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Medical endoscopy is used in a wide variety of diagnostic and surgical procedures. These procedures are renowned for the difficulty of orienting the camera and instruments inside the human body cavities. The small size of the lens causes radial distortion of the image, which hinders the navigation process and leads to errors in depth perception and object morphology. This article presents a complete software-based system to calibrate and correct the radial distortion in clinical endoscopy in real time. Our system can be used with any type of medical endoscopic technology, including oblique-viewing endoscopes and HD image acquisition. The initial camera calibration is performed in an unsupervised manner from a single checkerboard pattern image. For oblique-viewing endoscopes the changes in calibration during operation are handled by a new adaptive camera projection model and an algorithm that infer the rotation of the probe lens using only image information. The workload is distributed across the CPU and GPU through an optimized CPU+GPU hybrid solution. This enables real-time performance, even for HD video inputs. The system is evaluated for different technical aspects, including accuracy of modeling and calibration, overall robustness, and runtime profile. The contributions are highly relevant for applications in computer-aided surgery and image-guided intervention such as improved visualization by image warping, 3-D modeling, and visual SLAM.