Centering split equation

Just replace the split environment by gathered, then you don't have to suppress spurious equation numbers:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\begin{gathered}
a = \frac{M_{20}}{M_{00}} - x^{2}_{c}, \;\;\;
b = 2(\frac{M_{11}}{M_{00}} - x_{c}y_{c}), \;\;\;
c = \frac{M_{02}}{M_{00}} - y^{2}_{c}, \\
\quad x_{c} = \frac{M_{10}}{M_{00}}, \;\;\;
y_{c} = \frac{M_{01}}{M_{00}}, \\
\quad M_{ij} = \displaystyle\sum\limits_{x} \displaystyle\sum\limits_{y} x^i y^j I(x,y)
\end{gathered}
\end{equation}
\end{document}

\documentclass{minimal}
\usepackage{amsmath}

\begin{document}

\begin{gather}
a  = \frac{M_{20}}{M_{00}} - x^{2}_{c},\quad
b = 2(\frac{M_{11}}{M_{00}} - x_{c}y_{c}),\quad
c = \frac{M_{02}}{M_{00}} - y^{2}_{c},\nonumber\\[1ex]
\text{and}\nonumber\\[1ex]
x_{c} = \frac{M_{10}}{M_{00}},\quad
y_{c} = \frac{M_{01}}{M_{00}}, \nonumber\\
M_{ij} = \displaystyle\sum\limits_{x} \displaystyle\sum\limits_{y} x^i y^j I(x,y)
\end{gather}

\end{document}

\documentclass{article}   
\usepackage{amsmath}   
\begin{document}

\begin{gather}
a = \frac{M_{20}}{M_{00}} - x^{2}_{c}, \;\;\; 
    b = 2\left(\frac{M_{11}}{M_{00}} - x_{c}y_{c}\right), \;\;\;
    c = \frac{M_{02}}{M_{00}} - y^{2}_{c}, \nonumber\\
x_{c} = \frac{M_{10}}{M_{00}}, \;\;\; y_{c} = \frac{M_{01}}{M_{00}}, \\
M_{ij} = \sum_x\sum_y x^i y^j I(x,y)\nonumber
\end{gather}

\end{document}