Maximum Cramer-Rao Bound. Applications To: The Estimation Of Prior Probabilities, Image Restoration, And The Generation Of Quantum Mechanics

B. Roy Frieden

doi:10.1117/12.942100

18 January 1988 Maximum Cramer-Rao Bound. Applications To: The Estimation Of Prior Probabilities, Image Restoration, And The Generation Of Quantum Mechanics

B. Roy Frieden

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Proceedings Volume 0829, Applications of Digital Image Processing X; (1988) https://doi.org/10.1117/12.942100
Event: 31st Annual Technical Symposium on Optical and Optoelectronic Applied Sciences and Engineering, 1987, San Diego, CA, United States

Abstract

In this paper, we propose using maximum Cramer-Rao bound (MCRB) as a criterion for the estimation of a prior probability law p(x). This criterion has a useful interpretation: It is known that any estimate of x must have a mean-square error greater than or equal to the CRB. Hence, if one demands that p(x) have maximum CRB, he is demanding that p(x) be so non-informative, i.e., x be so random, that even the best estimate of x (attaining the CRB) will have maximal mean-square error. Hence, p(x) is made to describe a minimally knowable x. Because of the current interest in maximum entropy (ME) as an alternative approach, we compare results throughout this paper with those by ME. For example, if only the mean-value of x is known as prior information, the ME answer for p(x) is an exponential law, a smooth function. By contrast, MCRB gives the square of an Airy function, in fact an even smoother function. Comparisons are also made between ME and MCRB restorations, whereby p(x) is modeled to be the unknown object radiance distribution. Finally, MCRB may be applied to estimating the probability law describing particle position in a known potential energy field. With a constraint on average kinetic energy, the result is the Schroedinger wave equation.

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B. Roy Frieden "Maximum Cramer-Rao Bound. Applications To: The Estimation Of Prior Probabilities, Image Restoration, And The Generation Of Quantum Mechanics", Proc. SPIE 0829, Applications of Digital Image Processing X, (18 January 1988); https://doi.org/10.1117/12.942100

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KEYWORDS

Digital image processing

Image restoration

Particles

Differential equations

Error analysis

Image information entropy

Quantum mechanics

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