We propose an image-adaptive object detection method for adverse weather conditions such as fog and low-light. Our framework employs differentiable preprocessing filters to perform image enhancement suitable for later-stage object detections. Our framework introduces two differentiable filters: a B\'ezier curve-based pixel-wise (BPW) filter and a kernel-based local (KBL) filter. These filters unify the functions of classical image processing filters and improve performance of object detection. We also propose a domain-agnostic data augmentation strategy using the BPW filter. Our method does not require data-specific customization of the filter combinations, parameter ranges, and data augmentation. We evaluate our proposed approach, called Enhanced Robustness by Unified Image Processing (ERUP)-YOLO, by applying it to the YOLOv3 detector. Experiments on adverse weather datasets demonstrate that our proposed filters match or exceed the expressiveness of conventional methods and our ERUP-YOLO achieved superior performance in a wide range of adverse weather conditions, including fog and low-light conditions.