A photorefractive nonlinear joint-transform correlator based on the incoherent-to-coherent conversion is presented and analyzed. The nonlinearity of this incoherent-erasure joint-transform correlator (IEJTC) is tunable from the classical-matched to the phase-extraction limit. Correlation peak intensity, sharpness, and discrimination ability increase with the incoherent beam intensity. At easily achievable incoherent-to-coherent beam intensity ratios the IEJTC has its optimal performance, at which the IEJTC approaches the performance of the inverse filter for clean inputs and surpasses the inverse filter performance for noisy inputs. We examine this nonlinearity by using the transform method of analysis and computer simulations. Our study focuses on the effect of saturation on the correlation ability. Our results provide an explanation of why extending the severity of saturation by increasing the incoherent-to-coherent intensity ratio beyond a turning point results in lower optical efficiency, degraded correlation peak, and increased higher-order harmonics.