Based on an analogy between DNA replication and one dimensional nucleation-and-growth processes, various attempts to infer the local initiation rate I(x,t) of DNA replication origins from replication timing data have been developed in the framework of phase transition kinetics theories. These works have all used curve-fit strategies to estimate I(x,t) from genome-wide replication timing data. Here, we show how to invert analytically the Kolmogorov-Johnson-Mehl-Avrami model and extract I(x,t) directly. Tests on both simulated and experimental budding-yeast data confirm the location and firing-time distribution of replication origins.