This allows us to prioritize image patches that are expected to exhibit smaller alignment errors, especially when an image consists of hundreds of keypoints.

The rest of this paper is organized as follows: We first formally define the problem of subpixel refinement in Sec \[sec:prelim\]. In order to make the paper self-contained, we briefly review the IC-LK and ESM methods in Sec \[sec:energy\], followed by Linear Predictors in Sec \[sec:lin\_predictors\]. We describe our proposed Symbolic Linear Predictors in Sec. \[sec:symbolic\] and show experimental results in Sec \[sec:exp\_results\].

Subpixel Refinement {#sec:prelim}
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Throughout this paper, subpixel refinement is defined as a template matching problem. Given a pair of point correspondences $\mathbf{x}_1 \leftrightarrow \mathbf{x}_2$, where $\mathbf{x} = (x,y)^T$ is a pixel location, a template $\mathcal{T}$ is centered around $\mathbf{x}_1$ and another image patch $\mathcal{I}$ is centered around $\mathbf{x}_2$. The goal is to align the template with the image patch. Both patches are small and of equal sizes, typically ranging from $4 \times 4$ to $8 \times 8$ pixels. Further, different sampling methods can be used to select the pixels that will be used within the patch. We let $n$ denote the number of pixels used in a patch.

We let $\mathbf{M}$ define 