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The dynamic speckle is a non-destructive optical technique that has been used as a tool for the characterization of the biological activity and several studies are conducted to obtain for more information about the correspondence of the observed phenomena and their expressions in the interference images. Analysis in the frequency domain has been considered as powerful alternative, and although there are works using Fourier transform in the frequency analysis of the biospeckle signals, the majority presents the wavelet transform as tool for spectral analysis. In turn, there are still doubts if the Fourier transform is not enough for the analysis of the biospeckle, which would enable the reduction of processing time since an operation is computationally simpler. In this context, the present study aims to compare the constituents’ parts of the speckle signal according to Fourier and wavelet transforms for numerical analysis. The comparative analysis based on the absolute values of the differences technique (AVD) was carried out for performance evaluation of the Fourier and wavelet transforms, in which the speckle signals were decomposed spectrally and subsequently reconstructed with the elimination of specific frequency bands. Results showed that the wavelet transform allowed more information about signals constituents of the dynamic speckle, emphasizing its use instead of the Fourier transform, which in turn was restricted the situations in which the only interest is to know the spectral content of the data.

When a coherent light, such as laser, illuminates a rough surface, compared the wavelength of laser, it occurs a phenomenon of optical interference with the formation of light and dark regions, called speckle [

After applying the dynamic surface, there is a continuous formation of new and different speckles, and these random and dynamic interference patterns is called dynamic speckle or biospeckle, if the area concerned is biological. This technique allows extracting information about the structures movement of the illuminated material, making it an interesting tool in several knowledge areas [

The biospeckle has been used as a technique to measure detailed extensions of pine roots [

The biological activity expressed in the context of speckle does not present a clear definition of what phenomenon is creating, however can be understood as structural and molecular motions occurring in the material analysis [

In this context, the use of image processing techniques and signal analysis tools can be used in the biospeckle signal to understand better this optical phenomenon.

The interference patterns analysis can use graphical methods, which generate maps indicating the spatial variability of the biological activity, or a numerical interpretation of the temporal variation of patterns formed. An alternative of the graphical and numerical classifications is a signal analysis in the time domain or in the frequency domain [

The analysis of biospeckle signals in the frequency domain has been an alternative for many applications, allowing the filter and images contrast, beyond search of frequency markers of phenomena that contribute to the formation of the interference patterns in time, as described by [

Several studies have been conducted using the spectral analysis in the biospeckle signal, such as [

Although there are many papers applying spectral analysis in the biospeckle signal, the most journals use wavelet transform and there is no works evaluating if Fourier transform, which is simpler than wavelet transform. It’s enough in the frequency analysis of the dynamic speckle. In this context, the present study aims to compare the Fourier and wavelet transform in the spectral analysis of biospeckle signal.

The biospeckle is a nondestructive optical technique based on the analysis of the variations of the laser light scattered from material, and the biological activity presented reflects the state of the investigated object [

Follow a set of pixels of the images speckles in the time is a method of monitoring their time variations and consequently the biological activity of the studied object, and, in this context, [

The THSP is a two dimensional image that record a certain line or column of pixels in successive moments and arrange them vertically side by side. The x axis show information about the time evolution of the selected pixels and the y axis is the spatial distribution of the interference patterns [

The co-occurrence matrix was presented by [

which:

is the co-occurrence matrix, correspond the number of occurrences of an intensity value i, followed by an intensity value j to move through rows or columns of the time history.

Phenomenon that show low biological activities, their time variations of the speckle patterns are slow and present a THSP horizontally in the elongated shape and the co-occurrence matrix is characterized by small changes of the pixels intensity to i and j, as illustrated in the

One of the methods for analyzing of the speckle patterns is the technique of the absolute values of the differences (AVD), proposed by [

The AVD method is a statistics moment of first order which it is applied on the co-occurrence matrix and generates a number [

which:

AVD is a dimensionless value, i and j are coordinates of the row and column respectively, and M_{ij} is called of modified co-occurrence matrix and that is presented in Equation (3).

According [

Information of the biospeckle data in the frequency domain has been an alternative to the interpretation of the interference patterns [

Fourier transform can be understood as the mathematical technique that transforms a signal from the time domain to the frequency domain, and it is formed by a set of orthogonal functions, of period 2π [

which:

= amplitude of each component ω of the signal.

There is also the inverse Fourier transform, which is used to transform the signal from frequency domain to time domain with the reconstruction of the original function. Equation (5) presents the mathematical expression of the inverse Fourier transform.

The Fourier transform indicates the spectral information of the signal without providing the instant which these components happen, and in situations that to know when the frequencies occur are interesting precludes the use of Fourier transforms, unless if the series is stationary [

The wavelets are simply waves of duration adjusted with energy concentrated in variables intervals [

The continuous wavelet transform is defined as the convolution of with a scaled and translated version of [

which:

is the studied signal a scale parameter b translation value

is the mother function of wavelets

is the spectrum wavelets.

The scale is related to the frequency, in which high scales correspond to low frequencies and low scales correspond to high frequencies, whereas the translation is the displacement of the mother function about the studied signal [

The return of the signal from frequency domain to time domain, inverse wavelets transform, allows observe the behavior of the signal in specifics frequencies bands and also the reconstruction of the original function. According [

which:

is a factor that convert the wavelets transform in energy density,

;;; are specific constants of the base function used.

One of the major difficulties in wavelet analysis is the identification of the scales set used in the wavelet transform. Orthogonal wavelet, there is a limit and a discrete set of scales, as given by [

In this context, [_{j} is the lowest and J is the highest scale.

The s_{0} should be chosen so that the Fourier period is, and to the Morlet wavelet the largest value that can adjust the scale is of 0.5. For other wavelet functions can be used a larger value.

The sampling theorem describes the relationship between sampling frequency of a signal and the frequency maximum of the reconstructed signal. Below is transcript the sampling theorem as presented by [

“Theorem 1: If a function contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/2 seconds W apart”.

According to the theorem, the number of samples per unit time of a signal is called rate or frequency sampling (W), and half the sampling frequency corresponds to the frequency maximum of the signal which can be reproduced in full without aliasing error.

The sampling theorem is used in this work to define the highest frequently during the decomposition of signals.

It was conducted a comparison between Fourier and wavelet transforms using the time history of speckle patterns (THSP) relative to a paint drying process and presented by [

The database was formed by 8 THSP’s collected each 20 minutes during the paint drying using the back-scattering experimental setup. Each time history was made by a set of 128 images, resolution of 512 by 640 pixels, whose time acquisition between images was of 0.08 seconds (sampling frequency of 12.5 Hz).

The lines of the THSPs were concatenated creating a new signal that was decomposed into frequency spectra using Fourier and wavelet transforms with application posterior of the inverse transform. Some frequency bands were eliminated before the reconstruction of the signal in order to analyze the results of the speckle signal using a numerical method to measure the speckle activity. The selective filtering was conducted as well in order to create some frequency markers linked to the physical phenomena under monitoring.

According the sampling theorem the highest frequency that can be seen in the reconstruction process is 6.25 Hz, and using Equations (8) and (9) were calculated the number of frequency bands used in the transform. In addition, in the continuous wavelet transform was used mother function of Morlet, a damped complex exponential with a set of oscillation parameter that preserves an approximate relationship between the scale of the wavelet analysis and the frequency in a Fourier analysis, as described by [

The signal resulting of the inverse transform was converted to THSP format again and numerically analyzed using the technique of the absolute values of the differences (AVD) [