These authors contributed equally to this work.

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We have developed a computing method to determine the geometrical parameters of fibers and the textile periodical structure. This method combines two two-dimensional discrete fast Fourier transforms to calculate a diffraction pattern from a diffraction pattern image of material under consideration. The result is the same as that of observation of a diffraction pattern which is achieved by illuminating the diffraction pattern image of material by a beam of coherent monochromatic light. After the first transform we obtain the Fraunhofer diffraction picture with clearly visible elements of the periodical structure of material, but distances in this picture are reciprocally proportional to distances in the periodical structure of the source object so additional calculations are required. After the second transform we have a clear periodical structure of diffraction maximums where distances between them are equal to distances between repeating elements in the source material (fibers, knots, yarns,

The structure of textile materials consists of a network of repeating fibers. For instance, in plain weaving the structure repeats along warp and weft [

The diffraction method was claimed to analyze diffraction patterns produced after projecting the textile materials with a laser [

The idea arose to use the same diffraction method for the diffraction pattern image of the source material, which would form a new diffraction pattern periodical structure where distances between diffraction maximums would be equal to distances between elements in the source periodical structure.

The scheme of our experimental system is shown in

In the next stage of the experiment, we illuminate the image of the diffraction pattern

When we illuminate the piece of textile material with a beam of coherent light we have the Fraunhofer diffraction which can be described by the equation given by [

The intensity in point (^{*} is the complex conjugate of

If the object is illuminated by a monochromatic wave with non-equal complex amplitude

(

For a discrete digital image we can write that ξ =

Then we can write Equation (3) as a sum of discrete values [

Expression in Equation (4) is similar to the expression for the two-dimension discrete Fourier transform and can be calculated by FFT [

According to the Abbe theory (for example, see [

For instance, we consider a simple linear sequence {_{−2}, _{−1}, _{0}, _{1}} of four elements. The expression of Fourier transform is

After the second Fourier transform the result ^{2}_{−k}_{k}

This result confirms the Abbe theory. But if we substitute complex _{j}_{j}_{j}U_{j}^{*} after the first Fourier transform we achieve another result ^{~}U_{−1} after the second Fourier transform

That confirms the difference between our method and the optical system using two lenses described by two sequenced Fourier transforms.

In _{ξ} and _{η} between units in the periodical structure of material _{x}_{1} and _{y}_{1} are average distances between diffraction maximums along the

If distances between maximums in diffraction pattern _{x}_{2} and _{y}_{2} (

When we substitute Equation (10) for Equation (9) we have:

This means that distances between diffraction maximums in the image after the second Fourier transform are equal to distances between units of periodical structure in the source image of the material. To calculate _{ξ} and _{η} we only have to multiply _{x}_{2} and _{y}_{2} by a magnification coefficient of the digital camera taking the source photograph.

We have conducted a series of numerical experiments with different types of textile materials where we computed the diffraction pattern digital picture of the source image of the material and then we calculated the diffraction pattern image of this picture.

The

(

If we apply a Fourier transform to the digital image (_{ξ} is the distance between columns of knitting and _{η} is the distance between rows of knitting.

If we apply a Fourier transform to the digital image (_{ξ} = _{x}_{2}, _{η} = _{y}_{2}.

(

There is no difficulty in measuring the distances between diffraction maximums in

We have developed a program that automatically makes the necessary measurements (see screenshot,

On the left side of the program window, the graph shows the distribution of digital image pixel intensities along the vertical axis

Screenshot of the program automatically detecting distances between units in periodical structures.

The program has been developed in C++ with Microsoft .NET Framwork supporting.

Two additional examples of using our method are in _{ξ} and _{η}. After applying our method to this digital image we achieve the diffraction pattern (_{x}_{2} and _{y}_{2} between diffraction maximums using our program (_{ξ} = _{x}_{2} = 60 pixels and _{η} = _{y}_{2} = 52 pixels.

The digital microphotograph of the tartan is in _{ξ} and _{η}. After applying our method to this digital image we achieve the diffraction pattern (_{x}_{2} and _{y}_{2} between diffraction maximums using our program (_{ξ} = _{x}_{2}, _{η} = _{y}_{2}.

(

(

We offer a new method of textile material structure analysis which is based on principles of the Fraunhofer diffraction. The images for analysis can be achieved in the physical diffraction apparatus as in

This method is easier to use than the method determining distances from a diffraction pattern image after the first Fourier transform [

This method is also easier than methods which determine edges of complex forms in digital images such as the Hough transform (for example, see [

The authors are grateful for the support of the Department of Fibrous Materials Mechanical Technology, St. Petersburg State University of Technology and Design, 18 Bolshaya Morskaya St., St. Petersburg, Russia.

The authors declare no conflict of interest.