Let $(S_g,\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2)$ a Heegaard diagram of a Heegaard splitting $\Sigma=H_g \cup_{\phi_1\phi_2^{-1}}H_g$ of an integral homology sphere $\Sigma$, i.e. $S_g$ is a closed, orientable surface of genus $g$, $H_g$ is the $3$-dimensional $1$-handlebody of genus $g$, $\phi_1,\phi_2 \colon \partial H_g \rightarrow S_g$ are homeomorphisms and $\boldsymbol{\alpha}_i=\phi_i(\boldsymbol{\alpha})$ for $i=1,2$, where $\boldsymbol{\alpha}$ is the set of boundaries of a maximal system of meridians for $H_g$. I know that the Casson invariant of $\Sigma$ can be defined in terms of its Heegaard splitting, but I do not see clearly how to compute it from a Heegaard diagram. Is there a simple way to do that?

Let $(S_g,\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2)$ be a Heegaard diagram of a Heegaard splitting $\Sigma=H_g \cup_{\phi_1\phi_2^{-1}}H_g$ of an integral homology sphere $\Sigma$, i.e. $S_g$ is a closed, orientable surface of genus $g$, $H_g$ is the $3$-dimensional $1$-handlebody of genus $g$, $\phi_1,\phi_2 \colon \partial H_g \rightarrow S_g$ are homeomorphisms and $\boldsymbol{\alpha}_i=\phi_i(\boldsymbol{\alpha})$ for $i=1,2$, where $\boldsymbol{\alpha}$ is the set of boundaries of a maximal system of meridians for $H_g$. I know that the Casson invariant of $\Sigma$ can be defined in terms of its Heegaard splitting, but I do not see clearly how to compute it from a Heegaard diagram. Is there a simple way to do that?