Triangulation in Computer Vision

In cricket and tennis, we often see the ball’s position shown with the help of some technology such as hawk-eye. This technology helps in the Decision Review System, to make decisions. If we consider cricket, with the help of hawk-eye, we can track the ball’s trajectory and where it would hit the stumps. For this to be shown, a 3D model is created. The 3D model could be generated using photos, and videos or it could build manually. The 3D model represents 3D objects as a collection of three-dimensional points connected with geometric objects. Therefore, the initial step in modelling becomes collecting the cloud of 3D points. 3D point clouds are composed of hundreds of x, y, and z coordinates in a 3D space. We obviously, need a method to connect all such points to create a 3D model. Triangulation is one such method. The process of 3D modelling involves several steps like feature detection and matching, structure and motion recovery, stereo mapping and modelling.

Triangulation is nothing but the process of identifying a point in 3D space and its projection on given images. 3D modelling from a video is a process where the data is obtained from a video. The data is then processed into a point cloud data and this cloud point would be divided into regions before projection. A sequence of images of an object is a series of projections of the surface of that object into 2D space. Each image is dependent on the camera at that viewpoint, which in turn is dependent on the relevant camera projection matrix. The camera matrix is a 3 x 4 matrix describing the mapping of a pinhole camera from 3D points in the world to 2D points in an image. So, if the camera moves so, do the parameters of the camera and thus 2D projection, that is the image, therefore, giving a different view. There are many algorithms which have been developed for triangulating a surface. These can be categorized as structured and unstructured. A structured triangulation is an uniform approximation with a noticeable pattern whereas unstructured triangulation is exactly the opposite.

While generating a 3D model, high-speed graphics rendering typically relies on dividing curved surfaces into triangles for efficient handling. Therefore, a classic problem in computer graphics is decomposing a polygon into triangles. Again, there are various techniques to handle this issue, wherein the n-sided polygon is drilled down until only three vertices remain. Another problem is skinny triangles, for this, Delaunay triangulation is used which maximizes the minimum angle of all the angles of the triangles in the triangulation to avoid skinny triangulation.

The moment all the points are visible in the image being used as a basis for the triangulation, and the object contains no holes, then the surface will be topologically planar. If the triangulation represents the true surface in 3D, then it follows that the same connectivity of the 2D points in the other images will represent the same surface. Triangulation gives the minimum squared difference over all pixels between the actual and predicted image for all images and hence gives the best approximation of the surface given through the images, camera matrices and the points. Therefore, the 3D points can be projected into one of the images using the appropriate camera projection matrix and these triangulated 2D points.

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