Nanoobjects microscopy
Signal filtering with additive and multiplicative noise by orthogonal Legendre polynomials
Fitio V.M., Petrovska H.A.
Lviv Polytechnic National University. S. Bandera str., 12, Lviv-79013, Ukraine
E-mail: volodymyr.m.fito@lpnu.ua,  halyna.a.petrovska@lpnu.ua

Nowadays digital holographic interferometry is increasingly used to study macro-objects and micro-objects and phenomena. There is determination of surface roughness, study of micro-displacements, determination of part quality in MEMS, study of periodic microstructures, microscopy of living objects, and etc. In digital holographic interferometry, the first measuring stage is optical and the second is digital. The resolution of digital holographic microscopy can be limited by the parameters of the digital camera as well as the optical channel noise.
In this paper, the possibility of using orthogonal Legendre polynomials to filter noisy signals is investigated. It is possible to filter the signals in the interval [0, a] by scaling and shifting. Moreover, noise can be both the additive and the multiplicative. It is shown that Legendre polynomials can be successfully used for filtering interferograms and phase maps in digital holographic interferometry. Filtration is based on choosing the optimal number of polynomials at the signal approximation. Direct filtering of digital holograms (high spatial frequency of digital hologram signal) and phase maps (phase jump to π for certain coordinates) is impractical as several hundred polynomials must be used for filtering. It increases the time of numerical calculations. Therefore, in digital holographic interferometry it is necessary to directly filter the amplitudes of the fields calculated from the digital hologram. Further, interferograms and phase maps can be calculated from the filtered amplitudes of the object different states. 
It should be noted that interferograms also can be directly filtered by Legendre polynomials. If the minimum distance between adjacent local minima (maxima) for the real or imaginary part of the signal is equal to Δl, then for a satisfactory approximation of such a signal by Legendre polynomials requires 6/∆l polynomials. Multiplicative noise filtering requires more time for computer calculations than additive one. Legendre polynomial filtering efficiency is improved if the noise signal has harmonic components with a frequency that is higher than the frequency of the useful signal.