volatility - Proof of approximation formulas for implied volatilities

I am trying to calibrate a local volatility model to observed implied volatility smiles (not surfaces!, just a smile given for fixed maturity).

I ran into the following approximation, and thought I could plug in my implied volatility formula, and solve for local volatility;

$$\sigma_{i}^2(K,T) = \frac{\sigma^2(S(0), 0) + \sigma^2(K,T)}{2}.$$

$\sigma_i$ is implied volatility.

My question however is, where does this approximation come from?

I know Dupire's formula, you know the one that is a fraction of two integrals. I can't quite find it, but is it from that the approx is derived? If so, how?