Question: We will study measuring the orientation of a plane in an orthographic view, given the texture consists of points laid down by a homogenous Poisson point process. Recall that one way to generate points according to such a process is to sample the x and y coordinate of the point uniformly and at random. We assume that the points from our process lie within a unit square.
• Show that the probability that a point will land in a particular set is proportional to the area of that set.
• Assume we partition the area into disjoint sets. Show that the number of points in each set has a multinomial probability distribution. We will now use these observations to recover the orientation of the plane. We partition the image texture into a collection of disjoint sets.
• Show that the area of each set, back projected onto the textured plane, is a function of the orientation of the plane.
• Use this function to suggest a method for obtaining the plane's orientation.