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Recent studies have shown that diffusion models produce superior synthetic images when compared to Generative Adversarial Networks (GANs). However, their outputs are often non-deterministic and lack high fidelity to the ground truth due to the inherent randomness. In this paper, we propose a novel High-fidelity Brownian bridge model (HiFi-BBrg) for deterministic medical image translations. Our model comprises two distinct yet mutually beneficial mappings: a generation mapping and a reconstruction mapping. The Brownian bridge training process is guided by the fidelity loss and adversarial training in the reconstruction mapping. This ensures that translated images can be accurately reversed to their original forms, thereby achieving consistent translations with high fidelity to the ground truth. Our extensive experiments on multiple datasets show HiFi-BBrg outperforms state-of-the-art methods in multi-modal image translation and multi-image super-resolution.