In a manner similar to the re-formulation approach, the LP is then computed as $$\mathbf{A} = (\hat{\mathbf{D}}^T \hat{\mathbf{D}})^{-1} \hat{\mathbf{D}}^T$$ where, again, the size of the matrix to be inverted is $6 \times 6$.

Symbolic Linear Predictors {#sec:symbolic}
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From Sec. \[sec:energy\] and Sec. \[sec:lin\_predictors\], we see that both energy and learning-based methods consist of two steps: (1) a pre-computation step and (2) a run-time step. Although learning-based methods are faster during run-time, they also have a huge cost associated with the pre-computation step. The methods in Sec. \[subsec:efficient\_lin\_predictors\] reduce learning time through dimensionality reduction that results in a smaller matrix inversion. Although this results in computational savings for the application of planar target tracking, these savings do not translate to the problem of subpixel refinement.

Symbolic Error Matrix {#subsec:symb_err}
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We propose a once-off, pre-training step to learn a symbolic representation for LPs that is independent of pixel intensities. Once learned, it can be re-used on different image patches to learn the specific LP for that patch.

Our approach revolves around creating a representation for the error matrix $\mathbf{E}$ that is independent of pixel intensities. Fig. \[fig:symbolic\] illustrates how this is done. A bounding box, $\mathcal{B}$ (shown as the red 