Randomization of transfer functions in control systems via computer vision with pixel noises of order n

Complex-valued fractional Brownian motion of order n can be defined either as rotating Gaussian white noise on the net defined by the nth roots of the unity, or as the limit of a random walk in the complex plane. After a brief background, one shows that this stochastic process is quite relevant in computer vision, as the result of the definition of image in terms of pixels. It follows that in control systems involving numerical vision, one will have to consider disturbances in the form of fractional white noise of order n with independent increments, and the present paper deals with this problem. By using the central limit theorem, it is possible to consider the Laplace transform of such a process as a fractional Gaussian variable of order n of which the nth moment is the integral of the nth moment of the white noise. One can then use this result to carry on a statistical analysis of linear feedback systems subject to disturbing fractional white noises (or pixel noises) in control via computer vision. Stochastic optimal control of order n is outlined.