An analytical method to recover the two-dimensional probability density function (PDF) of a random light field from the PDF of photon counts is presented. Some illustrations of the inversion procedure are given with application to the problem of light intensity interferometry.
One ofthe major limitations of intensity interferometry both in space and time domain is that only the linearly filtered signals could be registered in photon-counting mode. However, this restriction can be removed by a prior processing ofthe registered signal in a simple way before image formation from the field correlation function. This is a typical example of ill-posed inverse problems for non-symmetric transforms. We investigate the implementation of eigenfunction method to the problem of correlation function restoration from the photocount data. The effectiveness of the restoration procedure for the typical examples of astronomic images is investigated by computer simulation, both for symmetric and non-symmetric objects with phase restoration by the application of incoherent reference source.
A numerical method of estimating the probability density function of light intensity fluctuations from photon counting data is described. The solution is found as an orthogonal series of the functions which are the eigen functions of left and right iterated kernels of Mandel transformation. Inverse operator in such representation has diagonal form and provides powerful regularization of inverse problem solution by simple restriction of eigen values taken into account. The efficiency of proposed method is illustrated with results of restoration of light intensity probability functions for some typical models of speckle formation.
Restoration of light intensity fluctuations from photon-counting data provides typical ill-posed inverse problem for non-Hermitian transforming operator. Such problems cannot be solved with the help of eigenfunction representation which diagonalizes operator and simplifies the regularization of quasi-solution. We describe an analytical method to solve this inverse problem which is based on the Poisson transform operator representation in mixed basis. The last one is determined by eigenfunctions of left and right iterated operators and diagonalize Poisson transform. On the base of proposed method we have performed the set of numerical experiments with typical intensity distributions. It has been shown that restoration procedure has necessary stability and may be used when the level of statistical errors is relatively high. In conclusion we describe the results of inversion procedure application to the processing ofphoton counts statistics experimental data.
411 analytical method to recover the twodimensional probability density function (PDF) of a randoin light field lroni the PDF of photon counts is presented. Some illustration of tlie inversion procedure are presented with application to tlie problem of light intensity interferometry.Poissoii transform in the optical speckleinterferometry and some other optical processing problems arises when iiiteiisity levels of analyzed field are so low that only few pliotoiis la11 on each pixel of interferogram. In this case the joint distribution of photon counts in two space-time points is given by the equation which is known as Mandel's equation aiid reprcseiits tcTio-dimensional Poisson transform of probability dciisity function (PDF) of iiitensity fluctuations P(I1 ,I2) in observed speckle-structure. As a resultant transform P(n,iii) is determined by two factors (the own field fluctuations which generate the speckle-structure, aiid tlie quantum fluctuations of photodetection), tlie deteriiiination of true radiation iiileiisity fluctualions becomes tlie problem o i real interest. It means the iiiversioni of transform (I), that is, tlie determination or two-dimensional PDF of intensity fluctuations from the photo-count experiment data.Single-dimensional version of similar problem was considered by niaiiy authors [1,2]. Tlie key feature of all these investigations is tlie solution instability which is caused by tlie fact that the problem is typically ill-posed. So, the resultant solutions are unstable and practically do not work even under the extraordinary small level of statistical error. Recently [3] the stable solution based on Padt-approxiiiiants method was proposed, aiid its generalizalioii 011 two-dinieiisioiial distributions was fulfilled [.I.]. This method gives tlie excellent results when tlie average iiuiiiber of photon counts per pixel is extremely small (about one or Icss), aiid Pad& approxiiiiants are based on tlie polynomials of tlie low order. In the iiiterniediate case of average photon counts, tlie nuniber of Pade-approxiinatioii coelficients becomes greater, and tlie problem of tlie solution correctness again takes on its iniportance.In this report we propose another approach to tlie problem of joint two-point iiitensity PDF determination which is based on the orthogonal expansion of tlie solutioin in a series of Poisson transform cigeilfuiictioiis and provides the generalization of correspoiiding one-poinl problem solution [SI. Tlie advantages of this solution are it's high stability, the simplicity of optimal regularization level deterniiiiation, as well as the ability of further extrapolalioli on tlie problems of higher diinensioiialities.Tlie niain idea of supposed method consists in lollowing. Let us define eigeilrunctioiis of oncdiinensioiial Poisson transform by the equation where { p, } aiid { 1; are orthogonal basises in PDF's PQj and P(11) spaces. respectively. It can bc sliowii [ 5 ] , that basises with described properties really exist and that they can be defined by the solutioiis or integral ...
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