Cramer-Rao bounds in functional form: theory and application to passive optical ranging

Area of Science:

• Statistical modeling
• Optical systems engineering
• Signal processing

Background:

• Passive optical ranging and imaging systems generate large datasets from CCD sensors.
• Statistical analysis of discrete pixel readouts presents computational challenges.
• Existing models may not fully capture the complexity of incoherent optical systems.

Purpose of the Study:

• To develop a functional statistical approach for modeling generalized incoherent passive optical systems.
• To enable analytical treatment of probability density functions and Fisher information.
• To derive bounds on estimator-error variances for system parameter estimation.

Main Methods:

• Approximating discrete pixel data as a continuous random function.
• Utilizing functional integrals for statistical averages in the Gaussian approximation.
• Applying the steepest-descent method to derive approximate probability density functions.
• Deriving Cramer-Rao bounds for scalar and functional parameters.

Main Results:

• A functional form for the joint probability density function (PDF) was established.
• Statistical averages and Fisher information were treated as functional integrals.
• Asymptotically Gaussian PDF was derived for the sum of Gaussian and Poisson variables.
• Cramer-Rao bounds were obtained for system parameters.

Conclusions:

• The functional approach provides a powerful analytical tool for passive optical system modeling.
• The method enhances the understanding and estimation of parameters in optical ranging and imaging.
• The developed framework is applicable to systems like passive optical rangefinders with advanced coding techniques.