Document Type


Publication Date



Bayesian field theory, Algorithms, Machine learning, Computer vision


In this work, Cox processes and Convolutional Neural Net classifiers (CNNs) are used to estimate the number of instances of an object in an image. Poisson processes are well suited to model events that occur randomly in space, such as the location of objects in an image or the enumeration of objects in a scene. Doubly Stochastic Poisson (or Cox) processes over increased flexibility, but their computational complexity and storage requirements do not easily scale with image size, typically requiring O(n3) computation time and O(n2) storage, where n is the number of pixels in an image. To mitigate this problem we employ the Kronecker algebra, which takes advantage of direct product structures. As the likelihood is non-Gaussian, the Laplace approximation is used for inference, employing the conjugate gradient and Newton's method. Our approach has then close to linear performance, requiring only O(n3/2) computation time and O(n) memory. The proposed algorithm consists of selecting a subset of bounding boxes in the image domain, and querying them for the presence of the object of interest by running a pre-trained CNN classifier. The resulting observations are then aggregated and a posterior distribution over the intensity of a Cox process is computed. This intensity function is summed-up, providing an estimator of the number of instances of the object over the entire image. Results are presented on simulated data and on images from the publicly available MS COCO dataset. We compare our counting results with the state-of-the-art detection method, Faster RCNN, and demonstrate superior performance.


This is a pre-print of an article under revision at Journal of Mathematical Imaging and Vision (JMIV).

Persistent Identifier

Included in

Mathematics Commons