Random Point Sources

This book provides a mathematical framework for sparse sensing and coherent recovery when the hidden object is a random marked point measure rather than a deterministic sparse vector. Point sources, scatterers, arrivals, paths, targets, and events are treated as random empirical fields observed through linear and coherent sensing operators. The book brings together point-process theory, random measures, inverse problems, signal processing, coherent spectra, matched filtering, sparse reconstruction, covariance geometry, Fisher information, and uncertainty quantification. Its central premise is that a sensor does not observe a point pattern directly: It observes a projection, blur, coherent superposition, sampled field, or noisy transformation of an underlying source measure. Recovery must therefore account not only for estimated atom locations and marks, but also for visibility, ambiguity, covariance, precision, phase coherence, and non-identifiable directions. The book develops this viewpoint from first principles. It begins with point processes and marked random measures and then develops linear observation operators, response atoms, Green operators, sampling maps, coherent fields, complex marks, Bartlett and coherent spectra, information geometry, matched-filter evidence, sparse recovery, and recovery reporting. Worked examples, exercises, and selected solutions are included to support advanced self-study and graduate-level use. The target audience includes researchers and doctoral students in signal processing, statistical sensing, inverse problems, communications, radar, sonar, spatial statistics, applied probability, and point-process theory.