By applying extra constraints on the above generalized 2x2 Hermitian matrix for satisfying positive semi-definiteness criteria, we can arrive at a generalized representation for 2x2 POVM operator, say $E_{i}$.
The constraint is that the Hermitian matrix should have only non-negative
eigenvalues [1].
Let $E_{i}$ be
$$
E_{i} =
\left[{\begin{array}{cc}
a_{i0}+a_{i3} & a_{i1}-ia_{i2} \\
a_{i1}+ia_{i2} & a_{i0}-a_{i3}
\end{array}}\right].
$$
The characteristic equation for above the matrix would be
$$
\left|{\begin{array}{cc}
\lambda-(a_{i0}+a_{i3}) & a_{i1}-ia_{i2} \\
a_{i1}+ia_{i2} & \lambda-(a_{i0}-a_{i3})
\end{array}}\right|=0
$$
The roots of the characteristic polynomial should be non-negative,
$$
\lambda^{2}-2a_{i0}\lambda+a_{i0}^{2}-a_{i3}^{2}-a_{i1}^{2}-a_{i2}^{2}=0\\
(\lambda-a_{i0})^{2}=a_{i3}^{2}+a_{i1}^{2}+a_{i2}^{2}\\
\lambda=\pm k+a_{i0}
$$
where $k=|\sqrt[]{a_{i1}^{2}+a_{i2}^{2}+a_{i3}^{2}}|$. Hence, we have the condition:
$$
a_{i0} \ge k
$$
because $\lambda_{min}=-k+a_{i0}$.
By definition, $\sum_{i}E_{i}=\mathbb{I_{2x2}}$. In total, we have (4+$i$) constraints:
- $\sum_{i}a_{i0}=1$
- $\sum_{i}a_{i1}=0$
- $\sum_{i}a_{i2}=0$
- $\sum_{i}a_{i3}=0$
- $\forall i, a_{i0} \ge |\sqrt[]{a_{i1}^{2}+a_{i2}^{2}+a_{i3}^{2}}|$
[1] Weisstein, Eric W. "Positive Semidefinite Matrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html
