performing a second-order Taylor expansion on the cost function (\[eq:lk\_cost\]): $$\label{eq:esm_2ndorder_tay}
    \mathrm{min}_{\Delta \mathbf{p}} || \mathcal{I}(\mathbf{p}) + \frac{\partial \mathcal{I}(\mathbf{p})}{\partial \Delta \mathbf{p}} \Delta \mathbf{p} + \frac{1}{2}\Delta \mathbf{p}^T \mathbf{H} \Delta \mathbf{p} - \mathcal{T}(\mathbf{0}) ||^2_2,$$ where $\mathbf{H}$ is the Hessian matrix. A first-order Taylor expansion is then performed on the steepest descent image: $$\label{eq:esm_1storder_tay}
    \frac{\partial \mathcal{I}(\mathbf{p})}{\partial \Delta \mathbf{p}} \approx \frac{\partial \mathcal{I}(\mathbf{0})}{\partial \Delta \mathbf{p}} + \mathbf{H} \Delta \mathbf{p}.$$ Substituting this first-order Taylor expansion (\[eq:esm\_1storder\_tay\]) into (\[eq:esm\_2ndorder\_tay\]) yields $$\mathrm{min}_{\Delta \mathbf{p}} || \mathcal{I}(\mathbf{p}) + \frac{1}{2}(\frac{\partial \mathcal{I}(\mathbf{0})}{\partial \Delta \mathbf{p}} + \frac{\partial \mathcal{I}(\mathbf{p})}{\partial \Delta \mathbf{p}}) - \mathcal{T}(\mathbf{0}) ||^2_2$$ where the Hessian matrix has now been approximated from two steepest descent images, one which is independent of the warp updates whereas the other has to be computed from the re-warped image.

Learning-based Methods {#sec:lin_predictors}
======================

In this section we briefly review the Linear Predictor (LP) method proposed by Jurie and Dhome [@Jurie2001] and its efficient variants [@Holzer2015]. Note that there are other variants [@Xiong2013; @Xiong2014; @Xiong2015] where multiple LPs are applied to learn a non-linear function, with each LP responsible for one iteration of the update step. These methods are beyond the scope of this paper.

Linear Predictors {#subsec:jd}
-----------------

The concept of a Linear Predictor (LP) was first proposed 