Retracted: The N-Localizer and Volume Imaging

The mathematics that were originally developed for the N-localizer apply to three N-localizers that produce three sets of fiducials in a planar, tomographic image. Recently, these mathematics have been extended to apply to three or more sets of fiducials in a planar, tomographic image. This paper discusses a further extension of the mathematics of the N-localizer that applies to volume images that are produced by magnetic resonance (MR) imaging. This extension of the mathematics applies to four or more N-localizers that are visualized in more than one planar section of the volume image. In addition, this extension provides a statistical measure of the quality of the volume image data that may be influenced by factors such as nonlinear distortion of MR images


Introduction
Recent review articles discuss the origin and mathematics of the N-localizer [1][2][3][4][5].The mathematics are summarized briefly in the remainder of this Introduction in preparation for the presentation of new developments in the Materials & Methods section of this article.
The N-localizer comprises a diagonal rod that extends from the top of one vertical rod to the bottom of another vertical rod (Figure 1).Assuming for the sake of simplicity that the two vertical rods are perpendicular to the tomographic section, the cross section of each vertical rod creates a fiducial circle and the cross section of the diagonal rod creates a fiducial ellipse in the tomographic image.The ellipse moves away from one circle and towards the other circle as the position of the tomographic section moves downward with respect to the N-localizer.The relative spacing between these three fiducials permits precise localization of the tomographic section relative to the N-localizer.The distance between the centers of circle and ellipse and the distance between the centers of circles and are used to calculate the ratio .This ratio represents the fraction of diagonal rod that extends from the top of vertical rod to the point of intersection of rod with the tomographic section.These geometric relationships are valid even if the vertical rods are not perpendicular to the tomographic section [6]; in this case, the cross sections of the vertical rods may be somewhat elliptical, depending on the degree of nonperpendicularity.The fraction is used to calculate the coordinates of the point of intersection between rod and the tomographic section (Figure 2).In this figure, points and represent the beginning and end, respectively, of the vector that extends from the top of rod to the bottom of rod .This vector coincides with the long axis of rod .The coordinates of the beginning point and the coordinates of the end point are known from the physical dimensions of the N-localizer.Hence, linear interpolation may be used to blend points and to obtain the coordinates of the point of intersection between the long axis of rod and the tomographic section The vector form of Equation 1shows explicitly the coordinates of points , , and Equation 1 or 2 may be used to calculate the coordinates of the point of intersection between the long axis of rod and the tomographic section.The point , which lies on the long axis of rod in the three-dimensional coordinate system of the Nlocalizer, corresponds to the analogous point , which lies at the center of ellipse in the two-dimensional coordinate system of the tomographic image.Hence, there is a one-to-one correspondence between a point from the N-localizer and a point from the tomographic image.The attachment of three N-localizers to a stereotactic frame (Figure 3) permits calculation of the , , and coordinates for the three respective points , , and in the three-dimensional coordinate system of the stereotactic frame.These three points correspond respectively to the three analogous points , , and in the two-dimensional coordinate system of the tomographic image.In the following discussion, the symbols , , and will be used as a shorthand notation for , , and .The symbols , , and will be used as a shorthand notation for , , and .The quadrilateral represents the tomographic section.The large oval depicts the base of the stereotactic frame.The vertical and diagonal lines that are attached to the large oval represent the nine rods.The centers of the six fiducial circles and the three fiducial ellipses that are created in the tomographic image by these nine rods are shown as points that lie in the tomographic section.The tomographic section intersects the long axes of the three diagonal rods at points , and that coincide with the respective centers , and of the three ellipses (Figure 4).The , and coordinates of the respective points of intersection , and are calculated in the three-dimensional coordinate system of the stereotactic frame using Equations 1 and 2. Because these three points determine the spatial orientation of a plane in three-dimensional space, the spatial orientation of the tomographic section is determined relative to the stereotactic frame.The target point lies in the tomographic section.The coordinates of this target point are calculated in the three-dimensional coordinate system of the stereotactic frame using Equation 5.
The three points, , , and , lie on the long axes of the three respective diagonal rods, , , and , and have respective coordinates , , and in the three-dimensional coordinate system of the stereotactic frame (Figure 3).
The analogous three points, , , and , lie at the centers of the three respective ellipses, , , and , and have coordinates , , and in the twodimensional coordinate system of the tomographic image (Figure 4).Because three points determine the orientation of a plane in three-dimensional space, the three coordinates, , , and , together with the three coordinates, , , and , determine the spatial orientation of the tomographic section relative to the stereotactic frame.This spatial orientation is specified using the matrix equation Equation 3 represents concisely a system of nine simultaneous linear equations that determine the spatial orientation of the tomographic section relative to the stereotactic frame.This equation transforms the , , and coordinates from the twodimensional coordinate system of the tomographic image to create , , and coordinates in the three-dimensional coordinate system of the stereotactic frame.
In Equation 3, the matrix elements , , , , , , , , and , as well as the matrix elements , , , , , and , are known.The matrix elements through are unknown; hence, Equation 3 may be inverted to solve for these unknown elements of the transformation matrix where the exponent "-1" indicates the inverse of the matrix that contains the elements , , , , , and .The inverse of this matrix exists and may be calculated to high precision if this matrix is non-singular, that is, if the three points , , and are not collinear [4].
Once the transformation matrix elements through are known, the coordinates of the target point may be transformed from the two-dimensional coordinate system of the tomographic image into the three-dimensional coordinate system of the stereotactic frame to obtain the coordinates of the analogous target point Equation 5 has been used for the past 36 years to calculate the coordinates of the target point in the three-dimensional coordinate system of the stereotactic frame [7][8].Despite its ubiquitous use, this equation applies to only three N-localizers.Some applications of the N-localizer have incorporated four N-localizers [9][10][11][12][13][14] and hence have required that one of the four N-localizers be ignored in order to apply Equation 5. Recently, however, the mathematics of the N-localizer have been extended to apply to three or more N-localizers and thus have eliminated the requirement to discard one of the N-localizers [5].A further extension of the mathematics of the N-localizer enables the application of the N-localizer to volume imaging, as discussed in the remainder of this article.

Materials And Methods
Magnetic resonance (MR) imaging differs from computed tomography (CT) imaging in the manner by which the images are obtained.CT obtains a volume of individual tomographic images of the patient by changing the position of the scanner bed between successive tomographic scans and hence is susceptible to errors in scanner bed positioning.MR obtains a volume image of the patient by applying magnetic field gradients [15] and thus does not require changing the position of the scanner bed.Indeed, MR may obtain a volume image of the patient directly without obtaining a series of planar, tomographic scans.Because MR is not susceptible to errors in scanner bed positioning, the spatial accuracy of a MR volume image ought to be greater than the spatial accuracy of a volume of successive CT images, provided that the patient does not move during the imaging procedure.
A volume image comprises individual volume elements, or voxels, that are identified via their coordinates in the same manner that the individual picture elements, or pixels, from a planar, tomographic image are identified via their coordinates.A planar section of these voxels is a subset of the voxels wherein one of the coordinates is held constant.An axial plane has constant and varying coordinates.A sagittal plane has constant and varying coordinates.A coronal plane has constant and varying coordinates.In this context, the term "planar section" or "plane" designates an axial, sagittal, or coronal plane, i.e., a subset of the voxels that one volume image comprises.Such a plane is to be distinguished from a tomographic image that comprises a set of pixels that are obtained via one planar, tomographic scan.
A MR localizer frame differs from a CT localizer frame in that the MR localizer frame is designed to create fiducials in sagittal and coronal planes in addition to axial planes [16].A MR localizer frame comprises five N-localizers that subtend the anterior, posterior, left, right, and superior faces of a cube that encloses the patient's head (Figure 5).In a manner analogous to Equation 3, the spatial orientation of MR voxel data may be determined relative to the stereotactic frame.The equation that applies to these voxel data requires four coordinates , , , and from the three-dimensional coordinate system of the stereotactic frame.This equation also requires four coordinates , , , and from the three-dimensional coordinate system of the voxel data.The coordinates are the centers of ellipses that are visualized in axial, sagittal, or coronal planes of the voxel data, similar to the approach that is discussed for an axial tomographic section in Figure 4.The coordinates are calculated from the coordinates of the centers of circles and and ellipses via Equation 1 or Equation 2.
The spatial orientation of the voxel data relative to the stereotactic frame is specified using the matrix equation [17] Equation 6 represents concisely a system of 12 simultaneous linear equations that determine the spatial orientation of the voxel data relative to the stereotactic frame.This equation transforms the , , , and coordinates from the three-dimensional coordinate system of the voxel data to create , , , and coordinates in the three-dimensional coordinate system of the stereotactic frame.
One important restriction applies to Equation 6.The four coordinates , , , and must not be coplanar; hence, these four coordinates must not be obtained from a single plane, such as an axial plane that comprises four fiducials.Thus, one of many acceptable sets of coordinates would comprise two coordinates from the left and right N-localizers that intersect an axial plane, plus two coordinates from the anterior and superior Nlocalizers that intersect a sagittal plane.This restriction is similar to the restriction that applies to Equation 3, i.e., three noncollinear coordinates must be used in Equation 3 [4].
In Equation 6, the matrix elements , , ; hence, it is possible (and tempting) to invert Equation 6 in order to solve for these unknown elements of the transformation matrix in a similar manner to the inversion of Equation 3 that yields Equation 4.However, a more useful solution may be obtained by applying the method of least squares to more than four sets of fiducials because the method of least squares minimizes the effect of errors in the voxel data [18].
For voxel data, a useful set of fiducials would comprise ten fiducials: four fiducials from an axial plane, three fiducials from a sagittal plane, and three fiducials from a coronal plane, where the target point is visualized in all three planes.The following equation transforms ten coordinates from the three-dimensional coordinate system of the voxel data to create ten coordinates in the three-dimensional coordinate system of the stereotactic frame where the subscript selects one of the ten fiducials.The equations that are required for leastsquares minimization are obtained by first expanding the matrix multiplication of Equation 7 and expressing the result for the matrix elements , and In the presence of error, Equation 8 may be modified to express the errors in , , and , respectively, as , , and In order to minimize these errors via the method of least squares, the equations for , , and are squared to obtain the error functions , , and The following discussion illustrates minimization of the error function ; minimization of the error functions and is performed in an analogous manner.At the minimum of a function, all of the derivatives are equal to zero.Evaluating the derivatives , , , and and setting the resulting expressions for these derivatives to zero yields Simplifying and rearranging the above equations for the derivatives yields a system of four simultaneous linear equations of the four unknowns , , , and where is the number of sets of fiducials; in this case, .These simultaneous linear equations may be solved using Cramer's rule [19] to yield the matrix elements , , , and that minimize the error function as follows.Each of the elements , , , and that are found in the first column of the transformation matrix and that minimize may be calculated as the ratio of two determinants wherein the denominator determinant contains the sums from Equation 12and wherein the numerator determinant for the calculation of , , , and , respectively, is obtained by replacing the first, second, third, and fourth columns of Equation 13with and the left-hand side is calculated as [21] The correlation coefficients and are calculated in an analogous manner.

Results
The author has access to neither a MR localizer frame nor a MR scanner, so this article discusses mathematics rather than experimental results.The reader is encouraged to read the Results and Discussion sections of "The Mathematics of Three or More N-Localizers for Stereotactic Neurosurgery" [5] that discuss the correlation coefficient that is calculated for a single tomographic image.The correlation coefficients, , , and , that are calculated for a volume image permit a similar analysis of image data.

Discussion
Equations 7-17 provide a method for transforming coordinates from the threedimensional coordinate system of voxel data, which are obtained via volume imaging, into the three-dimensional coordinate system of the stereotactic frame to produce coordinates.These equations require the use of four or more pairs of noncoplanar and coordinates.A useful set of and coordinates may be obtained from axial, sagittal, and coronal planes, in which the target point is visualized, by selecting the coordinates then calculating the coordinates via Equation 1or 2. Although the axial, sagittal, and coronal planes in which the target point is visualized would appear to be the most useful of the image planes, there is no requirement to include these particular planes in the calculation of elements, through , of the transformation matrix via Equations 7-16.Because the transformation matrix transforms coordinates from the three-dimensional coordinate system of the voxel data to produce coordinates in the three-dimensional coordinate system of the stereotactic frame, of the coordinates from the voxel data are transformed into coordinates, independent of the specific planes from which the coordinates are selected for application of Equations 1-2 and 7-17.
An alternative to Cramer's rule is Gauss elimination [22] that is more computationally efficient than Cramer's rule and that, in principle, results in less numerical error.However, in practice, Cramer's rule is sufficient for the purposes of this article so long as at least four noncoplanar points are used in Equation 7.
Equations 7-16 and 18-19 permit the calculation of the correlation coefficients , , and that provide a measure of the accuracy of the transformation.The accuracy of the transformation is degraded by nonlinear distortion to which MR data are susceptible.This nonlinear distortion may be caused by metallic elements of the stereotactic frame, inhomogeneity and temporal fluctuation of the magnetic field, and metallic equipment near the MR scanner [23][24][25][26][27].
In view of this susceptibility to nonlinear distortion, an assessment of the nonlinear distortion may be improved through the use of additional and coordinates that may be obtained from axial, sagittal, and coronal planes that do not include the target point .These additional planes could be chosen from throughout the volume image; their and coordinates would contribute to the calculation of the correlation coefficients , , and and thereby provide a measure of the presence of nonlinear distortion within the

FIGURE 1 :
FIGURE 1: Intersection of the tomographic section with the N-localizer Side view of the N-localizer.The tomographic section intersects rods , and .Tomographic image.The intersection of the tomographic section with rods , and creates fiducial circles and and fiducial ellipse in the tomographic image.The distance between the centers of circle and ellipse and the distance between the centers of circles and are used to calculate the ratio .This ratio represents the fraction of diagonal rod that extends from the top of rod to the point of intersection of rod with the tomographic section.

FIGURE 2 :
FIGURE 2: Calculation of the point of intersection between the rod B and the tomographic sectionThe long axis of rod is represented by a vector that extends from point at the top of rod

FIGURE 3 :
FIGURE 3: Representation of the tomographic section in the three-dimensional coordinate system of the stereotactic frame

FIGURE 4 :
FIGURE 4: Representation of the two-dimensional coordinate system of the tomographic

FIGURE 5 :
FIGURE 5: MR localizer frame and axial, sagittal and coronal planes A MR localizer frame comprises five N-localizers that subtend the anterior, posterior, left, right and superior faces of a cube.An axial plane (red) intersects the MR localizer frame at four Nlocalizers: anterior, posterior, left and right.A sagittal plane (green) intersects the MR localizer frame at three N-localizers: anterior, posterior and superior.A coronal plane (blue) intersects the MR localizer frame at three N-localizers: left, right and superior.