needed to perform FFT on the k-space signals of the 3D Shepp-Logan phantom. The transformed image domain signals are then compared with the analytical image domain signals, refer to the Matlab file below. In the revised SampleTest.m, I used imtool. So, please make sure that you click on a menu that allows you to change contrast (min=0.0 and max=3.0 should be fine and these values should be applied to both imtool panels), see the screen shot here.

If you have problem using STBB.jar within new version of MATLAB (2012R and above), try download the newer jar file compiled under JDK 1.7. Here is the link to the jar file.

SOURCE CODE in Java: 2D Shepp-Logan phantom and 3D Shepp-Logan phantom.

Finally, if you find the software useful and use the software in your research work, please spread the words by citation or word of mouth. Thanks.

NOTE: The 2D Shepp-Logan phantom in the image domain was first proposed in [2]. Later, the 3D Shepp-Logan phantom in the image domain was developed by Larry Shepp in [3]. The 3D Shepp-Logan phantom used in [1] was similar to that of [4], which was a simplified or reduced version of the original 3D phantom proposed in [3]. The 2D Shepp-Logan phantom in the Fourier domain (or K-space) can found in [4] and [5]. Due to the interest in 3D non-Cartesian K-space reconstruction, we felt the need to develop the 3D Shepp-Logan phantom in the Fourier domain, which was published in [1]. In the [1], we discussed several extensions to model relaxometry and diffusion within the existing model and also to model cardiac motion. Since then, the extension to model relaxometry in the phantom has been carried out in [6]. The Fourier transfrom of ellipsoids has been used in biophysical modeling, e.g., [7,8]. GPU computing is a very interesting and promising area and is likely to play a bigger role in the near future, see e.g., [9], which uses the 3D phantom for illustration.

References:

[1] Koay CG, Sarlls JE and Özarslan E. Three dimensional analytical magnetic resonance imaging phantom in the Fourier domain. Magnetic Resonance in Medicine. 2007; 58: 430-436.

[2]. Shepp LA, Logan BF. The Fourier reconstruction of a head section. IEEE Trans Nucl Sci 1974;NS-21:21–43.

[3]. Shepp LA. Computerized tomography and nuclear magnetic resonance. J Comput Assist Tomogr 1980;4:94–107.

[4]. Kak AC, Slaney M. Principles of computerized tomographic imaging. IEEE Press, New York, 1988.

[5]. Van de Walle R, Barrett HH, Myers KJ, Altbach MI, Desplanques B, Gmitro AF, Cornelis J, Lemahieu I. Reconstruction of MR images from data acquired on a general nonregular grid by pseudoinverse calculation. IEEE Trans Med Imaging 2000;19:1160–1167.

[6]. Gach HM, Tanase C, Boada F. 2D & 3D Shepp-Logan Phantom Standards for MRI. 19th International Conference on Systems Engineering, 521-526, 2008.

[7]. Özarslan E. Compartment shape anisotropy (CSA) revealed by double pulsed field gradient MR. Journal of Magnetic Resonance. 199 (1): 56-67, 2009.

[8]. Özarslan E, Koay CG and Basser PJ. Double-PFG diffusion-diffraction in ellipsoidal pores. In 9-th International Bologna Conference on Magnetic Resonance in Porous Media. 115, 2008.

[9]. Stone SS, Haldar JP, Tsao SC, Hwu WW, Sutton BP, Liang ZP. Accelerating advanced MRI reconstructions on GPUs, Journal of Parallel and Distributed Computing, 68(10): 1307-1318, 2008.