A novel Local feature descriptor using the Mercator projection for 3D object recognition

Point cloud processing is a rapidly growing research area of computer vision. Introducing of cheap range sensors has made a great interest in the point cloud processing and 3D object recognition. 3D object recognition methods can be divided into two categories: global and local feature-based methods. Global features describe the entire model shape whereas local features encode the neighborhood characteristics of the feature points. Local feature-based methods are more robust to occlusion and clutter which are often present in a real-world scene. They encode geometric information of the local surface around the keypoint into a feature descriptor. We introduce a new local feature descriptor based on the Mercator projection. The Mercator projection is one of the famous 3D to 2D projections that preserves true distance, direction, and relative longitude and latitude so, in the point cloud, it can preserve these properties between any two points. To validate the proposed method, it is compared with seven state-of-the-art descriptor methods. Experimental results show the superiority of the proposed method compared to existing methods in terms in terms of Root Mean Square Error (RMSE), Recall versus 1-Precision Curve (RPC) and registration correction, rotation and translation error that prove the proposed method has good descriptiveness power and it is robust to noise and varying mesh resolution. Introduction  In this paper, we propose a new local descriptor to provide robust and precise geometric features. The geometric features are extracted using the Mercator projection of the neighborhood sphere. Our contributions are as follows: (1) The proposed descriptor directly learns from the point clouds (2) using the proposed method, there is only one representation for each point so the problem of multiple representations of a point is addressed. Also, the Mercator projection has many properties that make it appropriate for data representations in a point cloud. (3) It can accurately describe the geometric properties around a point. (3) The Mercator projection is a conformal projection so it preserves true distances, directions, and relative longitudes and latitudes. (4) It keeps small element geometry, which means Mercator projection preserves the shapes of small regions.  The proposed method Given a query point p, a sphere of radius r is centered at p for determining the neighbor points. Then Mercator projection is used for mapping the sphere into a plane with considering the Local reference frame (LRF) as previously suggested by Tombaret al. (2010b). The Mercator projection is a cylindrical projection that was proposed by G. Mercator in 1569. In this projection, the surface of a sphere is mapped into a plane. It preserves true distances, directions, and relative longitudes and latitudes. The Mercator projection for each point is identified using two following equations: x=λ y=ln(tan(Φ)+π/4) For extracting images as the input of the Siamese network, we need ranges for achieved  x and  y. The variable x is in the interval [−π,  π] but range of y is different for the Mercator projection of each keypoint. As a result, the minimum and maximum of the variable y for all neighbor points are considered as the range of y, then a histogram 30 ×30 is measured. The Mercator projections of all neighbors are defined and the number of points  in each bin counted. Then we normalize the histogram by dividing each bin by the total number of neighbor points, it causes more robustness to noise and mesh resolution.   Results and discussion The performance of the proposed method is evaluated on the Bologna (Tombari  et  al.,  2010c) and John Burkardt  in terms of RMSE, RPC and registration correction rate, rotation and translation errors. The proposed outperforms other methods in term of RPC also the results show that the method is robust to noise, rigid transformation and varying mesh resolution.