Poissoniline

Poissoniline is a stochastic model used in plane geometry to describe a random collection of straight lines. It is formalized as a Poisson point process on the space of lines, typically parameterized by direction and offset. A common parameterization uses the direction angle theta in [0, pi) and the signed distance s from the origin, so a line is represented by (theta, s). The process is specified by an intensity measure mu(dtheta, ds) = lambda(theta) ds dtheta, where lambda(theta) is a nonnegative density on [0, pi). When lambda is constant, the model is stationary and isotropic; varying lambda introduces directional bias or anisotropy.

Construction and interpretation: In a given window, the lines from a Poissoniline process intersecting the window form a finite random set. The expected number of lines is the integral of the intensity over the region of the line space corresponding to lines that cross the window. Because the underlying mechanism is Poisson, the numbers in disjoint regions are independent, and the total counts follow a Poisson distribution with a parameter determined by mu.

Properties: The pattern is locally finite and can generate a planar line tessellation. If lambda is uniform, the resulting arrangement is directionally uniform, while a nonuniform lambda yields preferential directions. Statistical descriptors include the distribution of intersection points, polygonal cells formed by line segments, and moments of cell sizes.

Applications: Poissoniline models appear in stochastic geometry to study random line patterns in materials science (crack or grain structures), geology, and urban planning (road networks). They also support synthesis and analysis of linear features in image analysis and computer vision.

Relation: It is a variant or generalization of the Poisson line process, allowing directional inhomogeneity through lambda(theta). See also Poisson line process, stochastic geometry, and point processes.