Combining Conformal Deformation and Cook–Torrance Shading for 3-D Reconstruction in Laparoscopy

We propose a new monocular 3-D reconstruction method adapted for reconstructing organs in the abdominal cavity. It combines both motion and shading cues. The former uses a conformal deformation prior and the latter the Cook-Torrance reflectance model. Our method runs in two phases: first, a 3-D geometric and photometric template of the organ at rest is reconstructed in vivo. The geometric shape is reconstructed using rigid shape-from-motion while the surgeon is exploring-but not deforming-structures in the abdominal cavity. This geometric template is then used to retrieve the photometric properties. A nonparametric model of the light´s direction of the laparoscope and the Cook-Torrance reflectance model of the organ´s tissue are estimated. Second, the surgeon manipulates and deforms the environment. Here, the 3-D template is conformally deformed to globally match a set of few correspondences between the 2-D image data provided by the monocular laparoscope and the 3-D template. Then, the coarse 3-D shape is refined using shading cues to obtain a final 3-D deformed shape. This second phase only relies on a single image. Therefore, it copes with both sequential processing and self-recovery from tracking failure. The proposed approach has been validated using 1) ex vivo and in vivo data with ground-truth, and 2) in vivo laparoscopic videos of a patient´s uterus. Our experimental results illustrate the ability of our method to reconstruct natural 3-D deformations typical in real surgical procedures.