Characteristics in the Fourier spectrum of images associated with discontinuities such as cracks and fissures in concrete structures.

Increasing An algorithm is developed to identify differences between concrete slabs with some type of Crack and Fissure discontinuity. The Fourier transform is used as a basis. Different types of fil-ters were evaluated within the image processing algorithm, in order to adapt favorably to the discontinuity detection process, allowing the image data to be read in the frequency domain, generating a figure of merit to compare the image. where there are no discontinuities and those that have some alteration in their structure.


Introduction
Template Concrete is the most frequently used building material to date, contributing to the development of infrastructure worldwide.However, this material presents damage due to being subjected to stresses upper its resistance capacity, generating anomalies known as "cracks" and "fissures" type discontinuities.These phenomena severely affect the appearance, life, and intrinsic properties of concrete.[1].Various causes generate cracks and fissures, although they can be grouped into two types of actions: mechanical and spontaneous [2].
In addition, these discontinuities can be generated during the concrete setting or hardening stages, generating a greater or lesser effect on the hardness and resistance of the material [3].Triggering a variation in the material and causing the cracks and fissures to have a level of greater severity for the integrity of the concrete; evident in the dimensions of the discontinuity, especially in the width [4].Therefore, it is interesting to identify early, both discontinuity and causes, to keep impermeability and limit the damage.[5] [6].
Currently, crack detection is mostly done in two ways: applying destructive and non-destructive testing.[7].Non-destructive testing excels, due to its ease of application and its reliability in results.The use of visual tools, application of laser equipment, ultrasound [8], ray, infrared and thermal signals [9], They are part of some non-destructive tests that are used more frequently at the laboratory and industrial level [10].
Additionally, the methodology for detecting crack-type discontinuities and fissures is currently being applied through image processing.[11].Although this technique is rarely used, due to difficulties generated by various reasons such as the random shape and irregular size of the cracks, irregular lighting conditions, shadows, imperfections, and flaking in the concrete of the acquired images, it is projected as an alternative.for the detection of these types of anomalies [12] [13].
Consequently, the methods based on image processing are summarized and divided into four categories: integrated algorithm, morphological approach, method based on percolation and practical 1299 (2024) 012005 IOP Publishing doi:10.1088/1757-899X/1299/1/012005 2 technique.[14] [15].Although the use of image processing is rare, there are advances that seek to develop an artificial vision system that serves as a tool capable of enhancing images, previously taken, and create a new one in which the discontinuities present in the image are more clearly evident.initial image [16] [17] As in turn, other systems aim to improve certain aspects of the images, to achieve highlighting specific details [18].Thus, this document presents the development of an algorithm for the detection of discontinuities such as cracks and fissures, through the image processing technique applying the Fourier transform as an alternative for identification based on non-destructive tests of anomalies in structures of concrete.

Materials and methods
This work is develops using the analytical method, where the information of a series of images is observed and examined using the Fourier transform, as a method of converting the image information to the frequency domain; using the MATLAB system as a programming tool, which allows obtaining data sets in matrices where the information is stored and in turn, using this information for the development of the algorithm, for the comparison of possible alterations at the end of the method [19].By enhancing the con-version of the information, a comparative analysis was carried out between the results of an image where discontinuities were evident and one where the plate was free of these.
The information from the sample images is stored and compared with an original basis to determine differences and observe the effect it can have on other variables.Results are grouped and represented in a numerical way that shows the quantitative approach.
The purpose of the algorithm is to determine characteristics in the frequency domain.These features can be associated with discontinuities.We seek to obtain information that describes the phenomenon and not why it happens is. Figure 1 shows the process used in the algorithm that allows obtaining information generated by the discontinuities in the images.
For the development of the algorithm, we rely on the Fourier transform, known the frequency spectrum of a function.An example of this is what the human ear does since it receives waves and allows decomposition in different frequencies (that can to heard).The human ear perceives different frequencies as time passes.However, the Fourier transform contains all the frequencies contained in all the times in which the signal existed; that is, in the Fourier transform a single frequency spectrum obtained for the entire function [20].The Fourier transform is an application that maps a function (f), with complex values and defined on the line, with another function (g) defined as follows (equation 1): MATLAB was needed to be able to choose commands compatible with the Fourier transform and that in turn allows us to facilitate the reading of images and their treatment.The intention of this image processing is to increase the signal-to-noise ratio and accentuate the characteristics of the images using custom filters or predefined by MATLAB.Now, MATLAB defines the image as a two-dimensional function f (x) where (x) and (y) are the spatial coordinates, and the value of (f) in any pair of coordinates (x) is the intensity of the image at that point.An image can be continuous with respect to (x) and (y), and also in intensity (analog image).Converting this image to digital format requires both the coordinates and the intensity to be digitized.Digitizing the coordinates is called sampling while digitizing the intensity is called quantization.So, when all quantities are discrete, we call the image a digital image.[11] Initially, an analysis using Fast Fourier transformation (FFT) was performed.The objective is to compare these results using a series of commands that allow the information to storage in matrices made up of real numbers [21].This makes it easy to select the filter that best suits the needs of the algorithm; since this technique allows to improvement or modify an image.For example, you can filter images to emphasize certain entities or remove other entities.Image processing operations implemented with filtering include anti-aliasing, sharpening, and edge enhancement.[22].
The filter used in the algorithm is a two-dimensional filtering method that uses the inverse Fourier transformation to generate a response at the desired frequency.It is a matrix containing the desired frequency response at equally spaced points on the Cartesian plane.The equation 2 describes the filtering.[23]: Where: Hd is the filter frequency response.Finally, as a method of comparison of the elements obtained when filtering the images, the use of the mean square error (MSE) was used, this is defined as the way to evaluate the difference between an estimator and the real value of the quantity that you want to calculate.The MSE measures the average of the square of the "error", the error being the value by which the estimator differs from the amount to be estimated.Where, Ŷ is a predictor vector with n elements and Υ is the really values vector [24] (see equation 3).Finally, as a method of comparison of the elements obtained when filtering the images, the use of the mean square error (MSE) was used, this is defined as the way to evaluate the difference between an estimator and the real value of the quantity that you want to calculate.The MSE measures the average of the square of the "error", the error being the value by which the estimator differs from the amount to 1299 (2024) 012005 IOP Publishing doi:10.1088/1757-899X/1299/1/0120054 be estimated.Where, Ŷ is a predictor vector with n elements and Υ is the really values vector [24] (see equation 3).The information obtained through the Fourier transform in conjunction with the MSE improves the results obtained by filtering and thus achieves the best possible pattern classification [25].

Results
The algorithm employs the FFT to detect differences in features associated with cracks and fissures in concrete structures.Different Commands, which together allowed us to filter the images and to see their information in the frequency domain.In Figs 4-5, examples of the studied plates, and plate of reference are shown.With this information, the existence of discontinuities in them is evidence.However, the amount and diversity of information obtained in each resulting image did not identify the characteristics of the cracks or fissures.

Source: Author
Figure 3 shows the error value obtained when comparing the data obtained from the image that shows discontinuities and one in which it does not (which is the same for all comparisons).Each point refers to comparison, and the error value is interpreted as the deviation of the data with respect to the discontinuous image from the non-discontinuous one.Thanks to the results obtained by the algorithm, it was possible to determine the dis-continuities in the images.However, the information is not enough to determine a specific range to classify cracks or fissures.If we can group them by the resulting error (figure 3).It can be seen that plates 3 and 4 are the greater clarity, therefore with the greatest error.This is due to the fact that the algorithm compares each plate with the one that has no discontinuity and has a darker contrast.On the other hand, there are plates 1 and 6 which are similar in contrast and brightness to the plate of figure 4. (e), for which a smaller error is obtained.According to raise above, the quality and clarity of information are important features.A factor like the illumination, location of the capture, and the brightness generates variation in the error.The followings figures contain all information necessary for the study.

Discussion
The proposed hypothesis focuses on deciding whether the Fourier spectrum of the image can provide information that allows differentiating the types of structural discontinuities studied.Differences are observed that have been quantified by FFT, frequency filtering, and MSE calculation.With this methodology, it is possible to classify between structures without alterations and with alterations.However, the quantification of the damage (differentiating between crack or fissure) is not achieved with the present work.
Other works have implemented artificial intelligence tools, where the prediction capacity is evaluated but from the perspective of the classifier and not from its characteristics [4], [26] and [16].In our work, we present the calculation of one feature like MSE how a descriptor of the structural condition.However, it was observed that other factors affect the MSE such as lighting, location, and brightness that are associated with the way the image is captured.In previous works [17], [18] already exists some treatment employing other techniques like mechanic vibrations-wavelet, and imaging processing with special techniques that shown good results, but nor does it quantify the magnitude of the damage; in our work it is possible quantify the differences between both patterns (cracks and fissures).Although these same factors (lighting, location, and brightness) generate a data variation that does not allow to associate specific characteristics with cracks or fissures, when performing the process several times with different images, the result indicates that in images that have a similar contrast, the error can be classified into a determined range.
The same behavior occurs with better-quality images.In general, if the images that contain the discontinuities have a contrast similar to that of the base image, the error will be minimal, on the other hand, if it is of light tonality, contrary to the base image, the margin of error is high.These conditions are what make it possible to demonstrate the need to standardize the images and thus create the least alteration of the information.

Conclusions
An algorithm capable of extracting relevant information to quantify the differences between a plate with some alterations has been proposed.The standardization of test im-ages is left for future work, using digital image processing tools.
In fig 2 it can see three examples using the propose algorithm.

Figure 3 .
Figure 3. Results (plate without discontinuity vs others plates (calculations in the frequency domain).