box), encapsulates all possible pixel locations that can be reached by the $m$ number of of warps during pre-computation. An example warped template is shown in Fig. \[fig:symbolic\] as the black box, where each pixel from the template is transformed to a subpixel location.

For each sample warp, we first rasterize the warped template into an $n$-vector, $\mathbf{t}$, and the pixels in the bounding box into an $l$-vector, $\mathbf{u}$. An equivalent representation of the vector $\mathbf{t}$ is shown in Fig. \[fig:symbolic\] as a multiplication of a Warp Coefficient Matrix, $\mathbf{WCM}(\Delta \mathbf{p})$, and the vector $\mathbf{u}$. This matrix is a function of the warp update, where each row has 4 non-zero values representing bilinear interpolation coefficients.

For $m$ number of sample warps, we obtain $m$ number of $\mathbf{WCM}(\Delta \mathbf{p})$, which can be stacked up to create a tensor of size $n\times m \times l$ (shown as the blue cuboid). We also define a $\mathbf{WCM}(\mathbf{0})$ at the identity warp, whereby each row has 1 non-zero value with a coefficient of 1.0, representing a pixel location in the template. Stacking $m$ number of $\mathbf{WCM}(\mathbf{0})$ gives another tensor of size $n\times m \times l$ (shown as the green cuboid). Now, if we subtract the 