This paper describes a novel method for fast colonic polyp detection in colonoscopy images. Firstly, polyp detection is formulated as a similarity-based anomaly detection method, which formally involves non-dominated sorting based on multiple objectives. The chosen objectives rely on the main physical and visible differences, observed in colonoscopy images, between regions containing colonic polyps and the surrounding normal mucosa. These differences are defined primarily according to the contrast in shape, texture, and color. Secondly, as non-dominated sorting is of combinatorial nature and is costly to compute, it is replaced by a fast algorithm that approximates the sorting in the continuum limit. The fast algorithm involves numerical solutions to a particular Hamilton–Jacobi equation. The proposed similarity-based anomaly detection is thus reformulated into a fast polyp detection method. Several experiments were conducted with a proprietary medical data set, containing 1640 instances of 41 different polyps. The results show that the proposed Hamilton–Jacobi approach to non-dominated sorting speeds up the non-dominated sorting procedure, by more than 500%, and, when compared with other existing methods, it is also faster without lost of accuracy. Moreover, the tests conducted for streaming data, reveal an outstanding performance, in terms of sensitivity and specificity, as well as, a fast auto-adaptability, which demonstrate the power of the proposed approach towards a real-time and automatic detection, undoubtedly beneficial for clinical practice.