A Theory of Fermat Paths for 3D Imaging Sonar Reconstruction

Abstract

In this work, we present a novel method for reconstructing particular 3D surface points using an imaging sonar sensor. We derive the two-dimensional Fermat flow equation, which may be applied to the planes defined by each discrete azimuth angle in the sonar image. We show that the Fermat flow equation applies to boundary points and surface points which correspond to specular reflections within the 2D plane defined by their azimuth angle measurement. The Fermat flow equation can be used to resolve the 2D location of these surface points within the plane, and therefore also their full 3D location. This is achieved by translating the sensor to estimate the spatial gradient of the range measurement. This method does not rely on the precise image intensity values or the reflectivity of the imaged surface to solve for the surface point locations. We demonstrate the effectiveness of our proposed method by reconstructing 3D object points on both simulated and real-world datasets.