The copula: a tool for simulating speckle dynamics

Abstract

Use of a copula for generating a sequence of correlated speckle patterns is introduced. The chief characteristic of this algorithm is that it generates a continuous speckle sequence with a specified evolution of the correlation and does so with just two arrays of random numbers. Thus, physically realistic temporally varying speckle patterns with proper first- and second-order statistics are easily realized. We illustrate use of the algorithm for generating sequences with prescribed Gaussian, exponential, and equal-interval correlations and demonstrate how correlation times can be specified independently. This approach to generating sequences of random realizations with prescribed correlations should prove useful in modeling such phenomena as dynamic light scatter, flow-dependent laser speckle contrast, and propagation of spatial coherence.

Statistical studies on optical vortices in dynamic speckle fields

“…Sequences of 50 speckle patterns with user-defined correlation behaviors were simulated using the concept of a copula [28] in the MATLAB environment. Decorrelation rates are defined in terms of the number of frames it takes for the overall speckle pattern autocorrelation coefficient, Γ I , to reduce to a value of 1/e.…”

Simulation of speckle patterns with pre-defined correlation distributions

“…This equation has been used to generate speckle patterns, for example see the work of Kirkpatrick et al [1,2]. Since it is a standard Fourier transform, no spatial correlation mapping is preserved, as expected for a non-imaging free space field propagation.…”

“…Besides being experimentally investigated, simulation can be a powerful complement for understanding the properties of speckles, for evaluating data processing methods and for investigating the possibility of further applications of laser speckle phenomena. In many experiments, such as imaging circulation dynamics, the correlation properties of the observed area are not uniformly distributed, therefore the simulated speckle patterns are supposed to have a desired correlation distribution across the area of interest, which demands the simulation to have a few-to-one mapping [1] between the input matrix and the correlation distribution between simulated speckle patterns.…”

“…Since many applications of laser speckle are based on their correlation properties, simulation of speckle patterns with a predefined temporal correlation is desirable. Duncan and Kirkpatrick introduced the copula method to generate a sequence of speckle patterns, between which the correlation coefficient could be pre-defined [1,3]. With this method one can investigate the speckle properties of a dynamic object by generating a series of correlated speckle patterns [4,5].…”

Optical vortex behavior in dynamic speckle fields

“…6 Using this approach, we were able to generate fully developed, dynamic speckle patterns with very well-controlled first and second order statistics, and also with prescribed decorrelation behavior. Following the generation of the speckle sequences, the pseudo-phaseφ 7 of the speckle patterns was generated using a two-dimensional (2-D) Hilbert transformation of the speckle patterns.…”