## Three-dimensional analytical magnetic resonance imaging phantom

Venue: | in the Fourier domain, Magn. Reson. Med |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Koay_three-dimensionalanalytical,

author = {Cheng Guan Koay and Joelle E. Sarlls and Evren Özarslan},

title = {Three-dimensional analytical magnetic resonance imaging phantom},

booktitle = {in the Fourier domain, Magn. Reson. Med},

year = {},

pages = {430--436}

}

### OpenURL

### Abstract

This work presents a basic framework for constructing a 3D analytical MRI phantom in the Fourier domain. In the image domain the phantom is modeled after the work of Kak and Roberts on a 3D version of the famous Shepp-Logan head phantom. This phantom consists of several ellipsoids of different sizes, orientations, locations, and signal intensities (or gray levels). It will be shown that the k-space signal derived from the phantom can be analytically expressed. As a consequence, it enables one to bypass the need for interpolation in the Fourier domain when testing image-reconstruction algorithms. More importantly, the proposed framework can serve as a benchmark for contrasting and comparing different image-reconstruction techniques in 3D MRI with a non-Cartesian k-space trajectory. The proposed framework can also be adapted for 3D MRI simulation studies in which the MRI parameters of interest may be introduced to the signal intensity from the

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Citation Context ...cal quadrature algorithm for MR image reconstruction from a set of parallel plane measurements. Later, a simplified version of this 3D head phantom was used by Kak and Roberts (18) and Kak and Slaney =-=(19)-=- to test cone beam reconstruction algorithms. In such studies, the advantage of using ellipses or ellipsoids is that the projection through these objects can be analytically expressed. In MRI simulati... |

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Citation Context ...psoid” section above. From Eqs. [1] and [2] we have where cos(�) � cos(�1)cos(�2) � sin(�1)sin(�2)cos(�1 ��2) (22). A useful integral representation of elementary spherical wave functions is given by =-=(23)-=-: j n�2�k˜ r�P n�cos�� 2�� � ��i�n 4� �0 2� �0 2� e�i2�k ˜ r cos���Pn�cos��1��sin��1�d�1d�1 [A9] G�k x, k y, k z� ���� �� �� �� ��� ��� � g�x, y, z�e �i2��kxx�kyy�kzz� dxdydz [A1] where j n is the sph... |

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Citation Context ...then it reduces to the FT of a sphere. Second, at the center of k-space, G(0, 0, 0) is proportional to the volume of the ellipsoid. That is, sin�2�K� � 2�K cos�2�K� G�0, 0, 0� � limK30 �abc� 2�2K3 �, =-=[4]-=- FT of an Ellipsoid Under a Constant Nonsingular Affine Transformation The general case is no more difficult than the example discussed above. However, the key concept in this subsection is that of a ... |

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Citation Context ...splacement vector. The vector or matrix transposition is denoted by a superscript T. Rewriting x, y, and z in terms of p x, py, and pz, Eq. [8] is transformed to the following expression: r � Ap � �, =-=[9]-=- where r � [x, y, z] T , and p � [px, py, pz] T . Another means of expressing the integral in Eq. [6] is to state the region of integration, which is written as G�k x, ky, kz� � � ��� e�i2��kxx�kyy�kz... |

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Citation Context ...dies, a common and flexible analytical phantom in the Fourier domain is needed. A simulated head section, which consists of several overlaying ellipses, was first proposed and used by Shepp and Logan =-=(16)-=- to compare techniques for reconstructing an image from projections through the image. In another important paper by Shepp (17), a 3D version of the SheppLogan head phantom containing 17 ellipsoids an... |

3 |
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Citation Context ...e.wiley.com). Published 2007 Wiley-Liss, Inc. † This article is a US Government work and, as such, is in the public domain in the United States of America. 430 � g�x, y, z�e �i2��kxx�kyy�kzz� dxdydz, =-=[1]-=- and let g( x, y, z) be defined as follows: g�x, y, z� �� � �x/a�2 � �y/b� 2 � �z/c� 2 � 1 0 �x/a� 2 � �y/b� 2 � �z/c� 2 � 1 . [2] It is shown in Appendix A that the FT of g( x, y, z) defined in Eq. [... |

3 |
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Citation Context ...dering of the 3D version of the Shepp-Logan head phantom. The top portion of the skull is removed to show the smaller ellipsoids.432 Koay et al. and � p x p y p z � � A�1�� x y � z �� �x �y � z �� , =-=[8]-=- where A is a constant nonsingular 3 � 3 matrix, and � � [� x, �y, �z] T is a constant translation or displacement vector. The vector or matrix transposition is denoted by a superscript T. Rewriting x... |

3 |
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Citation Context ... [x, y, z] T , and p � [px, py, pz] T . Another means of expressing the integral in Eq. [6] is to state the region of integration, which is written as G�k x, ky, kz� � � ��� e�i2��kxx�kyy�kzz�dxdydz, =-=[10]-=- R where R is the ellipsoidal region defined by ( px/a) 2 � ( py/b) 2 � ( pz/c) 2 � 1. Performing a change of variables from ( x, y, z) to(px, py, pz), the integral above takes the following form: G�k... |

2 |
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Citation Context ...olume of a sphere of radius r. It should be noted here that the formula for the FT of a sphere can be found in a previous work by Bracewell (21). ��� ��� � g�p x, p y, p z�e �i2��kxx�kyy�kzz� dxdydz, =-=[6]-=- where px, py, and pz are functions of x, y and z. Specifically, we have the following expressions: g�p x, p y, p z� �� � �p x/a� 2 � �p y/b� 2 � �p z/c� 2 � 1 0 �p x/a� 2 � �p y/b� 2 � �p z/c� 2 � 1 ... |

2 |
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Citation Context ...here px, py, and pz are functions of x, y and z. Specifically, we have the following expressions: g�p x, p y, p z� �� � �p x/a� 2 � �p y/b� 2 � �p z/c� 2 � 1 0 �p x/a� 2 � �p y/b� 2 � �p z/c� 2 � 1 , =-=[7]-=- FIG. 2. a: An x-y plane cross section of the 3D version of the Shepp-Logan head phantom at z � –0.25. b: An x-z plane cross section of the 3D version of the Shepp-Logan head phantom at y � 0.125. c: ... |

2 |
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Citation Context ...ring of the head phantom after a virtual hemicraniectomy procedure. By defining k˜ � ATk, Eq. [12] reduces to G�k x, ky, kz� � ��det�A��e�i2�kT ���� R R � dp xdp ydp z. [12] e �i2�k˜ T p dpxdp ydp z. =-=[13]-=- Note that the integral in Eq. [13] is exactly the FT of an ellipsoid under the condition discussed in previous subsection if we replace the vector k in Eq. [3] by the vector k˜ in Eq. [13]. Therefore... |

2 |
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Citation Context ...eral overlaying ellipses, was first proposed and used by Shepp and Logan (16) to compare techniques for reconstructing an image from projections through the image. In another important paper by Shepp =-=(17)-=-, a 3D version of the SheppLogan head phantom containing 17 ellipsoids annotated with relevant anatomical structures (e.g., nose, eyes, blood clots, ventricles, tumors, and many others) was proposed a... |

1 |
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Citation Context ...n in b, c, and d, respectively. The effect of the displacement vector (�x ��0.25, �y � 0.55, �z � 0.55) on the rotated ellipsoid is shown in e. sin�2�K� � 2�K cos�2�K� G�k x, ky, kz� � �abc� 2�2K3 �, =-=[3]-=- where K � ((ak x) 2 � (bky) 2 � (ckz) 2 ) 1/2 . Before moving on to the general setting, it is instructive to reflect on two limiting cases. First, if all axes of the ellipsoid are of the same length... |

1 |
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Citation Context ...hemselves functions of the original coordinates. Let the 3D FT of a function g( p x, py, pz) be �� G�k x, k y, k z� ���� �� �� � lim K30 �abc� 8 3 �3 K 3 � 16 15 �5 K 5 � ··· 2� 2 K 3 � � � 4 3 �abc. =-=[5]-=- In Eq. [5] the denominator inside the bracket was expanded in terms of Taylor series about K � 0. In the case of a sphere, G(0, 0, 0) �� 4 3 �r3 , which is a product of � and the volume of a sphere o... |

1 |
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Citation Context ...( px/a) 2 � ( py/b) 2 � ( pz/c) 2 � 1. Performing a change of variables from ( x, y, z) to(px, py, pz), the integral above takes the following form: G�k x, ky, kz� � � ��� e�i2�kT�r�det�A��dpxdpydpz, =-=[11]-=- R where k � [kx, ky, kz] T , and �det(A)� denotes the absolute value of the determinant of A. Substituting Eq. [9] into Eq. [11], this yields G�k x, ky, kz� � � ��� e�i2�kT�Ap����det�A��dpxdpydpz, R ... |

1 |
Design and analysis of a practical 3D cones trajectory. Magn Reson Med 2006:55;575–582. Koay et al
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Citation Context ..., whereas Fig. 2b shows a vertical cross section of the head phantom at y � 0.125. Figure 2c is a 3D rendering of the head phantom after a virtual hemicraniectomy procedure. By defining k˜ � ATk, Eq. =-=[12]-=- reduces to G�k x, ky, kz� � ��det�A��e�i2�kT ���� R R � dp xdp ydp z. [12] e �i2�k˜ T p dpxdp ydp z. [13] Note that the integral in Eq. [13] is exactly the FT of an ellipsoid under the condition disc... |

1 |
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(Show Context)
Citation Context ...ing Eq. [9] into Eq. [11], this yields G�k x, ky, kz� � � ��� e�i2�kT�Ap����det�A��dpxdpydpz, R � ��det�A��e�i2�kT ���� e�i2��ATk�Tp G�k x, k y, k z� � �abc�det�A��e �i2�kT �� sin�2�K� � 2�K cos�2�K� =-=[14]-=- 2� 2 K 3 � where K � ((ak˜ x) 2 � (bk˜ y) 2 � (ck˜ z) 2 ) 1/2 and k˜ � A T k. FT of an Ellipsoid Under a Rigid-Body Transformation In this case the general nonsingular 3 � 3 matrix A is replaced by a... |

1 |
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Citation Context ...� RT (r � �). Based on these properties, the expression of G(kx, ky, kz) under a rigid-body transformation can further be simplified to: G�k x, ky, kz� � �abce�i2�kT��sin�2�K� � 2�K cos�2�K� 2�2K3 �, =-=[15]-=- where K � ((ak˜ x) 2 � (bk˜ y) 2 � (ck˜ z) 2 ) 1/2 and k˜ � R T k. 3D Phantom The 3D version of the Shepp-Logan head phantom used in Refs. 18 and 19 was adapted for testing purposes. The specificatio... |

1 |
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Citation Context ... to investigate a numerical quadrature algorithm for MR image reconstruction from a set of parallel plane measurements. Later, a simplified version of this 3D head phantom was used by Kak and Roberts =-=(18)-=- and Kak and Slaney (19) to test cone beam reconstruction algorithms. In such studies, the advantage of using ellipses or ellipsoids is that the projection through these objects can be analytically ex... |

1 |
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Citation Context ...e k-space signal derived from these objects. In the 2D case the Fourier transform (FT) of an ellipse can be analytically expressed, as found in the work of Kak and Slaney (19) and Van de Walle et al. =-=(20)-=-. In this paper a common and flexible 3D analytical phantom in the Fourier domain is proposed. In the image domain the phantom is modeled after the one described in Refs. 18 and 19. It is shown that t... |

1 |
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Citation Context ...e, G(0, 0, 0) �� 4 3 �r3 , which is a product of � and the volume of a sphere of radius r. It should be noted here that the formula for the FT of a sphere can be found in a previous work by Bracewell =-=(21)-=-. ��� ��� � g�p x, p y, p z�e �i2��kxx�kyy�kzz� dxdydz, [6] where px, py, and pz are functions of x, y and z. Specifically, we have the following expressions: g�p x, p y, p z� �� � �p x/a� 2 � �p y/b�... |

1 |
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(Show Context)
Citation Context ...llipsoid under the simplest condition discussed in the “Basic Example of the FT of an Ellipsoid” section above. From Eqs. [1] and [2] we have where cos(�) � cos(�1)cos(�2) � sin(�1)sin(�2)cos(�1 ��2) =-=(22)-=-. A useful integral representation of elementary spherical wave functions is given by (23): j n�2�k˜ r�P n�cos�� 2�� � ��i�n 4� �0 2� �0 2� e�i2�k ˜ r cos���Pn�cos��1��sin��1�d�1d�1 [A9] G�k x, k y, k... |