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Ray tracing (RT) and perspective projection (PP) using fiducial-based registration can be used to determine points of interest in biplanar X-ray imaging. We sought to investigate the implementation of these techniques as they pertain to X-ray imaging geometry. The mathematical solutions are presented and then implemented in a phantom and actual case with numerical tables and imaging. The X-ray imaging is treated like a Cartesian system in millimeters (mm) with a standard frame-based stereotactic system. In this space, the point source is the X-ray emitter (focal spot), the plane is the X-ray detector, and fiducials are in between the source and plane. In a phantom case, RT was able to predict locations of fiducials after moving the point source. Also, a scaled PP matrix could be used to determine imaging geometry, which could then be used in RT. Automated identification of spherical fiducials in 3D was possible using a center of mass computation with average Euclidean error relative to manual measurement of 0.23 mm. For PP, RT projection or a combinatorial approach could be used to facilitate matching 3D to 2D points. Despite being used herein for deep brain stimulation (DBS), utilization of this kind of imaging analysis has wide medical and non-medical applications.
In 1895, Wilhelm Roentgen discovered X-rays, which subsequently had an important impact in medicine for over 100 years [1]. As a major diagnostic tool, X-rays revolutionized medicine by empowering clinicians to diagnose various conditions such as pneumonia, bony fractures, and dental cavities. Despite the long history of X-ray use, developments primarily have included advances in speed, quality, and, more recently, digitization. Computational X-ray analysis has not been the standard despite the wide use of X-rays around the world. Rather, advancements in computed tomography (CT) and magnetic resonance imaging (MRI) have been utilized for 3D (three dimension) localization. Nevertheless, X-rays represent a simple, low cost, low radiation dose technique for which the utility can be optimized.
With the addition of more than a single X-ray plane, stereotactic intraoperative location (STiL) during deep brain stimulation (DBS) surgeries has been demonstrated to provide accurate 3D positions using ray tracing (RT) and perspective projection (PP) [2]. RT and PP assume a Cartesian-based system where from a point source (X-ray emitter focal spot), rays are created through objects (fiducials) to a detector plane (X-ray cartridge), which are then presented on a display plane (the X-ray image). Computationally, RT requires knowledge of the entire imaging geometry, whereas PP requires knowledge of the 3D points and their associated projection points in 2D (two dimension). Using these techniques, RT and PP represent important tools that offer critical analysis of X-rays.
In this paper, we discuss the mathematics involved with RT and PP as well as provide numerical samples and imaging of each. We analyze a phantom case, a real case, explore automations and error propagation. These methods effectively compute 3D to 2D (3D-2D) and, for STiL, 2D to 3D (2D-3D). We explore these techniques using point analysis, but similar techniques can be expanded using volumes.
Technical Report
In stereotactic neurosurgery, a Cartesian coordinate system can be generated using anatomical structures in the brain or using a frame-based apparatus. Anatomical points include the anterior commissure (AC), posterior commissure (PC), and a midline structure to generate the coordinate system, generally from the middle of AC-PC (MidACPC). Frame-based coordinate systems typically use an N-localizer apparatus calibrated on a CT or MRI [3-5]. Generally, these coordinate systems are considered linear and are scaled in millimeters. In the following, we implement mathematical solutions for point-based registration from 3D (x, y, z) to 2D (U, V) using RT and PP. These techniques assume that there are no geometrical distortions (scattering or diffraction), that the focal spot of the X-ray emitter is a point source, that the X-ray detector can be treated like a plane, and that the center of spheres or pin tips/bases can be treated as point objects in the system (fiducials). The centers of spheres are used in the majority of cases as the center is geometrically ideal, but we also utilize pin tips and pin bases in addition to other stable objects when available, which depends on the X-ray exposure. All objects are mathematically utilized as points. For the display plane (X-ray image), pixel spacing may be present from DICOM metadata, but this is often not present in fluoroscopic images. In some instances, calibration may be needed to determine this scaling and ensure no inhomogeneity in the image. Also, to maintain a single frame-based coordinate system, the subsequent tables utilize CRW/BRWLF (Cosman-Roberts-Wells/Brown-Roberts-Wells Localizer Frame, Radionics CRW Stereotactic System, Integra LifeSciences Corporation, Plainsboro, New Jersey). Fiducial positions were calculated using CT with the incorporation of important points fused from MRI in iPlan 3.0 Stereotaxy (Brainlab Inc, Feldkirchen, Germany). Also, the numerical tables utilize AP (Antero-Posterior), LAT (Lateral), VERT (Vertical) rather than x, y, z. Finally, many of the solutions provided assume biplanar imaging (two views or poses), but the same mathematical solutions can allow many more.
Background mathematical solutions
In RT, the geometry of the point source (focal spot size) (P0) to a fiducial (P1 or P2) forms a ray that intersects a detector (Pp1 or Pp2) plane, which is seen on a display (Puv1 or Puv2) plane (Figure 1) [6]. If the stereotactic coordinate system utilized contains an axis orthogonal to the detector plane, the system is coherent on-axis (COA). Having a COA system makes the determination of display points more simple, such that the two variable coordinates are translated, rotated and scaled to the display plane. The display plane can also be rotated separately. If the detector plane is not COA, then a change of basis transformation is needed to normalize an axis. For example, in a COA antero-posterior (AP) view, the AP value of all plane intersections with the AP plane is the same, which allows straightforward utilization of the other dimensions, lateral (LAT) and vertical (VERT), where the intersections occur. However, a system that is not COA yields 3D coordinates for which all values have variation, requiring a normalization prior to projection to the display plane.
The X-ray detector can be considered as a mathematical plane (equation 1). The X-ray emitter then functions as a point source that generates rays which intersect fiducials, generating lines in 3D parametric form (equations 2, 3, 4). When these lines intersect the X-ray detector (Step 1, Figure 1), there are the points in the detector plane (equations 5, 6, 7). Because the display plane would vary in presentation, added translation/rotation is required for pixel location, which can be computed in matrix form incorporating this 2D rotation and translation (equation 8). Here, the 2D presentation of the planar image expects a dimension of the 3D intersection to be a constant (COA). When not COA, the normalization procedure (Step 2, Figure 1) is simplified by the use of unit vectors (equation 9) and dot-products between the two coordinate systems thereby identifying the matrix components.
$$Ax + By +Cz + D = 0\tag{1}$$
$$x = x_{1} + a_{1}\cdot r_{1}\tag{2}$$
$$y = y_{1} + b_{1}\cdot r_{1}\tag{3}$$
$$z = z_{1} + c_{1}\cdot r_{1}\tag{4}$$
$$x = x_{1} - \frac{a_{1}(Ax_{1}+By_{1}+Cz_{1}+D)}{Aa_{1}+Bb_{1}+Cc_{1}}\tag{5}$$
$$y = y_{1} - \frac{b_{1}(Ax_{1}+By_{1}+Cz_{1}+D)}{Aa_{1}+Bb_{1}+Cc_{1}}\tag{6}$$
$$z = z_{1} - \frac{c_{1}(Ax_{1}+By_{1}+Cz_{1}+D)}{Aa_{1}+Bb_{1}+Cc_{1}}\tag{7}$$
$$\begin{bmatrix} U'\\ V' \\ 1 \end{bmatrix} = \begin{bmatrix} r_{11} & r_{12} & t_{1}\\ r_{21} & r_{22}& t_{2} \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} U\\ V \\ 1 \end{bmatrix}\tag{8}$$
$$\begin{bmatrix} x_{1}'\\ y_{2}'\\ z_{3}' \end{bmatrix} = \begin{bmatrix} i_{1}'\cdot i_{1} & i_{1}'\cdot i_{2} & i_{1}'\cdot i_{3}\\ i_{2}'\cdot i_{1} & i_{2}'\cdot i_{2} & i_{2}'\cdot i_{3}\\ i_{3}'\cdot i_{1} & i_{3}'\cdot i_{2} & i_{3}'\cdot i_{3} \end{bmatrix} \cdot \begin{bmatrix} x_{1}\\ y_{2}\\ z_{3} \end{bmatrix}\tag{9}$$
In perspective projection (PP) a 3D point in space is associated with a 2D point on the display plane (equation 10) [7]. Importantly, a 2D point is generated (Step 2, Figure 1) by dividing the third component (t) with u or v (equations 11-12). A 4x3 conversion matrix is used wherein 11 of the 12 elements are unknowns initially. The three columns represent the projection matrix components (equations 13-15). Each of these three column components has four numerical values, but the last value (C43) is scaled to 1 in this homogenous formulation (equations 16-19). The system is then arranged as a set of overdetermined equations generally, such that each instance of 3D-2D points generates two equations (equation 20), one for U and another for V. Therefore, 5.5 3D-2D points are needed to solve the 11 matrix elements, but generally this system would be solved via a least squares formulation to optimize the matrix and minimize errors. Lastly, the complete PP matrix is 4x3 (equation 21), but will be subsequently utilized as its transpose in 3x4 (equation 22).
$$\left ( x,y,z,1 \right )\cdot C = (u,v,t)\tag{10}$$
$$U = u/t\tag{11}$$
$$V = v/t\tag{12}$$
$$u = (x,y,z,1)\cdot C_{1}\tag{13}$$
$$v = (x,y,z,1)\cdot C_{2}\tag{14}$$
$$t = (x,y,z,1)\cdot C_{3}\tag{15}$$
$$C_{1} = \begin{bmatrix} C_{11}\\ C_{21}\\ C_{31}\\ C_{41} \end{bmatrix}\tag{16}$$
$$C_{2} = \begin{bmatrix} C_{12}\\ C_{22}\\ C_{32}\\ C_{42} \end{bmatrix}\tag{17}$$
$$C_{3} = \begin{bmatrix} C_{13}\\ C_{23}\\ C_{33}\\ C_{43} \end{bmatrix}\tag{18}$$
$$C_{43} = 1\tag{19}$$
$$\begin{bmatrix} x_{1} & y_{1} & z_{1} & 1 & 0 & 0 & 0 & 0 & -U_{1}x_{1} & -U_{1}y_{1} & -U_{1}z_{1}\\ 0 & 0 & 0 & 0 &x_{1} & y_{1} & z_{1} & 1 & -V_{1}x_{1} & -V_{1}y_{1} & -V_{1}z_{1}\\ \vdots & & & & & & & & & & \vdots\\ x_{n} & y_{n} & z_{n} & 1 & 0 & 0 & 0 & 0 & -U_{n}x_{n} & -U_{n}y_{n} & -U_{n}z_{n}\\ 0 & 0 & 0 & 0 &x_{n} & y_{n} & z_{n} & 1 & -V_{n}x_{n} & -V_{n}y_{n} & -V_{n}z_{n} \end{bmatrix} \cdot \begin{bmatrix} C_{11}\\ C_{21}\\ \vdots\\ C_{33} \end{bmatrix} = \begin{bmatrix} U_{1}\\ V_{1}\\ \vdots\\ U_{n}\\ V_{n} \end{bmatrix}\tag{20}$$
$$C = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33}\\ C_{41} & C_{42} & C_{43} \end{bmatrix}\tag{21}$$
$$ C^{T} = \begin{bmatrix} a_{11} & a_{21} & a_{31} & a_{41}\\ a_{12} & a_{22} & a_{32} & a_{42} \\ a_{13} & a_{23} & a_{33} & a_{43} \end{bmatrix}\tag{22}$$
Interpretation of the PP model is generally widespread in computer vision [7]. Next, we utilize this technique to gain insight into the geometric interpretation of the X-ray system. The essential elements of the complete PP matrix, here now interpreted as 3x4, include a 3x3 intrinsic parameter and the extrinsic parameters include a 3x3 rotation and a 3x1 translation (equation 23). The components of the intrinsic parameter are important to note, which include scalar dimensions, shear, and translation (equation 24). The five intrinsic components are alpha and beta (which should be equal), theta, and the principal point (U0, V0). Of note, a skew angle is computed as the absolute value of theta minus pi/2. Again, to calculate these values for geometric interpretation, the scale of the imaging system (detector plane and display plane) should be the same or undergo a conversion. For example, the imaging plane may generate uncalibrated pixels and the stereotactic frame (or anatomic frame) is in millimeters. If the U and V coordinates on the imaging plane are converted to a millimeter system, interpretation of the components of the intrinsic parameters is facilitated. Notably, this scaling is not required for STiL using PP. Next, the vector components (equations 25-27) of the PP matrix are utilized to calculate components of these intrinsic parameters (equations 28-31). Of note, the expected result of equation 31 is 0, as in an ideal setting the imaging system would be non-skew (theta = pi/2 (approximately 1.57) radians or 90 degrees). The next elements calculated are utilized to generate the point source and the imaging plane. To formulate relevant information, the negative inverse (-a-1) of the vector component (3x3) of the PP matrix is performed and multiplied by a translation (equations 32-25). Visually, we see from Figure 1 that a 2D point on the display plane can be projected back to a plane that intersects the point source and is parallel to the detector plane. First, by using the translation component of the original PP matrix (b) and multiplying that with the negative inverse of the vector component, the point source (Omega) is then determined (equation 36). Picking two more unique 2D points (c, d) in the display plane generates two more points in the plane that intersect the point source (equations 37-38). Using these three points in the image source plane, two vectors can be created (equations 39-40). Forward or reverse cross-products of these vectors (equation 41-42) yields the normal of the image source plane, which is parallel to the detector plane (equation 43). This result may be expressed as a plane equation wherein the last element (sigma) is not yet known (equation 44). However, sigma can be determined using components of the intrinsic matrix for focal length, which in X-ray imaging is equivalent to the Source Image Distance (SID) (equations 45-46). It is possible, and even likely, under many circumstances that the imaging system may be slightly skew, so taking the average of each scalar component (alpha and beta) may yield a good approximation (equation 47). Furthermore, the relationship between focal length, the point source, and the detector plane tie all these elements together (equation 48). If the focal length is known, then one can readily calculate the missing component (sigma) of the detector plane equation.
$$C = K^{3,3} \begin{bmatrix} R^{3,3} & \textbf{b}^{3,1} \end{bmatrix}\tag{23}$$
$$K = \begin{bmatrix} \alpha & -\alpha cot\theta & U_{0}\\ 0 & \beta /sin\theta & V_{0}\\ 0 & 0 & 1 \end{bmatrix}\tag{24}$$
$$\textbf{a}_{1} = \begin{bmatrix} a_{11} & a_{21} & a_{31} \end{bmatrix}\tag{25}$$
$$\textbf{a}_{2} = \begin{bmatrix} a_{12} & a_{22} & a_{32} \end{bmatrix}\tag{26}$$
$$\textbf{a}_{3} = \begin{bmatrix} a_{13} & a_{23} & a_{33} \end{bmatrix}\tag{27}$$
$$\rho = \pm {1}/\left | \textbf{a}_{3}\right |{}\tag{28}$$
$$U_{0}=\rho ^2(\boldsymbol{a}_{1}\cdot \boldsymbol{a}_{3})\tag{29}$$
$$V_{0}=\rho ^2(\boldsymbol{a}_{2}\cdot \boldsymbol{a}_{3})\tag{30}$$
$$cos\theta = (\boldsymbol{a}_{1} \times \boldsymbol{a}_{3})\cdot (\boldsymbol{a}_{2} \times \boldsymbol{a}_{3}) /\left | \boldsymbol{a}_{1} \times \boldsymbol{a}_{3} \right |\cdot \left | \boldsymbol{a}_{2} \times \boldsymbol{a}_{3} \right |\tag{31}$$
$$-a^{-1} = \begin{bmatrix} -a_{11} & -a_{21} & -a_{31}\\ -a_{12} & -a_{22} & -a_{32} \\ -a_{13} & -a_{23} & -a_{33} \end{bmatrix}^{-1}\tag{32}$$
$$\textbf{b} = \begin{bmatrix} a_{41}\\ a_{42}\\ a_{43} \end{bmatrix}\tag{33}$$
$$\textbf{c} = \begin{bmatrix} U_{i}\\ V_{i}\\ 1 \end{bmatrix}\tag{34}$$
$$\textbf{d} = \begin{bmatrix} U_{n}\\ V_{n}\\ 1 \end{bmatrix}\tag{35}$$
$$\Omega = -a^{-1} \cdot \textbf{b}\tag{36}$$
$$\Delta = -a^{-1} \cdot \textbf{c}\tag{37}$$
$$\Psi = -a^{-1} \cdot \textbf{d}\tag{38}$$
$$\vec{v_{1}} = \Delta -\Omega\tag{39}$$
$$\vec{v_{2}} = \Psi -\Omega\tag{40}$$
$$\vec{n_{p}} = \vec{v_{1}} \times \vec{v_{2}}\tag{41}$$
$$\vec{n_{p}} = \vec{v_{2}} \times \vec{v_{1}}\tag{42}$$
$$\vec{n_{p}} = {(\lambda ,\mu ,\kappa)}\tag{43}$$
$$\lambda x + \mu y + \kappa z + \sigma = 0\tag{44}$$
$$\alpha = \rho^2\left | \textbf{a}_{1}\times \textbf{a}_{3} \right | sin\theta\tag{45}$$
$$\beta = \rho^2\left | \textbf{a}_{2}\times \textbf{a}_{3} \right | sin\theta\tag{46}$$
$$\phi \approx (\alpha +\beta )/2\tag{47}$$
$$\phi = \left | \lambda x_{0} + \mu y_{0} + \kappa z_{0} + \sigma \right |/\sqrt{\lambda^2 + \mu^2 + \kappa^2}\tag{48}$$
Further evaluation of the geometry reveals that the relationship required for solving the sigma of the detector plane equation can be alternatively solved using a quadratic solution (Figure 1) as explained below. First, consider two sets of parametric equations of 3D lines that emerge from the point source through 3D points (equations 49-54). These lines then intersect the plane at some points (equations 55-60). Interestingly, PP allows one to determine the 2D scalar distance, which is a Euclidean distance, on the display plane using a projection of two 3D (we used {1,1,1} and {100, 100, 100}) points using equation 10. This scalar distance is equal to zeta in equation 61 and could alternatively be measured manually on the display plane. The final formation when squared reveals an equation that has a single unknown, sigma (equation 62). Finally, sigma can be solved here via the quadratic method (equations 63-64).
$$a_{1} = x_{1} - x_{0}\tag{49}$$
$$b_{1} = y_{1} - y_{0}\tag{50}$$
$$c_{1} = z_{1} - z_{0}\tag{51}$$
$$a_{2} = x_{2} - x_{0}\tag{52}$$
$$b_{2} = y_{2} - y_{0}\tag{53}$$
$$c_{2} = z_{2} - z_{0}\tag{54}$$
$$x_{p1} = x_{1} - ( a_{1}*( \lambda*x_{1} + \mu*y_{1} + \kappa*z_{1} + \sigma)/ (\lambda*a_{1} + \mu*b_{1} + \kappa*c_{1}) )\tag{55}$$
$$y_{p1} = y_{1} - ( b_{1}*( \lambda*x_{1} + \mu*y_{1} + \kappa*z_{1} + \sigma)/ (\lambda*a_{1} + \mu*b_{1} + \kappa*c_{1}) )\tag{56}$$
$$z_{p1} = z_{1} - ( c_{1}*( \lambda*x_{1} + \mu*y_{1} + \kappa*z_{1} + \sigma)/ (\lambda*a_{1} + \mu*b_{1} + \kappa*c_{1}) )\tag{57}$$
$$x_{p2} = x_{2} - ( a_{2}*( \lambda*x_{2} + \mu*y_{2} + \kappa*z_{2} + \sigma)/ (\lambda*a_{2} + \mu*b_{2} + \kappa*c_{2}) )\tag{58}$$
$$y_{p2} = y_{2} - ( b_{2}*( \lambda*x_{2} + \mu*y_{2} + \kappa*z_{2} + \sigma)/ (\lambda*a_{2} + \mu*b_{2} + \kappa*c_{2}) )\tag{59}$$
$$z_{p2} = z_{2} - ( c_{2}*( \lambda*x_{2}+ \mu*y_{2} + \kappa*z_{2} + \sigma)/ (\lambda*a_{2} + \mu*b_{2} + \kappa*c_{2}) )\tag{60}$$
$$\zeta_{12} = \sqrt( (x_{p1} - x_{p2})^2 + (y_{p1} - y_{p2})^2 + (z_{p1} - z_{p2})^2) = \sqrt( (U_{p1} - U_{p2})^2 + (V_{p1} - V_{p2})^2 )\tag{61}$$
$$\zeta_{12}^2 = ( (x_{p1} - x_{p2})^2 + (y_{p1} - y_{p2})^2 + (z_{p1} - z_{p2})^2) \tag{62}$$
$$0 = i\sigma^2 + j\sigma + k\tag{63}$$
$$\sigma = ( -j \pm \sqrt(j^2 - 4ik)/2i)\tag{64}$$
Fiducial matching from 3D to 2D
A time-consuming and potentially error-prone task is matching 3D to 2D fiducials for PP. One method is to utilize knowledge of the geometric setup and perform RT-assisted-matching (RTAM) for PP. Other options include applying identifiable characteristics or geometry to the fiducials. Lastly, after matching approximately six points (minimum 5.5), the PP matrix itself can be utilized to identify further matches.
A fully automatic matching for PP represents an important improvement in the workflow as it does not require precise projection knowledge between the 3D and the 2D points. It does require 3D knowledge of the fiducials and identification of the fiducials on 2D, but their precise correlation between 3D to 2D is not needed. Rather, a combinatorial optimization matching method (COMM) was implemented for this step to eliminate matching errors.
Mathematically, for a case comprising N (3D fiducials) and K (2D fiducials), where K <= N, the space of all possible combinations of K in N is:
$$N! / (K! * (N-K)!)\tag{65}$$
For each combination, there are K! permutations that must be searched to reach the best match. Therefore, given that unique combinations where order does need to match both 3D and 2D, the resulting numerical possibilities to investigate are:
$$N! / (N-K)!\tag{66}$$
As mentioned above, the solution of the PP transformation matrix derives from an overdetermined system that should be solved as a least squares error computation. Without additive information for each candidate match, the sum of the squares of the residuals of its solutions can be used as a metric of the matching quality. In principle, the automatic matching selection reduces to computation of all possible combinations and permutations keeping the one with the minimum least-squares error from its PP matrix solution. The previous crude approach produces very long computation times especially when the numbers K and N grow.
Singular value decomposition (SVD) is the most robust linear solution for PP matrix computation, but the suitability of QR-factorization, Gaussian Elimination, and Cholesky Factorization were explored. The methods were used in several test cases and each selected the same optimum match when minimizing mean square error (MSE). Cholesky factorization achieved convergence in less than a quarter of the time needed by SVD. Additional measures of the quality of the PP matrix can be utilized. For example, matrix condition number (CN) represents a ratio of the largest to smallest singular values (via SVD), which can eliminate singular or ill-conditioned systems. Theta, which should be 90 degrees (~1.57 radians), determines if the resultant vectors of the PP matrix are distorted as a measure of skew. Further, geometric constraints can be added if the estimated point source or detector plane are known. Generally, the fastest is MSE, which is not dependent on other factors or knowledge, but the addition of other thresholds ensures a consistent optimum result. Due to the nature of this approach and the prevalence of multicore CPUs/GPUs, a parallel implementation of the search process can also be implemented. For fiducials of n=8 and k=6, we get 20,160 possible combinations, and in combination with multiple thresholds, this represents a reasonable minimum for most CPUs.
RT fiducial tracking by moving focal spot location
As discussed above, the solution for RT is determined by the geometric properties of X-ray acquisition. If only some elements are known, a solution might be accomplished iteratively by changing the point source location and a tentative magnification multiplier until good overlap with fiducials is accomplished. If the geometry is largely unknown, but PP is possible, then PP can be used to compute the point source and image plane locations, which can subsequently be used in RT. This could be particularly useful when measurements are difficult to establish, but a relatively fixed setup is present.
If the X-ray setup is fixed (Source, Object(s), and Detector), then known changes in geometry should be compensated quickly using RT. For example, we demonstrate a case of a fixed X-ray setup with a phantom skull and widely distributed geometric fiducials. In this example, PP is used to determine the origin point for RT in both AP and LAT views. Then the source position is displaced linearly by 20 mm and a new image is obtained. From PP, the prediction of the point source by Euclidean Error of the two exposures was 19.7 mm and 20.0 mm for AP and LAT, respectively. In RT, simple displacement of 20 mm produced points wherein the fiducials overlapped without added adjustments (Figures 2-5).
PP and orthogonal and non-orthogonal RT
The following tables demonstrate the mathematical solutions for RT and PP. Starting with orthogonal imaging (COA) in RT, point source and planar data can be combined with objects (fiducials) and line-detector plane intersections (Table 1). In this case, the source image distance (SID) is 1168 mm and the pixel spacing is 388 micrometer (um)/pixel when using 2D fluoroscopy mode using the O-arm2 (Medtronic, Dublin, Ireland). Computing the ray-plane intersections for all the relevant objects reveals the AP dimension to be constant at -515 (COA), which allows the use of the LAT and VERT coordinates to be translated directly to image (Table 2). Similarly, the LAT image has a constant LAT dimension of 513 (Table 3).
The following tables (Tables 4-8) demonstrate the PP solution using 3D coordinates combined with scaled (in millimeters) U and V screen coordinates for analysis in both AP and LAT views. The complete PP matrices are computed using SVD with 22 and 26 equations in AP and LAT, respectively (Table 6). Applying the matrices to the images for all the hyperdense objects (fiducials, pins, electrode tips) shows overlap in the AP and LAT image (Figures 6, 7). The next step is to use PP to develop the geometry and compare the results to RT (Table 7). Finally, we apply biplanar imaging to localize the 3D position (STiL) in both RT and PP. The solution in RT can be derived from a line-line intersection problem using parametric equations for the lines to solve for the parameters r, which describes the closest point of intersection (equations 67-69). After r is solved, the values for x, y, and z can be readily calculated by plugging the value back into each parametric equation. A skew ray (line-line) intersection computation generating the closest points of intersection on each ray allows for a Euclidean distance between those points serving as a quality check between the two images. The biplanar point solution for PP can be derived by first generating complete matrices in AP and LAT (equations 70, 71). Then the U and V screen coordinates are used in the four planar systems of equations to solve x, y, and z simultaneously (equations 72-75). The results for the left electrode tip, converted to MidACPC coordinates demonstrate a 3D Euclidean error of 0.48 mm and 0.30 mm relative to immediate postoperative CT for RT and PP, respectively (Table 8).
$$x_{AP} + a_{AP}*r_{AP} = x_{LAT} + a_{LAT}*r_{LAT}\tag{67}$$
$$y_{AP} + b_{AP}*r_{AP} = y_{LAT} + b_{LAT}*r_{LAT}\tag{68}$$
$$z_{AP} + c_{AP}*r_{AP} = z_{LAT} + c_{LAT}*r_{LAT}\tag{69}$$
$$\begin{bmatrix} AP_{11} & AP_{12} & AP_{13} \\ AP_{21} & AP_{22} & AP_{23}\\ AP_{31} & AP_{32} & AP_{33}\\ AP_{41} & AP_{42} & AP_{43} \end{bmatrix}\tag{70}$$
$$\begin{bmatrix} LAT_{11} & LAT_{12} & LAT_{13} \\ LAT_{21} & LAT_{22} & LAT_{23}\\ LAT_{31} & LAT_{32} & LAT_{33}\\ LAT_{41} & LAT_{42} & LAT_{43} \end{bmatrix}\tag{71}$$
$$x(AP_{11}-AP_{13}U_{AP}) + y(AP_{21}-AP_{23}U_{AP}) + z(AP_{31}-AP_{33}U_{AP}) - U_{AP}AP_{43} + AP_{41} = 0 \tag{72}$$
$$x(AP_{12}-AP_{13}V_{AP}) + y(AP_{22}-AP_{23}V_{AP}) + z(AP_{32}-AP_{33}V_{AP}) - V_{AP}AP_{43} + AP_{42} = 0 \tag{73}$$
$$x(LAT_{11}-LAT_{13}U_{LAT}) + y(LAT_{21}-LAT_{23}U_{LAT}) + z(LAT_{31}-LAT_{33}U_{LAT}) - U_{LAT}LAT_{43} + LAT_{41} = 0 \tag{74}$$
$$x(LAT_{12}-LAT_{13}V_{LAT}) + y(LAT_{22}-LAT_{23}V_{LAT}) + z(LAT_{32}-LAT_{33}V_{LAT}) - V_{LAT}LAT_{43} + LAT_{42} = 0 \tag{75}$$
Next, we consider the situation of uncertain correlation between 3D points to 2D points. For this, the same patient case discussed above now includes an initial oblique image with the N-localizer positioned (Figure 8). We start the solution by using a combinatorial approach. Various thresholds can be analyzed, but here we assume no specific knowledge and use MSE and the matrix condition number (CN), but include the theta angle result (Table 9). The combination of low MSE, low CN, and theta closest to 1.57 radians identify the result. The next step is to use the geometric analysis of the point source and plane location (Table 10). These data are then placed in RT, but require normalization along the AP axis to make COA (Table 11). The final result produced by RT shows excellent overlap (Figure 9). Note that the projection can also be performed using PP. Because both methods project the scalar distance on the detector plane, data from RT and PP should be similar. Here, we randomly chose the AC point to Midline point yielding a distance in RT of 89.604 mm to PP of 89.694 mm. Finally, in RT, the result between detector plane intersection and display plane (normalized) is exactly the same (89.604 mm).
Error propagation with PP
While RT is computationally deterministic, PP utilizes a least squares optimization. Therefore, error propagation with PP is an important study. First, we evaluate automatic sphere detection on 3D images (CT) using a "center of mass" calculation and then evaluate potential errors. Hounsfield units and spherical size were used as thresholds for automatic sphere detection. This method would expedite acquisition of 3D points from CT imaging. Here, a comparison between this automation and manual selection is presented (Table 12). While this is an important step, error propagation in PP becomes relevant. We therefore introduced Monte Carlo simulation evaluating root mean square (RMS) as a function of the number of references when introducing random 1 mm errors into 1 million simulations (Figure 10). This reveals that error propagation is reduced by adding more data, which facilitates the least squares computation. In our physical setup, the vertical (VERT) distance between spheres had the least geometric variance. Fixing these points at 8, but increasing this VERT distance between them, causes a significant reduction in worst-case calculation of a 3D point when a 2-mm artificial error is introduced (Figure 11). This suggests that a wider geometric spread of fiducials reduces 3D error propagation while using PP.
X-ray imaging is widespread in various aspects of medicine. Single or multiple-plane computations, however, have not been widely utilized in standard X-rays imaging. While some efforts have been reported previously, herein, we have attempted a more comprehensive, usable, and illustrative approach [8-12]. We also explored various automations and error propagation. Further, enhanced computing power may facilitate these computations. Coincidentally, computer vision for 3D-2D rendering applies many similar principles to those used in X-ray imaging described above [7]. In computer vision, the task is generally to produce a 2D screen image from a dynamically changing 3D pose, but many modern applications include 3D-3D such as with virtual reality. Consideration of augmented reality also applies similar projection techniques superimposed on real-time analysis of images. Unlike our description for X-rays, the use of perspective projection with camera lenses may yield radial inhomogeneity (pincushion or barrel distortions) that require other steps or non-linear solutions.
Herein, we describe some of the mathematical principles of RT and PP applied to X-ray images and implement them in phantom and actual cases. Importantly, these methods have wide applications whenever planar images are obtained. Depending on the use case, simply increasing the number of reference points and the geometric spread ensures a quality result. However, RT depends more on knowledge of the imaging geometry. Also, application of scaling on the display image allows an enhanced understanding of the exposed geometry using PP. Three-dimensional imaging using CT or MRI images has largely overtaken 2D planar imaging when 3D positions are needed. However, CT/MRI incurs more expense and complexity, in addition to increased radiation (for CT) dose, without a significant improvement in resolution for a single target point. Further, the pixel size of an X-ray or fluoroscopy apparatus in our case were 0.125 mm and 0.388 mm, respectively, whereas the pixel size for 3D imaging is at best about 0.625 mm-0.8 mm for most current CT or MRI systems. Moreover, 3D imaging requires image fusion and introduces some artifacts by the nature of the acquisition and reconstruction techniques. Even though potentially millions of pixels are processed in 3D on CT/MRI, a single pixel displacement can result in a 0.625 mm error. In 2D imaging, for RT and PP, image point analysis is also required, but the 2D determination is at a higher resolution such that a single pixel displacement renders a 0.125 mm error. Therefore, this kind of point resolution to determine points in space, such as for deep brain stimulation (DBS) leads, can be sufficiently optimized using these simple X-ray techniques, which correlate well with preoperative CT/MRI imaging. In addition, as previously published, DBS lead rotation may be computed from projection X-ray data [2].
Important use cases for X-ray/fluoroscopy imaging include DBS, spine surgery, vascular interventions, non-vascular interventions, orthopedic interventions, radiosurgery and dental procedures as well as numerous non-medical applications or basic science laboratories, all of which can be enhanced with improved computing capacity [13 -16]. Because these systems can be designed using cost-effective materials, computational enhancements for these use cases could continue to grow. In addition, current X-ray and fluoroscopy technologies perform well with RT and PP, even though they were not designed for their use. Considering the mathematics and the physical nature of the X-ray systems, some assumptions should be considered carefully before applying RT or PP. Assumptions include that the system behaves like a Cartesian system, that the focal spot of the X-ray is a point source, that the X-ray detector is a plane, and that there is no significant scatter, or inhomogeneity in the image. We also assume X-ray and fluoroscopic imaging behave similarly. Some of these assumptions may not be completely accurate. For example, the focal spot of an X-ray can vary in size between different machines. Also, some fluoroscopic detectors may produce inhomogeneity. We also observed 2D images on some CT scanners, which were not a projection but rather a reconstruction. Therefore, care must be considered when applying these techniques to an imaging system.
Ray tracing (RT) and perspective projection (PP) are useful tools for geometric imaging computation. RT or PP performs well for X-ray imaging analysis and may be used independently or together. PP can serve as an initial calibration for RT in a fixed setup when the geometry is unknown or difficult to measure. Because the pixel resolution of X-ray images is generally greater than CT/MRI, which also have reconstructive artifacts, these X-ray techniques offer excellent precision for point analysis. These tools have great importance in functional neurosurgery, such as with DBS, but can be extended to other medical or non-medical applications.
- Howell JD: Early clinical use of the X-ray. Trans Am Clin Climatol Assoc. 2016, 127:341-349.
- Sedrak M, Sabelman E, Pezeshkian P, et al.: Biplanar X-ray methods for stereotactic intraoperative localization in deep brain stimulation surgery. Oper Neurosurg. 2019, [Published online ahead of print]:10.1093/ons/opz397
- Brown R, Nelson J: The invention and early history of the N-localizer for stereotactic neurosurgery. Cureus. 2016, 8:e642. 10.7759/cureus.642
- Brown R: A computerized tomography-computer graphics approach to stereotaxic localization. J Neurosurg. 1979, 50:715-720. 10.3171/jns.1979.50.6.0715
- Brown R: A stereotactic head frame for use with CT body scanners. Invest Radiol. 1979, 14:300-304. 10.1097/00004424-197907000-00006
- Glassner AS: An Introduction to Ray Tracing. Elsevier, 1989.
- Forsyth DA, Ponce J: Computer Vision: A Modern Approach. Pearson, 2002.
- Henderson J, Hill B: Fluoroscopic registration and localization for image-guided cranial neurosurgical procedures: a feasibility study. Stereotact Funct Neurosurg. 2008, 86:271-277. 10.1159/000147635
- Hunsche S, Sauner D, Majdoub F, Neudorfer C, Poggenborg J, Goßmann A, Maarouf M: Intensity-based 2D 3D registration for lead localization in robot guided deep brain stimulation. Phys Med Biol. 2017, 62:2417-2426. 10.1088/1361-6560/aa5ecd
- Bock A, Lang N, Evangelista G, Lehrke R, Ropinski T: Guiding deep brain stimulation interventions by fusing multimodal uncertainty regions. IEEE Xplore. 2013, 97-104. 10.1109/PacificVis.2013.6596133
- Aouadi S, Sarry L: Accurate and precise 2D-3D registration based on X-ray intensity. Comput Vision Image Understanding. 2008, 110:134-151. 10.1016/j.cviu.2007.05.006
- Choo A, Oxland T: Improved RSA accuracy with DLT and balanced calibration marker distributions with an assessment of initial-calibration. J Biomech. 2003, 36:259-264. 10.1016/s0021-9290(02)00361-5
- Willson R, Shafer S: Perspective projection camera model for zoom lenses. Proc SPIE. 1994, 2252:10.1117/12.169831
- Novins K, Sillion F, Greenberg DP: An efficient method for volume rendering using perspective projection. VVS '90. 1990, 11:95-102. 10.1145/99307.99329
- Cox D, Papanastassiou A, Oreper D, Oreper D, Andken BB, DiCarlo JJ: High-resolution three-dimensional microelectrode brain mapping using stereo microfocal X-ray imaging. J Neurophysiol. 2008, 100:2966-2976. 10.1152/jn.90672.2008
- Kakadiaris I, Toderici G, Evangelopoulos G, et al.: 3D-2D face recognition with pose and illumination normalization. Comput Vision Image Understanding. 2017, 154:137-151. 10.1016/j.cviu.2016.04.012
Applied Mathematics of Ray Tracing and Perspective Projection in Fiducial-Based Registration of X-Ray Images
Ethics Statement and Conflict of Interest Disclosures
Human subjects: All authors have confirmed that this study did not involve human participants or tissue. Animal subjects: All authors have confirmed that this study did not involve animal subjects or tissue. Conflicts of interest: In compliance with the ICMJE uniform disclosure form, all authors declare the following: Payment/services info: All authors have declared that no financial support was received from any organization for the submitted work. Financial relationships: Armando Alaminos-Bouza, Msc declare(s) employment from Mevis Informatica Medica LTDA. Partial owner of Stereotactic Software MNPS. Other relationships: All authors have declared that there are no other relationships or activities that could appear to have influenced the submitted work.
Maureen Sedrak, MHA; Gary Heit, PhD, MD; Bruce Hill, PhD; Siddharth Srivastava PhD; Eric Sabelman, PhD; Patrick Pezeshkian, MD; Ivan Bernstein, PA-C; Diana Bruce, PA-C, DMS; Yan-Bin Jia, PhD; Matthew Gardner, PhD
Article Information
Cite this article as:
Sedrak M, Alaminos-Bouza A L (April 30, 2020) Applied Mathematics of Ray Tracing and Perspective Projection in Fiducial-Based Registration of X-Ray Images. Cureus 12(4): e7904. doi:10.7759/cureus.7904
Publication history
Received by Cureus: April 06, 2020
Peer review began: April 11, 2020
Peer review concluded: April 22, 2020
Published: April 30, 2020
© Copyright 2020
Sedrak et al. This is an open access article distributed under the terms of the Creative Commons Attribution License CC-BY 4.0., which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Applied Mathematics of Ray Tracing and Perspective Projection in Fiducial-Based Registration of X-Ray Images
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Neighborhood satisfaction is an important component of individuals' quality of life, and is associated with social capital, community involvement, and residential stability. Although research has found that neighborhood satisfaction is linked with homeownership, this correlation appears to diminish for the poorest homeowners. Little is known about the association between homeownership status and neighborhood satisfaction for low- and moderate-income (LMI) households. Given that LMI households are more likely to purchase houses in lower-income, distressed neighborhoods, it is important to discover whether they still experience higher rates of neighborhood satisfaction than LMI renters. This study aims to examine the role of homeownership among LMI households in explaining neighborhood satisfaction. Further, this study examines individual and neighborhood characteristics to explore important predictors of neighborhood satisfaction among LMI households.
The data for this study come from the 2007 Community Advantage Secondary Market Program (CAP) survey. CAP is a secondary mortgage market pilot program for LMI households, operated in 49 states. This study uses a quasi-experimental design of LMI homeowners (n=689); a comparison panel of LMI renters (n=764); and U.S. Census tract information to represent neighborhood characteristics. Given the nested data, we use hierarchical generalized linear modeling (HGLM) to assess the differences between homeowners and renters on two dichotomous neighborhood satisfaction outcomes: rating of neighborhood and recommending of neighborhood. Individual-level covariates (level 1) include demographic characteristics and household income; social capital measured by resource generation scale (alpha=0.81); and new neighborhood status. Census tract covariates (level 2) include the neighborhood economic disadvantage scale (alpha=0.91); the neighborhood stability scale (alpha=0.68); and the proportion of non-White population. All variables are grand mean centered for analysis.
First, homeownership is significantly related to neighborhood satisfaction for the LMI sample population. Controlling for individual and neighborhood characteristics, the odds of LMI homeowners positively rating and recommending their neighborhood are 2.3 times (<.001) and 2.7 times (<.001) higher, respectively, than for LMI renters. Second, we demonstrate that all neighborhood characteristics (level 2) are significantly related to neighborhood satisfaction. For example, neighborhood economic disadvantage is associated with less likelihood of positively rating (<.001) and recommending (<.001) the neighborhood. Third, among individual level variables, social capital and movement to a new neighborhood are significant predictors for neighborhood satisfaction among LMI households. For example, a one point increase on the social capital scale is associated with a 15% (<.001) and a 19% (<.001) increase in the odds of positively rating and recommending the neighborhood, respectively.
This study makes several unique contributions, most notably the focus on LMI households, which are under investigated in neighborhood satisfaction research. Increased knowledge of the impacts of homeownership for LMI families is of critical importance to the field of social work. Moreover, we account for nested data (individual within neighborhood) with HGLM, correcting for biases generated when researchers use uncorrected models such as ordinary least squares. Finally, we discuss the importance of using experimental and quasi-experimental designs and sophisticated statistical methods, to advance the rigor of social work research.


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Discover Blue Lake an oasis in the Cocotal Golf & Country Club in Punta Cana!
Tags: Caribbean dream homes, Punta Cana, Blue Lake, Cocotal Golf & Country Club
It is 2020! One of my New Year's resolutions for this year is, to keep showing you the most amazing projects in Punta Cana. Yes, my love for Punta Cana is just as strong as it was in 2019. It amazes me how this town continues to grow positively! One of the projects that helps this great tendency is Blue Lake.
I have introduced you to some great projects in Punta Cana. In my blogs you got to know Ocean Bay, Coral Bay, Cana Pearl, and Reserva Real. Now it is time to introduce you to Blue Lake, which is a little different than the other properties.
What makes Blue Lake different is that it is located in Cocotal Golf & Country Club. This is a gated community located in Bávaro, Punta Cana in the Dominican Republic, near the Punta Cana International Airport. It is a peaceful community where you can walk around in beautiful surroundings and meet friendly people.
Blue Lake was created to have you living in a space that is not only tastefully designed, but also has the highest level of comfort. It was made to give a sense of freedom. The 2- and 3-bedroom condos are spacious and have everything you need to make your Caribbean Dream life come true.
At Blue Lake you can wake up with a stunning view of the Golf Course and the lake that different bird species call their home. It is truly an oasis.
Enjoy exclusive access to the gym, beach and pools of the Melia Caribe. Play some paddle and tennis in Cocotal and hop over to the private beach club at Playa Coral.
But you can also enjoy the infinity pool with lakeview that is part of Blue Lake.
This development also benefits from CONFOTUR, an incentive from the Dominican government for investors which provides a 15-year property tax and transfer tax exemption.
This could be your new vacation, permanent or part-time home in the sun! Get in touch with me, so I can give you more information about floorplans, interiors, prices and delivery dates!
Nick Caccavella


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Welcome to Kimpton Taconic Hotel. At this time, we are open and accepting guests.
We're continuing to monitor developments and following guidance provided by the Centers for Disease Control and Prevention (CDC), World Health Organization (WHO) and the State of Vermont. To help safeguard your health and best serve you during your stay with us, we've made necessary modifications to our amenities and services. In addition to our rigorous standard housekeeping routines, we've increased the frequency of our cleaning procedures and have put in place additional measures to enhance our disinfection protocol in public areas.
We have temporarily modified/suspended the following amenities:
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- In-room dining is available for breakfast and dinner. Limited menu + limited service delivery
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2 years ago
Great vid, tfs.


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Microsoft SharePoint is an integrated solution that utilizes browser-based technology to promote teamwork and knowledge sharing by providing policy-based control of content, workflow efficiency, and consolidated functions for finding company-wide business data. SharePoint is closely connected with familiar Office applications, Microsoft Exchange, and browsers to offer an intuitive user experience that streamlines the way people work with business data and with one another. With Microsoft SharePoint your company can quickly build safe and dependable portals that assemble and display important company information from an array of locations via built-in mechanisms that include Wikis, social sites, and Web Parts. SharePoint Server's Report Center includes a powerful publication database for Excel workbooks, SQL Reporting Services reports, and a variety of other classes of business reports.
Microsoft SharePoint Server is built on a standardized, extensible architecture with support for a library of functions and interface specifications such as XML and the SOAP protocol. SharePoint Server features rich programming interfaces and event handlers for lists and mixed-media documents. These capabilities enable incorporation into existing systems and offer the flexibility to work with non-Microsoft technology.
Progent's Microsoft-certified engineers can provide Albuquerque, New Mexico organizations affordable remote and onsite help for planning, deploying, and maintaining productive SharePoint Server installations that will dramatically improve important areas of your company's information system such as intranet portals, enterprise search, forms-based business processes, delivery of business intelligence, and content control. Progent's Power BI integration experts can show you how to use SharePoint information to build visualizations to produce actionable Power BI analytical reports and dashboards.
Progent can assist your IT team you to design, deploy and manage SharePoint environments that incorporate local and cloud hosted resources. (Find out about Progent's Microsoft Azure enterprise hybrid cloud solutions.)
New Mexico State Contact Information:
Progent offers consulting services and onsite technical support throughout New Mexico.
|6300 Riverside Plaza Lane NW, Suite 100
Albuquerque, NM 87120
|Albuquerque Network Support Services
SharePoint 2019 Server and SharePoint Online Consulting
SharePoint Server 2019 delivers much of the modern look and feel of SharePoint Online, providing a crossover platform that bridges the classic UI of SharePoint 2016 and the new SharePoint environment. Progent's certified SharePoint Server 2019 and SharePoint Online consultants offer economical online and onsite consulting, application development, and debugging services for businesses of all sizes who want to migrate to SharePoint Server 2019 or SharePoint Online from prior releases of SharePoint. Progent can help customers plan and execute a cost-effective upgrade to SharePoint 2019 on premises, SharePoint Online, or a hybrid environment that incorporates local and cloud resources into a seamless intranet solution. To learn more, visit SharePoint 2019 Server and SharePoint Online Services.
SharePoint 2016 Migration and Hybrid Integration Services
SharePoint Server 2016, designed for on-premises deployments of SharePoint, offers integration with Microsoft SharePoint Online and Office 365 to enable hybrid network models that seamlessly unify on-premises and cloud resources. Other improvements include fast provisioning with the new MinRole feature, increased scale, higher performance, tighter security and compliance, enterprise analytics via Power BI integration, and a modernized client interface for collaboration. Progent's Microsoft-certified consultants can provide affordable remote and on-premises expertise, software development, and technical support services for organizations of all sizes who intend to migrate to SharePoint Server 2016 or SharePoint Online from prior versions of SharePoint. To find out more, see SharePoint 2016 Migration Consulting.
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Progent has been providing SharePoint support since the initial product was released over a decade ago, and Progent has the breadth and depth of technical knowledge to assist businesses of any size to design and implement a SharePoint Server 2013 installation and develop SharePoint 2013 apps that add significant business value to your information network. Progent's Microsoft-certified consultants can assist you to evaluate the risks and benefits of migrating to SharePoint Server or SharePoint Online from legacy releases of SharePoint or from another collaboration environment; determine whether an on-premises, online or hybrid deployment is most sensible for your busi9ness; and select the most appropriate version of SharePoint 2013 for your purposes. Progent offers full project management support or can assist in individual jobs related to your SharePoint 2013 deployment. To learn more, see SharePoint Server 2013 Consulting and Support.
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Microsoft SharePoint 2010 includes major enhancements in all key areas such as web infrastructure, communities, enterprise search, content, business analytics, and composites. Progent's Microsoft-certified consultants offer cost-effective online technical support and development services to help you to design, implement, manage, and troubleshoot SharePoint 2010 environments that can feature intranets, extranets, document and media management tools, wikis, data warehouses, business analytics, enterprise search engines, workflows, and other tools that enhance teamwork. To learn more, see SharePoint 2010 Consulting and Development.
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The primary goal of SharePoint Server 2007 is to gather together, in a productive way, all of the many stores of knowledge available inside an organization. Microsoft Office SharePoint Server 2007 includes major enhancements for connecting workers, customers, workgroups, and stakeholders with the data they've developed in a solution that makes business information convenient to locate, retrieve and apply. To find out more details, go to Progent's support for Microsoft SharePoint 2007 Server.
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To ask Progent about professional expertise for MS SharePoint Server technology, call
Progent's Consulting Services for Microsoft-powered Platforms
For companies of any size in Albuquerque New Mexico, Progent's Microsoft-authorized experts offer low-cost online computer help and professional consulting services for the whole family of Microsoft .NET Enterprise Servers. Progent offers Windows Server 2019 migration consulting, Windows Server 2016 hybrid cloud integration, Windows Server 2012 R2 design and deployment help, Microsoft Windows Server 2008 consulting, and Windows Server 2003 technical support. Progent's migration, integration, update, and consulting services cover system design, configuration, and administration outsourcing for project management and documentation, local and remote IT support and system troubleshooting, Help Desk Outsourcing Support, comprehensive security expertise, full-service outsourcing, and ProSight Virtual Hosting services.
Progent's System Center 2016 consultants include System Center 2016 Operations Manager experts, Microsoft System Center 2016 Configuration Manager consultants and System Center 2016 Virtual Machine Manager (SCVMM 2016) hybrid cloud management consultants who can assist you to design and deploy Microsoft's platform for unified management of hybrid cloud environments that combine on-premises resources and cloud-based infrastructure and services.
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Immediate Sorts Synthesis: Students read a number of texts about a matter and create an argument that synthesizes at least three of the sources to help their thesis. You have to to affix our Edmodo class for the summer work. The be a part of group code and URL can be supplied on the 'summer work' meeting for AP English Literature. 2. "Thou Blind Man's Mark" AP Lit poetry immediate. Fastidiously annotate poem, and make a brief define of how you would answer this question.
Eliza Cole is a master in International History with the achieved greater schooling. She has a grasp degree and prime skills from the American College and totally different programs of the applied specifications. Apart from knowing every thing about international historical past and affairs, Eliza is fond of reading basic literature, thus, boasts of good writing expertise and just limitless vocabulary. Just like all other writers in our staff, Eliza has achieved a course on educational writing and paper formatting. So she knows how a top quality tutorial paper should appear like, be formatted, and structured. APA, MLA, Turabian, Chicago codecs won't be an issue for her. Since writer's pursuits are broad, you can easily assign to her papers on Literature, Administration, Psychology, Culturology, and Religion. A top quality result's a hundred% assured.
When reading, take care to go slowly and reread necessary or advanced sections. Pause usually to think about which means, context, and intent. Become an energetic reader, underlining and taking notes as you go. Keep in mind that the importance of the textual content comes not only from the writer, but also from how the text affects you, the reader. Pay attention to how you're feeling and why you are feeling that method. Consider whether or not your response is the one intended by the writer. If you don't imagine it is, ask yourself what the writer might have meant you to feel, and why. Go to the Faculty Board's Reading Study Expertise for extra data.
Painless essay samples Methods Across The Usa
Even earlier than the ghost showed up, although, Hamlet's relationship with Claudius was additionally strained because of Hamlet's depression. Hamlet already hated that Gertrude remarried so shortly. And it's worse that she remarried Claudius, since he was the king's brother. Claudius doesn't like the way in which Hamlet was performing so down in the dumps, or the way in which he continually makes nasty remarks towards Gertrude. But once Hamlet actually begins performing loopy and placing on this show to investigate, Claudius actually tries to have him killed. He seems to see through the act more than everybody else, who all think Hamlet is barely appearing this way as a result of Ophelia rejected him.
Locating Swift Methods For literature essay samples
Choose 3+ particular passages from To the Lighthouse that you feel may help create bridges to particular elements of Hamlet. You need not find specific allusions, per se, however essay on their eyes are watching god can think about common issues of theme, battle, and normal humanistic questions raised. Be ready to discuss your concepts in brainstorming groups.
Hamlet would not have a relationship together with his uncle. I mean, how do you have a relationship with the guy who killed your dad and married your mother? He wants to kill his uncle for many of the book however he always chickens out or can't do it at the right time.
10. Take a look at Yourself: Essentially the most helpful and efficient solution to put together for the multiple-selection portion of the AP English Literature exam is by testing yourself. Put together early within the semester for the examination. Periodically, take practice a number of-selection tests on the content material you have learned thus far. There are a number of websites out there devoted to serving to you quiz yourself for the AP Literature exam. One in all these is , which allows you to test your skills on practically each concept lined within the AP English Literature course.
The A response not solely provides one other example of the daddy-daughter inverse relationship — the extra he helps her fit in, the extra estranged they turn out to be — but also ends the place the writer began: although they're physically reunited, they're nonetheless emotionally separated. Without repeating it verbatim, the student returns to the thesis assertion on the finish. This return and recap reinforce the main target and management of the argument when the entire previous paragraphs efficiently proved the thesis assertion.
Considering Aspects In essay sample
22. Take 10-12 minutes to read and deconstruct the immediate, annotate the poem or passage and develop a thesis earlier than you begin writing the essay. That considering and planning time will allow you to stay focused which is able to make sure that your essay is obvious and cohesive. Thanks for the tip from Erin M. at Mercy County Senior High.


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The Mon 29 Jun 2020 Set For Life draw results: first was the twenty, then second the twenty six, the three went to him twenty nine.
The four went to him thirty three and number five thirty seven.
The life Ball: three.
Draw number one hundred thirty five, with the machine Excalibur 2 and using the set-ball SFL3.
Match five plus Life Ball: zero winners, with a prize of ten thousand pounds each month for thirty years.
Match five: zero winners, with a prize of ten thousand pounds each month for one year.
Match four plus Life Ball: thirty five winners, with a prize of two hundred and fifty pounds each one. Which is a total of eight thousand seven hundred fifty pounds.
Match four two hundred twenty eight winners, with a prize of fifty pounds each one. A total of eleven thousand four hundred pounds.
Match three plus Life Ball: one thousand two hundred fifty winners, with a prize of thirty pounds each one. That is a total of thirty seven thousand five hundred pounds.
Match three: ten thousand four hundred fifty three winners, with a prize of twenty pounds each one. Totaling two hundred nine thousand sixty pounds.
Match two plus Life Ball: eighteen thousand five hundred thirty one winners, with a prize of ten pounds each one. Total: one hundred eighty five thousand three hundred ten pounds.
Match two: one hundred forty eight thousand five hundred ten five winners, with a prize of five pounds each one. This is a total of seven hundred forty two thousand five hundred seventy five pounds.
The total of the winners is: one hundred seventy nine thousand twelve.
The total of the prizes is: one million one hundred ninety four thousand five hundred ninety five.
These were the Set For Life results from the draw 135 of the date Mon 29 Jun 2020.
This website is "COOKIE'S FREE!".
We do not obtain any information from each visitor, their navigation is completely anonymous.
The content of this website is only for information purpose, it's not possible to take a game in this website. Our mayor effort is focussed in to serve concrete and quikly information from each draw, so anybody can see the latest Set For Life draw results from the UK National Lottery. This website is not responsible for any defects that may be found to exist in and these results should not be taken as official.
The content and of this website have not been approved or endorsed by Camelot UK Lotteries Limited or the National Lottery Commission.
Crytpocurrencies - Quiniela Nacional


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The goal of socialism is Communism - Vladimir Lenin
Libs going that way anyway
Side Score: 7
Red scare in full effect
Side Score: 3
Socialism and Communism are one in the same. They share the same philosophical base justifying its existence: The individual is an end unto the state, as opposed to being an end unto self.
Centralized government authority fails (socialism):
Nazi Germany
Soviet Russia
Today's North Korea
Today's Venezuela
Today's Cuba
The covered up mass murder and confiscation of land in South Africa
The endless poverty, starvation and disease in many African nations
The latter era of the Roman Empire
The latter era of Ancient Greece
All brought to as the result of Socialism >> A centralized state of authority where the individual is not an end unto self but rather a end unto the state, to the king, to the tribe, to the gang leader thug, to the Furher, to the common good, to the moral cause and whatever else you can fill the blank, that usurps an individual's right to his/her own terms to self-existence.
Side: Libs going that way anyway
"The goal of socialism is communism."
-Vladimir Lenin
Side: Libs going that way anyway
Obviously you have a lot more free time than I do and I'm not going to let you waste my time going through your list of disinformation.
Give ONE single CREDIBLE source for that quote.
You Reich wing nuts love to pass BS back and forth. Get a grip.
Side: Red scare in full effect


text: A unique place to experience masterful blending, powerful botanicals and herbs, rich, smooth flavours.Imagine yourself stepping through what appears to be the simple door of an old red brick building into a very magical almost mystical place of elegance, refinement and magnificence! Enter a world of herbs, rich botanicals, special blends, where spirit reigns supreme and you experience the bounty of perfection. Here you will discover fine Gins, beautiful Vodkas; elegant, rich Herbal Liqueurs and stately, distinguished Ports. The atmosphere will astound you and tastings will "knock your socks off!". Each and every product stands alone providing unique, beautiful flavours. The Gins will delight each in their own way; Sweet Jennifer is slightly sweet with a little kick; Lady Susanne is bold with a slight citrus, London Dry style. Navy Strength, the International Award Winner, is a true experience not to be missed. Springwater Vodka is surprisingly smooth with a hint of 'something' as only a master blender can create. Equally smooth and rich in flavour is our Espresso Vodka, perfect for coffee lovers everywhere! The Herbal Liqueurs are a kaleidoscope of adventure and taste. Eight different blends, unique and stylish in temperament and design tempt the palate in so many different ways. And no added sugar! Then there is the port! With five different blends to choose from; Malt Whisky, Bourbon, Rum, Brandy and Double Pot Still, the greatest challenge is deciding which ones to purchase! Of course, the shining light in the range of ports is the Award Winning Smoked Malt Whisky Tawny which came third place in the International Wine and Spirit show in London. Whatever your desire, whatever your pleasure, fine flavour and pure adventure is yours to experience when you visit Herbal Lore and Terra Australis Distillery. herballoreliqueurs.com award-winning Daylesford drink gin Herbal Liqueurs Herbal Lore herbs PORT powerful botanicals smooth flavours. Terra Australis Distillery. vodka Whisky Share this articleWhile You're Here Gin's the ThingSliding into the limelight with a bouquet of botanicals and a swag of awards are two boutique gin…Eat & Drink Wine online (or over the phone)We've put together a list of our wineries that can be purchased (and subsequently consumed) from…Eat & Drink Turn Over a New LeafYou may have seen the 'documentary' (and I use the word very loosely) Game Changers on Netflix. For…Eat & Drink Something's Brewing in WoodendBrewer Paul Holgate stands proudly in his brand new taproom. The sweet smell of malted barley and…Eat & Drink Market HeroesThe sun is setting on a warm Saturday night, and farmer Florian Hofinger is bringing in fat, ripe…Eat & Drink Wheelie Fun SummerSummertime is almost here, and that means the best of the food truck scene will heat up in the…Eat & Drink Interview with Stockroom KynetonALL ARTICLESThe Ancient Incas called it "the Fibre of the Gods" - Alpaca Passion


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Free Cholesterol Screenings - INHS Caregivers Only
health@work is offering free cholesterol screenings for INHS Caregivers at the below dates and times. These screenings are first-come, first-served, so please register early as we anticipate all appointments to fill.
At this screening you will receive instant results for total cholesterol (TC), HDL, TC/HDL ratio, LDL and Triglycerides. Fasting is not required to obtain all numbers.
Registration is required.
Make-up screenings will not be available.


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Great news … you can use your Stadia controller wirelessly on your PC.
Taken from TPU … The Google Stadia controller launched alongside Google Stadia 6 months ago and was touted as the controller for all platforms with integrated ultra-low latency features. However, PC users could not take advantage of the ultra-low latency features which connects the Stadia controller directly to your Wi-Fi router. PC users have instead had to connect their controller to their PC with a USB Type-C cable. Some users report that this will increase compatibility on obscure systems by offloading the input handling to Stadia's servers.
"Starting this week, you can now use the Stadia Controller to play wirelessly on your laptop and desktop. You no longer need to connect your Stadia Controller physically to your computer to play your favorite games. You'll be able to link your Stadia Controller and Stadia.com through your Wi-Fi network and play without a USB-C cable connection."
Source: TPU


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martimm0290 wrote on the wall for Bring the Rain:
Wow, I beat everyone in every class except engineer which i dropped because I dont use it often. Im the best pilot here tho xD. If anyone here wants to go rape in some conquest toss an invite/friend request or message me.


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Your Local St. Mobi Road Carpenters
Our St. Mobi Road carpenters are skilled Uran Build qualified craftsmen right on top of their profession. By using the most effective tradesmen in the industry our reputation has been established as the ' most likely to' company when high quality and service issues. Their craftsmanship can be seen in a few of city 's most exclusive addresses yet expenses are constantly tailored to fit the task whether it be floors, stairs cabinets or total repairs. We could even offer a ' group' of multi competent tradespersons for huge jobs with each member being a specialist within their very own profession.
Available 24 hrs our carpenters additionally supply emergency situation services for such things as doors, home windows, and burglary or unintentional damage. Our tradespersons will constantly ensure minimal disruption which they leave your premises clean and tidy following their see.
Professional Carpenters St. Mobi Road
carpenters Services are open 24 hours a day for carpenters within 1 hour
"Call Us Now For A Fixed Free Quote"
Carpentry Work we do in
- Bathrooms
- Boarding up & securing
- Ceilings
- Cupboards
- Design & project management
- Door locks
- Doors – internal & external
- Fencing & gates
- Fitted furniture
- Flooring installation
- Garden sheds
- Kitchens
- Shelving
- Wardrobes
- Worktops
Every little thing you need to learn about finding carpenters
carpenters construct, put up or fix frameworks that are made of wood, steel or wood substitutes. They could develop the structure of your house, wall surfaces and roofing system; mount doors, home windows, floors, cabinets, ceiling ceramic tiles; put up scaffolding; complete various other structural job such as sub-floor framing; and could also repair damaged furnishings. Carpenters may specialise in a particular location of the trade, so make certain your picked expert has the ability to finish your details project.
Duty of a carpenters:
Layout/design: carpenters should select the materials for the project and measure and mark them to guarantee that they are the correct measurements needed.
Cutting and signing up with products: Cutting may be finished with power or hand devices. The reducing phase likewise consists of any kind of shaping that needs to be done. Products are signed up with using nails, glue, screws or screws and leaders. Degrees and various other framework devices are employed to make certain accuracy in joining.
Estimation: carpenters approximate the quantity of products that will certainly be called for to complete a job.
Analysis: carpenters translate and read blueprints and other plans to figure out exactly what will be needed for a project.
Follow building standards: carpenters are needed to follow regional and nationwide building ordinance and criteria when working on any type of project.St. Mobi Road
Projects Carpenters Can Assist With
Doors: carpenters mount, repair and replace all kinds of doors from basic bedroom doors to difficult ornate doors. If the old door structure is not adequate, they can install a completely brand-new frame relying on your needs. See carpenters – doors to find out more.
Windows: As well as doors, carpenters can set up new home windows and fixing or change old windows. They can likewise mount window devices such as joints and openers. For more information see carpenters – windows.
Formwork: This is required when concrete is layed on your home or business. A carpenter is the ideal specialist for making, structure and taking down formwork and ensuring it is durable and able to support the framework.
Handrails: These are necessary to guarantee staircases, verandas, patios and decks are secure and safe and secure. carpenters are competent at the workplace with lumber, so could build a simple or ornate handrail to suit your house style and tastes. For additional information see carpenters – handrails.
Structure outdoor decking: A carpenter could design and construct an attractive deck, so you could appreciate your outdoor area year-round. They can additionally install required devices such as stairs, ramps and handrails. See building decking for more.
Architraves: These are an attractive attribute in any type of home, as they go around windows or doors to hide gaps between the home window or door structure and the wall surface (see architraves). carpenters could assist select the excellent architrave and the suitable kind of fixing to match.
House structures: As the house structure is the skeleton of the residence, it is necessary to ensure it is erected properly. Carpenters can put up home frames for brand-new residences and extensions or additions, be it flat roofings, gable roofs or various other kinds of home structures. For more informatoin see carpenters – home structures.
Ramps: carpenters have the ability to build any type of kind of ramp you need, consisting of mobility device and access ramps, skate ramps, boat ramps and so forth. They could likewise assist you obtain the needed licenses for the ramp, and create a ramp that is secure and appropriate. See carpenters – ramps for more details.
Improvements: Whether you are carrying out tiny or large renovations in your home, carpenters could aid throughout the process. They could assist mount kitchen cabinets, add an expansion or deck, install new floorings, closets and more. They could additionally help you acquire preparing permissions and allows where called for and aid with the layout of your renovation. For more information see carpenters – renovations.
Skirtings: Skirtings are a decorative attribute that could assist finish off the look of a space. Carpenters could mount brand-new skirtings or change old skirtings easily using wood, MDF or plaster. See carpenters – skirting for more choices.
What does it cost? Does a carpenters Cost?
carpenters will commonly charge anywhere from EUR30 to EUR50 per hr, depending upon your place and their experience. This will certainly include their time and you must confirm if materials are part of any kind of quote. Much more expensive hardwood woods, for instance, will certainly lug a cost premium over soft timbers such as ache. carpenters can recommend you which kind of hardwood is best suited to your project. If you are building a big structure such as a hardwood deck, any type of quote should include itemised labour and products costings.
Carpenter Licensing
In some cities carpenters are called for to be licensed to do work over a certain worth. This is a legal demand and guarantees your task is done by a qualified tradesperson. There is more info on licensing of carpenters on www.nationalguild.ie.
Useful links: National Guild. Construction Industry Register, Passive House Association.
Nearby areas we cover:carpenters near Ekm Mont
joiners near Marlboro Street
joiners near St. Mobi Road
carpenters near Lord Edward Street
joiners near Temple Bar
carpenters near South Great Georges Street
carpenters near College Green
carpenters near D2
carpenters near Four Courts
carpenters near Phoenix Park
Urban Build Carpenters St. Mobi RoadUrbanBuild is a St. Mobi Road based carpentry, joinery & fitting firm with years of experience. Get your furniture solutions designed by reliable carpenters carpenters.8267 Cherry Street
St. Mobi Road, L EI33
Dublin 9, Leinster (L)Email: info.St. Mobi firstname.lastname@example.com: 0877090265


text:
Sep 24
Community Event
Light the Night - Leukemia & Lymphoma Society Walk
09/24/20 (All day)
Save the Date - Join the Presque Isle CPCU Chapter to support the Leukemia & Lymphoma Society (LLS) for the annual Light the Night Walk on Thursday, September 24, 2019.
More details about registration will be shared soon! For questions and details about walking, or to help support the LLS, please contact Cindy Kerchoff or Lindsay Filley. Thank you so much for your support!


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What's more, because the LG LT has easy-to-master controls, you can easily adjust the color balance, aspect ratio, and brightness just the way you like it. Stand Base detachable Yes. For Consumer For Business. Voice Assistant Control select LG products with your voice. Monitors by Lori Grunin Oct 15,
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LG LT-BF Monitors – 19 Standard Series LCD Monitor – LG Electronics CA_EN
Additional Product Features Monitor Color. Warranty Info Refer to warranty info.
Because the pg select is handily placed on the monitor I was able to do double duty with one of the displays. Cyber Monday Newegg deals: No time to chat?
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These high-performance, feature-packed monitors are the ultimate multitaskers. Newegg's Cyber Monday deals are available now, including 43 percent off a inch LG gaming monitor. Where to Buy Where to Buy. It's also practical and convenient.
LG LT LCD Monitor | eBay
Learn more about some the features that help put LG flat-screen monitors in a class of their own. Weight Kg Set with Stand 3.
Product Registration Get the most out of your new purchase. Common Spec Product Type Office. Brightness, contrast, gamma correction, phase, clock.
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Laptops by Justin Jaffe Nov 26, Details shown on product images may vary by region, country, or model. The LED backlighting delivers deep black levels flayron rich colours for an amazing contrast and colour detail. Boasting true-to-life colours, IPS computer monitors lessen colour loss and help to keep colours consistent at virtually any angle. With features like a built-in digital tuner and Dolby Surround sound, LG TV monitors can bring your favorite movies and TV shows to life, along with your favorite online content and games.
In order to get the best flatrron experience on LG. About this product Product Information This inch LG monitor is designed for taking your gaming experience up a notch and bringing you outstanding graphics as you add to your system's performance. An inexpensive general-purpose monitor deserving of its popularity An inexpensive general-purpose monitor deserving of its popularity by Lori Grunin. I bought two to use as a p1942t display on a PC.
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Product Code: FELLQUJ
Style Tip: Choose next size up if wearing a sweater underneath
Lightweight and easy to wear everyday quilted jacket by Alan Paine. Water repellent outer with a pretty paisley lining featuring a two-way front zip with studded placket. Finished with mock suede trims throughout.
• Classic Fit
• 100% Polyester Water Repellent Outer
• Two-Way Centre Front Zipper With Studded Placket
• Luxurious Quilted Outer
• Lined With Striking Scarlet Paisley Lining
• Two Zipped Pockets
• Mock Suede Binding Throughout Front Edge & Sleeve Cuffs
Delivery Country:
|UK Delivery (2 to 3 days)
|FREE for orders over £49, £3 on orders below
|European Delivery (3 to 4 days)
|FREE for orders over €59, €6 on orders below
|USA Delivery (3 to 4 days)
|Delivery charge is $20
|Rest Of The World Delivery (5 to 6 days)
|Delivery charge is £30+
*Sweden Excluded - 120SEK
+£70 delivery charge for delivery to the United Arab Emirates
During this period of uncertainty, we're extending our returns policy from 30 to 60 days if it's unwanted or unsuitable. There's a free label included with every box for all UK orders, and we promise to refund any products that you are not entirely happy with provided that they come back to us in a saleable condition. Please visit see our Delivery & Returns page for more details.
Call: +44 (0)1623 339002
(Mon - Thur 8.45 am - 5pm & Fri 8.45am - 3.45pm)