to the warp parameters. As the cost function is non-linear, the LK algorithm is applied iteratively, with the update computed as $\mathbf{p} \leftarrow \mathbf{p} + \Delta \mathbf{p}$. However, the steepest descent image must be computed on the re-warped image at every iteration, thus making the LK algorithm computationally demanding.

Baker and Matthews [@Baker2001] proposed a computationally efficient variant of the LK algorithm, known as the inverse compositional method (IC-LK). The IC-LK algorithm is derived by swapping the roles of the input image and the template, thus minimizing the cost function $$\label{eq:iclk_cost}
    \mathrm{min}_{\Delta \mathbf{p}} || \mathcal{I}(\mathbf{p}) - \mathcal{T}(\Delta \mathbf{p}) ||^2_2.$$ Proceeding in a manner similar to the LK algorithm, the cost function (\[eq:iclk\_cost\]) is linearized: $$\mathrm{min}_{\Delta \mathbf{p}} || \mathcal{I}(\mathbf{p}) - \mathcal{T}(\mathbf{0}) - \frac{\partial \mathcal{T}(\mathbf{0})}{\partial \Delta \mathbf{p}} \Delta \mathbf{p} ||^2_2,$$ and the warp update is computed as $$\label{eq:iclk_up}
    \Delta \mathbf{p} = \frac{\partial \mathcal{T}(\mathbf{0})}{\partial \Delta \mathbf{p}}^{\dagger} (\mathcal{I}(\mathbf{p}) - \mathcal{T}(\mathbf{0})),$$ where the subscript $\dagger$ denotes the pseudo-inverse operator. The advantage of this formulation is that the Jacobian and the pseudo-inverse are independent of $\Delta \mathbf{p}$, and hence can be pre-computed and re-used during subsequent iterations.

Efficient Second-order Minimization
-----------------------------------

The Efficient Second-order Minimization (ESM) algorithm was proposed by Benhimane [@Benhimane2004]. It is derived by 