Local volatility recovery from call prices (${\sigma }^2=2(C_T+rK{C}_K)/(K^2{C}_{KK})$)

Where $C_{KK}>0$, rearranging the Dupire equation gives $\sigma (T,K{)}^2=2(C_T+rK{C}_K)/(K^2{C}_{KK})$. The strike curvature represents discounted density, while the adjusted maturity derivative gives the local diffusion contribution.