Could someone please confirm for me the partial derivatives of this function: $\dfrac{\mathrm{\partial}L}{\mathrm{\partial}U}$ and $\dfrac{\mathrm{\partial}L}{\mathrm{\partial}V}$:
$L = \dfrac{1}{N} \sum_{i, j}^{N} \Bigg( \Big(I_2(i + U, j + V) - I_1(i, j)\Big)^2 + \epsilon^2\Bigg)^\alpha$
Thank you in advance.
\begin{align} \frac{\partial L}{\partial U} &= \dfrac{1}{N} \frac{\partial}{\partial U}\sum_{i, j}^{N} \Bigg( \Big(I_2(i + U, j + V) - I_1(i, j)\Big)^2 + \epsilon^2\Bigg)^\alpha\\[0.1in] &=\dfrac{1}{N} \sum_{i, j}^{N} \frac{\partial}{\partial U} \Bigg( \Big(I_2(i + U, j + V) - I_1(i, j)\Big)^2 + \epsilon^2\Bigg)^\alpha\\[0.1in] &=\dfrac{\alpha}{N} \sum_{i, j}^{N} \Bigg( \Big(I_2(i + U, j + V) - I_1(i, j)\Big)^2 + \epsilon^2\Bigg)^{\alpha-1} \frac{\partial}{\partial U}\Bigg( \Big(I_2(i + U, j + V) - I_1(i, j)\Big)^2 + \epsilon^2\Bigg)\\[0.1in] &=\dfrac{2\alpha}{N} \sum_{i, j}^{N} \Bigg( \Big(I_2(i + U, j + V) - I_1(i, j)\Big)^2 + \epsilon^2\Bigg)^{\alpha-1} \Big(I_2(i + U, j + V) - I_1(i, j)\Big)\frac{\partial}{\partial U}I_2(i + U, j + V) \end{align}
The partial derivative with respect to $V$ can be evaluated in exactly the same way.