You can convert the implied volatility to local volatility using this formula:

$\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$

Where y is the money-ness, defined as $y=\ln \left(\frac{ K}{F} \right)$, and w is the transformation of Black Scholes implied vol $w=\sigma_{BS}^2\,T$

So the conversion part is easy. The challenge then is: we have implied vol quotes for only a limited number of strikes and maturities, we can certainly fit surfaces through these points and get the local vol surface at as granular level as we like, but there is an infinite number of functional forms that will fit the finite number of data points we got. So you have to bring in some constraints, these could be in the form of specifying the function itself (e.g., cubic spline, SVI etc), or putting in some regularisation (e.g., smooth function to be preferred), so that's how the regulation aspect comes into play.

Hope this helps!