Bayesian blocks in two or more dimensions: Image segmentation and cluster analysis

I describe an extension, to higher dimensions, of the Bayesian Blocks algorithm for estimating signals in noisy time series data [17, 18]. We seek the partition of the data space with the maximum posterior for a model consisting of a homogeneous Poisson process in each partition element. Model M-n attributing the data within region it of the data space to a Poisson process with a fixed event(1) rate lambda(n), has a global posterior depending on only N, the number of data points in the region, and V, its volume: P(Mn) = Phi(N,V) = Gamma(N + 1)Gamma(V - N + 1)/Gamma(V + 2) = N!(V - N)/(V + 1)!.