Find Pareto front of a two-objective function of a two-dimensional variable of different dimmension.

Question

I need to find Pareto front of two objective functions $J_g(g,h)$ and $J_h(g,h)$ where $g$ is a matrix (image) and $h$ is a small convolution (blur) matrix \begin{equation} J_{g}(g, h)=\frac{1}{2}\|h * g-z\|_{L^{2}(\Omega)}^{2}+\int_{\Omega} \alpha_1|D g| d x \label{7} \end{equation} \begin{equation} J_{h}(g, h)=\frac{1}{2}\|h * g-z\|_{L^{2}(\Omega)}^{2}+\int_{\Omega}\alpha_2|D h| d x \label{8} \end{equation} The problem is those two functions depend on each other and are of two different dimmension. I faced the same problem while trying to minimize them simultaneously (i derived the two and used a gradient method to solve the equations, in the first step i fix $h$, solve $\frac{\partial J_{g}}{\partial g}=0$, i obtain a value $g$ that i use when fixing $g$ in $\frac{\partial J_{g}}{\partial g}=0$ and repeat till convergence). $z$ is a given blurred image.
\begin{equation} \frac{\partial J_{g}}{\partial g}=h(-x,-y) *(h * g-z)-\alpha_1 \nabla \cdot \frac{\nabla g}{|\nabla g|}=0 \quad x \in \Omega \end{equation} and \begin{equation} \frac{\partial J_{h}}{\partial h}=g(-x,-y) *(g * h-z)-\alpha_2 \nabla \cdot \frac{\nabla h}{|\nabla h|}=0 \quad x \in \Omega \end{equation} None of the predefined functions in Matlab (paretosearch,gamultiobj) worked for me.