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Documentation for taking the derivatives of innerProducts

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rowanc1
2014-03-02 23:34:31 -08:00
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@@ -188,12 +188,80 @@ Which is nice and easy to invert if necessary, however, in the fully anisotropic
Taking Derivatives
------------------
TODO: Take the derivatives of the tensors.
We will take the derivative of the fully anisotropic tensor for a 3D mesh, the other cases are easier and will not be discussed here. Let us start with one part of the sum which makes up \\\(\\mathbf{M}^f_\\Sigma\\\) and take the derivative when this is multiplied by some vector \\\(\\mathbf{v}\\\):
.. math::
\left[\begin{smallmatrix}0.5 \sigma_{1} & 0 & 0.25 \sigma_{3} & 0.25 \sigma_{3}\\0 & 0.5 \sigma_{1} & 0.25 \sigma_{3} & 0.25 \sigma_{3}\\0.25 \sigma_{3} & 0.25 \sigma_{3} & 0.5 \sigma_{2} & 0\\0.25 \sigma_{3} & 0.25 \sigma_{3} & 0 & 0.5 \sigma_{2}\end{smallmatrix}\right]
\mathbf{P}^\top \boldsymbol{\Sigma} \mathbf{Pv}
Here we will let \\\( \\mathbf{Pv} = \\mathbf{y} \\\) and \\\(\\mathbf{y}\\\) will have the form:
.. math::
\mathbf{y} = \mathbf{Pv} =
\left[
\begin{matrix}
\mathbf{y}_1\\
\mathbf{y}_2\\
\mathbf{y}_3\\
\end{matrix}
\right]
.. math::
\mathbf{P}^\top\Sigma\mathbf{y} =
\mathbf{P}^\top
\left[\begin{matrix}
\boldsymbol{\sigma}_{1} & \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{5} \\
\boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{2} & \boldsymbol{\sigma}_{6} \\
\boldsymbol{\sigma}_{5} & \boldsymbol{\sigma}_{6} & \boldsymbol{\sigma}_{3}
\end{matrix}\right]
\left[
\begin{matrix}
\mathbf{y}_1\\
\mathbf{y}_2\\
\mathbf{y}_3\\
\end{matrix}
\right]
=
\mathbf{P}^\top
\left[
\begin{matrix}
\boldsymbol{\sigma}_{1}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{4}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{5}\circ \mathbf{y}_3\\
\boldsymbol{\sigma}_{4}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{2}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_3\\
\boldsymbol{\sigma}_{5}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{3}\circ \mathbf{y}_3\\
\end{matrix}
\right]
Now it is easy to take the derivative with respect to any one of the parameters, for example, \\\(\\frac{\\partial}{\\partial\\boldsymbol{\\sigma}_1}\\\)
.. math::
\frac{\partial}{\partial \boldsymbol{\sigma}_1}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right)
=
\mathbf{P}^\top
\left[
\begin{matrix}
\text{diag}(\mathbf{y}_1)\\
0\\
0\\
\end{matrix}
\right]
Whereas \\\(\\frac{\\partial}{\\partial\\boldsymbol{\\sigma}_4}\\\), for example, is:
.. math::
\frac{\partial}{\partial \boldsymbol{\sigma}_4}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right)
=
\mathbf{P}^\top
\left[
\begin{matrix}
\text{diag}(\mathbf{y}_2)\\
\text{diag}(\mathbf{y}_1)\\
0\\
\end{matrix}
\right]
These are computed for each of the 8 projections, horizontally concatenated, and returned.
The API
-------