Medical imaging is an important tool to make accurate medical diagnosis and disease intervention. Current medical image reconstruction algorithms mainly run on Si-based digital processors with von Neumann architecture, which faces critical challenges to process massive amount of data for high-speed and high-quality imaging. Here, we present a memristive image reconstructor (MIR) to greatly accelerate image reconstruction with discrete Fourier transformation (DFT) by computing-in-memory (CIM) with memristor. To implement DFT on memristor arrays efficiently, we proposed a high-accuracy quasi-analogue mapping (QAM) method and generic complex matrix transfer (CMT) scheme, to improve the mapping precision and transfer efficiency, respectively. With these two strategies, we used MIR to demonstrate high-fidelity magnetic resonance imaging (MRI) and computed tomography (CT) image reconstructions, achieving software-equivalent qualities with peak signal-to-noise ratios (PSNR) of 40.88 dB and 22.38 dB, respectively. The reconstructed images were then segmented using a popular nnU-Net algorithm to further evaluate the reconstruction quality. For the MRI task, the final DICE scores were 0.979 and 0.980 for MIR and software, respectively; while for the CT task, the DICE scores were 0.977 and 0.985 for MIR and software, respectively. These results validated the feasibility of using memristor-reconstructed images for medical diagnosis. Furthermore, our MIR also exhibited more than 153× and 79× improvements in energy efficiency and normalized image reconstruction speed, respectively, compared to graphics processing unit (GPU). This work demonstrates MIR as a promising platform for high-fidelity image reconstruction for future medical diagnosis, and also largely extends the application of memristor-based CIM beyond artificial neural networks.