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STYLE: Replace \bf with \textbf
In modern LaTeX, you should not use the old two-letter command \bf. Instead, you should use the recommended \textbf{...} command or the \bfseries declaration for producing bold text.
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\begin{minipage}[t][1.0cm][b]{\textwidth}
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\begin{center}
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Printed and produced in the United States of America.\\
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\bf{ISBN 978-1-930934-35-1}
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\textbf{ISBN 978-1-930934-35-1}
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\end{center}
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\end{minipage}
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\begin{minipage}[t][1.0cm][b]{\textwidth}
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\begin{center}
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Printed and produced in the United States of America.\\
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\bf{ISBN 978-1-930934-36-8}
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\textbf{ISBN 978-1-930934-36-8}
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\end{center}
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\end{minipage}

SoftwareGuide/Latex/DesignAndFunctionality/DemonsRegistration.tex

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of the displacement is derived from the instantaneous optical flow equation:
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\begin{equation}
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\bf{D}(\bf{X}) \cdot \nabla f(\bf{X}) = - \left(m(\bf{X}) - f(\bf{X}) \right)
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\textbf{D}(\textbf{X}) \cdot \nabla f(\textbf{X}) = - \left(m(\textbf{X}) - f(\textbf{X}) \right)
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\label{eqn:OpticalFlow}
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\end{equation}
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In the above equation, $f(\bf{X})$ is the fixed image, $m(\bf{X})$
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is the moving image to be registered, and $\bf{D}(\bf{X})$ is the displacement
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In the above equation, $f(\textbf{X})$ is the fixed image, $m(\textbf{X})$
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is the moving image to be registered, and $\textbf{D}(\textbf{X})$ is the displacement
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or optical flow between the images. It is well known in optical flow
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literature that Equation \ref{eqn:OpticalFlow} is insufficient to specify
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$\bf{D}(\bf{X})$ locally and is usually determined using some form of
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$\textbf{D}(\textbf{X})$ locally and is usually determined using some form of
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regularization. For registration, the projection of the vector on the
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direction of the intensity gradient is used:
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\begin{equation}
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\bf{D}(\bf{X}) = - \frac
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{\left( m(\bf{X}) - f(\bf{X}) \right) \nabla f(\bf{X})}
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\textbf{D}(\textbf{X}) = - \frac
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{\left( m(\textbf{X}) - f(\textbf{X}) \right) \nabla f(\textbf{X})}
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{\left\| \nabla f \right\|^2 }
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\end{equation}
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re-normalizes the equation such that:
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\begin{equation}
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\bf{D}(\bf{X}) = - \frac
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{\left( m(\bf{X}) - f(\bf{X}) \right) \nabla f(\bf{X})}
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{\left\| \nabla f \right\|^2 + \left( m(\bf{X}) - f(\bf{X}) \right)^2 / K }
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\textbf{D}(\textbf{X}) = - \frac
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{\left( m(\textbf{X}) - f(\textbf{X}) \right) \nabla f(\textbf{X})}
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{\left\| \nabla f \right\|^2 + \left( m(\textbf{X}) - f(\textbf{X}) \right)^2 / K }
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\label{eqn:DemonsDisplacement}
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\end{equation}
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value of the pixel spacings. The inclusion of $K$ ensures the force computation is
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invariant to the pixel scaling of the images.
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Starting with an initial deformation field $\bf{D}^{0}(\bf{X})$, the demons
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Starting with an initial deformation field $\textbf{D}^{0}(\textbf{X})$, the demons
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algorithm iteratively updates the field using Equation
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\ref{eqn:DemonsDisplacement} such that the field at the $N$-th iteration is
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given by:
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\begin{equation}
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\bf{D}^{N}(\bf{X}) = \bf{D}^{N-1}(\bf{X}) - \frac
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{\left( m(\bf{X}+ \bf{D}^{N-1}(\bf{X}))
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- f(\bf{X}) \right) \nabla f(\bf{X})}
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\textbf{D}^{N}(\textbf{X}) = \textbf{D}^{N-1}(\textbf{X}) - \frac
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{\left( m(\textbf{X}+ \textbf{D}^{N-1}(\textbf{X}))
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- f(\textbf{X}) \right) \nabla f(\textbf{X})}
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{\left\| \nabla f \right\|^2 + \left(
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m(\bf{X}+ \bf{D}^{N-1}(\bf{X}) )
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- f(\bf{X}) \right)^2 }
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m(\textbf{X}+ \textbf{D}^{N-1}(\textbf{X}) )
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- f(\textbf{X}) \right)^2 }
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\label{eqn:DemonsUpdateEquation}
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\end{equation}
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Reconstruction of the deformation field is an ill-posed problem where
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matching the fixed and moving images has many solutions. For example, since
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each image pixel is free to move independently, it is possible that all
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pixels of one particular value in $m(\bf{X})$ could map to a single image
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pixel in $f(\bf{X})$ of the same value. The resulting deformation field may
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pixels of one particular value in $m(\textbf{X})$ could map to a single image
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pixel in $f(\textbf{X})$ of the same value. The resulting deformation field may
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be unrealistic for real-world applications. An option to solve for the field
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uniquely is to enforce an elastic-like behavior, smoothing the deformation
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field with a Gaussian filter between iterations.
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used in this case is
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\begin{equation}
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\bf{D}(\bf{X}) = - \frac
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{2 \left( m(\bf{X}) - f(\bf{X}) \right) \left( \nabla f(\bf{X}) + \nabla g(\bf{X}) \right) }
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{\left\| \nabla f + \nabla g \right\|^2 + \left( m(\bf{X}) - f(\bf{X}) \right)^2 / K }
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\textbf{D}(\textbf{X}) = - \frac
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{2 \left( m(\textbf{X}) - f(\textbf{X}) \right) \left( \nabla f(\textbf{X}) + \nabla g(\textbf{X}) \right) }
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{\left\| \nabla f + \nabla g \right\|^2 + \left( m(\textbf{X}) - f(\textbf{X}) \right)^2 / K }
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\label{eqn:DemonsDisplacement2}
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\end{equation}
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SoftwareGuide/Latex/DesignAndFunctionality/ImageMetrics.tex

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In the following sections, we describe the ITKv4 metric types in detail.
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You can check ITK descriptions in doxygen for details about ITKv3 metric classes.
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For ease of notation, we will refer to the fixed image $f(\bf{X})$
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and transformed moving image $(m \circ T(\bf{X}))$ as images $A$ and $B$.
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For ease of notation, we will refer to the fixed image $f(\textbf{X})$
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and transformed moving image $(m \circ T(\textbf{X}))$ as images $A$ and $B$.
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\subsection{Mean Squares Metric}
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\label{sec:MeanSquaresMetricv4}

SoftwareGuide/Latex/DesignAndFunctionality/LevelSetsSegmentation.tex

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The paradigm of the level set is that it is a numerical method for tracking
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the evolution of contours and surfaces. Instead of manipulating the contour
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directly, the contour is embedded as the zero level set of a higher
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dimensional function called the level-set function, $\psi(\bf{X},t)$. The
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dimensional function called the level-set function, $\psi(\textbf{X},t)$. The
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level-set function is then evolved under the control of a differential
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equation. At any time, the evolving contour can be obtained by extracting
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the zero level-set $\Gamma(\bf{X},t) =
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\{\psi(\bf{X},t) = 0\}$ from the output. The main advantages of using level
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the zero level-set $\Gamma(\textbf{X},t) =
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\{\psi(\textbf{X},t) = 0\}$ from the output. The main advantages of using level
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sets is that arbitrarily complex shapes can be modeled and topological
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changes such as merging and splitting are handled implicitly.
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those inputs will be cast to an image of appropriate type when the filter is
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executed.
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Most filters require two images as input, an initial model $\psi(\bf{X},
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Most filters require two images as input, an initial model $\psi(\textbf{X},
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t=0)$, and a \emph{feature image}, which is either the image you wish to
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segment or some preprocessed version. You must specify the isovalue that
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represents the surface $\Gamma$ in your initial model. The single image

SoftwareGuide/Latex/DesignAndFunctionality/Registration.tex

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Let's begin with a simplified typical registration framework where its components
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and their interconnections are shown in Figure \ref{fig:RegistrationComponents}.
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The basic input data to the registration process are two images: one is defined
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as the \emph{fixed} image $f(\bf{X})$ and the other as the \emph{moving} image
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$m(\bf{X})$, where $\bf{X}$ represents a position in N-dimensional space.
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as the \emph{fixed} image $f(\textbf{X})$ and the other as the \emph{moving} image
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$m(\textbf{X})$, where $\textbf{X}$ represents a position in N-dimensional space.
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Registration is treated as an optimization problem with the goal of finding the
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spatial mapping that will bring the moving image into alignment with the fixed image.
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\label{fig:RegistrationComponents}
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\end{figure}
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The \emph{transform} component $T(\bf{X})$ represents the spatial mapping of
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The \emph{transform} component $T(\textbf{X})$ represents the spatial mapping of
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points from the fixed image space to points in the moving image space. The
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\emph{interpolator} is used to evaluate moving image intensities at non-grid
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positions. The \emph{metric} component $S(f,m \circ T)$ provides a measure of

SoftwareGuide/Latex/DesignAndFunctionality/Transforms.tex

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indicates for a point in the input space how much its mapping on the output
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space will change as a response to a small variation in one of the transform
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parameters. Note that the values of the Jacobian matrix depend on the point in
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the input space. So actually the Jacobian can be noted as $J(\bf{X})$, where
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${\bf{X}}=\{x_i\}$. The use of transform Jacobians enables the efficient
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the input space. So actually the Jacobian can be noted as $J(\textbf{X})$, where
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${\textbf{X}}=\{x_i\}$. The use of transform Jacobians enables the efficient
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computation of metric derivatives. When Jacobians are not available, metrics
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derivatives have to be computed using finite differences at a price of $2M$
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evaluations of the metric value, where $M$ is the number of transform
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\begin{equation}
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\begin{array}{lccccccc}
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\mbox{Point } & \bf{P'} & = & T(\bf{P}) & : & \bf{P'}_i & = & \bf{P}_i \cdot S_i \\
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\mbox{Vector} & \bf{V'} & = & T(\bf{V}) & : & \bf{V'}_i & = & \bf{V}_i \cdot S_i \\
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\mbox{CovariantVector} & \bf{C'} & = & T(\bf{C}) & : & \bf{C'}_i & = & \bf{C}_i / S_i \\
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\mbox{Point } & \textbf{P'} & = & T(\textbf{P}) & : & \textbf{P'}_i & = & \textbf{P}_i \cdot S_i \\
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\mbox{Vector} & \textbf{V'} & = & T(\textbf{V}) & : & \textbf{V'}_i & = & \textbf{V}_i \cdot S_i \\
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\mbox{CovariantVector} & \textbf{C'} & = & T(\textbf{C}) & : & \textbf{C'}_i & = & \textbf{C}_i / S_i \\
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\end{array}
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\end{equation}
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where $\bf{P}_i$, $\bf{V}_i$ and $\bf{C}_i$ are the point, vector and covariant
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vector $i$-th components while $\bf{S}_i$ is the scaling factor along dimension
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where $\textbf{P}_i$, $\textbf{V}_i$ and $\textbf{C}_i$ are the point, vector and covariant
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vector $i$-th components while $\textbf{S}_i$ is the scaling factor along dimension
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$i-th$. The following equation illustrates the effect of the scaling transform
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on a $3D$ point.
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\textbf{Restrictions} \\
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\hline\hline
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Represents a $3D$ rotation and a $3D$ translation. The rotation is specified as a
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quaternion, defined by a set of four numbers $\bf{q}$. The relationship
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between quaternion and rotation about vector $\bf{n}$ by angle $\theta$ is as
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follows: \[ \bf{q} = (\bf{n}\sin(\theta/2), \cos(\theta/2))\] Note that if the
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quaternion, defined by a set of four numbers $\textbf{q}$. The relationship
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between quaternion and rotation about vector $\textbf{n}$ by angle $\theta$ is as
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follows: \[ \textbf{q} = (\textbf{n}\sin(\theta/2), \cos(\theta/2))\] Note that if the
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quaternion is not of unit length, scaling will also result. &
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7 &
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The first four parameters defines the quaternion and the last three parameters

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