Detection of Cracks and Deformations Through Moment Transform Techniques ^†

Abstract

Ensuring the structural integrity of mechanical components is a key challenge in industries such as automotive, aerospace, and energy. Conventional techniques for defect identification, including non-destructive testing (NDT) and the Finite Element Method (FEM), offer reliable solutions—yet FEM often requires intensive modeling work and high computational cost. To streamline the detection process, this study proposes a method based on orthogonal moment transforms applied to digital images. This fast and automated technique is particularly suited for integration into industrial vision systems. The approach consists in encoding the visual features of a component using continuous orthogonal moments (e.g., Zernike, Chebyshev, or Fourier), and analyzing the extracted descriptors to identify irregularities associated with surface cracks or structural flaws.

1. Introduction

The Finite Element Method (FEM) has emerged as a cornerstone in computational mechanics, particularly for addressing engineering challenges involving intricate geometrical shapes and complex boundary interactions. Its core idea lies in segmenting a given structure into a mesh of discrete subdomains—finite elements—within which the local behavior is governed by physical equations. These localized computations are then combined to estimate the system’s overall response. Typically, the modeling begins with the creation of a digital representation of the component, often developed using CAD software like CATIA V5 [1].

Once the digital model is constructed, the next critical phase involves assigning material properties, such as elasticity modulus, Poisson’s ratio, yield strength, and fracture resistance. These parameters are crucial to ensure that the numerical simulations reflect realistic physical behavior. The geometry is then discretized into smaller cells through meshing, with finer meshes placed strategically in zones prone to stress accumulation—such as near geometric notches or crack-prone regions—to improve result accuracy [2].

Following meshing, external forces and boundary conditions are incorporated into the model. This setup enables the numerical solver to compute various physical fields, including stress, strain, and displacement across the domain. From these outputs, regions with elevated stress—commonly termed as “hot spots”—can be identified, which are often precursors to structural failures. Failure assessments generally rely on theoretical models such as the Von Mises or Tresca criteria [3].

In scenarios requiring advanced fracture analysis, artificial cracks can be introduced into the model, and their progression monitored using the Extended Finite Element Method (XFEM). This method offers the flexibility of modeling discontinuities without the need for re-meshing, enabling efficient simulations of crack growth and propagation [4].

Despite its effectiveness, FEM has some limitations. High-resolution 3D simulations, especially those involving dynamic fracture propagation, can be extremely demanding in terms of computational resources. Additionally, the precision of FEM outcomes heavily relies on mesh quality; poor meshing in sensitive regions can distort the stress field interpretation. Implementing FEM also necessitates deep knowledge of numerical analysis, mechanics of materials, and post-processing interpretation. Moreover, advanced techniques like XFEM are often complex to apply and require specific expertise. Experimental validation is usually essential to support FEM results, which may further increase project time and cost.

To address these drawbacks, the present work introduces an alternative methodology based on orthogonal moment transforms, primarily used in image-based analysis. This approach enables rapid, automated detection of structural defects such as cracks, with significantly reduced computational requirements compared to FEM. Moments like Zernike and Chebyshev offer the additional advantage of geometric invariance—resilience to translation, rotation, and scaling—making them ideal for robust anomaly detection across variable part orientations. The effectiveness of this technique will be evaluated through objective image quality metrics, including Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR), allowing direct comparison between damaged and undamaged structures.

The remainder of this paper is structured as follows: Section 2 outlines the families of orthogonal moments examined and their mathematical basis. Section 3 describes the experimental design and presents the obtained results. Section 4 concludes the study and highlights possible future extensions.

2. Orthogonal Moments

2.1. Zernike Moments

Zernike moments are commonly used in image analysis and pattern recognition to extract global shape features from images. First introduced into the field of computer vision by Teague [5], these moments have shown superior performance compared to other moment-based descriptors [6,7] due to their strong representational capacity and their resilience to noise and geometric distortions.

In recent developments, researchers have continued to investigate Zernike moments, focusing particularly on improving computational speed and enhancing accuracy. A Zernike moment is a complex-valued coefficient that contains both a magnitude and a phase component, each contributing different aspects to the representation of shape and structure.

The underlying Zernike functions are complex orthogonal polynomials defined over the unit circle. An image’s Zernike moments are obtained by projecting its intensity values onto this set of basic functions. These functions exhibit three key properties that make them especially effective: they are orthogonal, invariant to rotation, and capable of compressing meaningful information—where lower-order moments capture most of the image’s low-frequency content.

Mathematically, Zernike moments are expressed in terms of a radial order p and an angular repetition q, and are defined over the unit disk as follows:

$$Z_{pq}={\displaystyle \frac{p+1}{\pi }}{\displaystyle \underset{x^2+y^2\le 1}{\iint }{V}_{pq}^{\ast }}(x,y)f(x,y)dxdy$$

(1)

where $V_{pq}^{\ast }$ denotes the complex conjugate of $V_{pq}$, which is defined as:

$$V_{pq}(\rho ,\theta )=R_{pq}(\rho )e^{iq\theta }$$

(2)

$$R_{pq}(\rho )={\displaystyle \sum _{\begin{array}{l}k=\left|q\right|\\ |p-q|pair\end{array}}^p\frac{{(-1)}^{\frac{p-k}{2}\frac{p+k}{2}!}}{\frac{p-k}{2}!\frac{k-q}{2}!\frac{k+q}{2}!}}{\rho }^k$$

(3)

Based on Equations (1)–(3), the Zernike moments corresponding to an image that has been rotated by an angle α around its center (in polar coordinates) can be represented as follows:

$$Z_{pq}^{\alpha }=Z_{pq}{e}^{iq\alpha }$$

(4)

Equation (4) confirms that the magnitude of Zernike moments is invariant under rotation, since a rotation of the image only affects the phase component of the complex moments, leaving their magnitude unchanged. This property is particularly useful in pattern recognition and shape analysis, where orientation should not affect feature descriptors.

Furthermore, due to the orthogonality of the Zernike polynomials, an image can be reconstructed as a linear combination of these basis functions, each weighted by its corresponding Zernike moment:

$$\stackrel{\sim }{f}(x,y)={\displaystyle \sum _{(p,q)\in D}{\displaystyle \sum Z_{pq}}}{V}_{pq}(x,y)$$

(5)

2.2. Tchebichef Moments

Discrete orthogonal moments have been developed to address the approximation errors that arise when applying continuous orthogonal moments—such as Zernike or Legendre moments—to digital images. These errors stem from the discretization process required to adapt continuous mathematical formulations to the inherently discrete nature of digital images, leading to inaccuracies in image representation and analysis.

To mitigate these issues, discrete orthogonal moments, like the Tchebichef moments, are employed. These moments are based on discrete orthogonal polynomials, which are inherently defined over a finite set of points, making them more suitable for digital image processing [8,9].

The Tchebichef polynomials satisfy the following recurrence [10] relation:

$$\begin{gathered} (n+1)t_{n+1}(x)-(2n+1)(2x-N+1)t_n(x)+n(N^2-n^2)t_{n-1}(x)=0 \\ n=1,2,.........N-1 \end{gathered}$$

(6)

The Tchebichef moment is defined by the following expression:

$$\begin{gathered} T_{nm}={\displaystyle \frac{1}{\rho (n,N)\rho (m,N)}}{\displaystyle \sum _{x=0}^{N-1}{\displaystyle \sum _{y=0}^{N-1}{t}_n}}(x)t_m(x)f(x,y) \\ n,m=0,1,2,......N-1 \end{gathered}$$

(7)

2.3. Pseudo-Jacobi-Fourier Moments

The pseudo-Jacobi–Fourier moments are defined as follows [11,12]:

$$M_{m,n}={\displaystyle {\int }_0^1{\displaystyle {\int }_0^{2\pi }f}}(\rho ,\theta )R_m(\rho )e^{-jn\theta }\rho d\theta d\rho$$

(8)

where $R_m(\rho )$ is the radial polynomial given by:

$$R_m(\rho )=\sqrt{{\displaystyle \frac{2m+4}{(m+3)(m+1)}}(\rho -{\rho }^2)}\times {\displaystyle \sum _{k=0}^m\left({(-1)}^{m+k}\frac{(m+k+3)!}{(m-k)!k!(k+2)!}{\rho }^k\right)}$$

(9)

and $f(\rho ,\theta )$ is the image to be analyzed.

3. Materials and Methods

This study focuses on a two-dimensional problem involving a rectangular plate modeled using CATIA, with a mesh size of 5 mm. The plate exhibits various types of deformations, as shown in Figure 1, Figure 2 and Figure 3. To analyze these defects, we apply three families of orthogonal moments: Zernike moments, which are continuous, and two discrete counterparts—Chebyshev moments and pseudo-Jacobi-Fourier moments (PJFM).

The main objective is to enable automatic detection of anomalies, such as cracks or holes, by evaluating the performance of each method both qualitatively and quantitatively.

The adopted methodology includes a standardized preprocessing procedure applied to each image: grayscale conversion, resizing to a fixed size of 256×256 pixels, followed by the extraction of orthogonal moment vectors either at a fixed order (e.g., order 10) or over a progressively increasing range up to order 50. For each order, three key metrics were computed: descriptors extracted at order 10, the Euclidean distance between moment vectors of healthy and damaged images (to measure structural dissimilarity), and the cumulative energy of the moments, which reflects the total amount of information encoded. This approach provides a robust framework for detecting mechanical defects in structural components.

4. Simulation Results

Figure 4, Figure 5 and Figure 6 present the descriptors obtained from Tchebichef, Pseudo-Jacobi-Fourier, and Zernike moments, while Figure 7, Figure 8 and Figure 9 illustrate the corresponding variations of energy and Euclidean distance with respect to the moment order; Figure 10 and Figure 11 finally report the image quality metrics (MSE and PSNR) for the comparison between defect-free and defective cases.

The Euclidean distance between the moment vectors of the two images was calculated as follows [13]:

$$D={‖|{\overrightarrow{M}}_1|-|{\overrightarrow{M}}_2|‖}_2=\sqrt{{\displaystyle \sum _{i=1}^N{\left(\left|M_{1,i}\right|-\left|M_{2,i}\right|\right)}^2}}$$

(10)

$M_{1,i}$ and $M_{2,i}$ are are the i-th moments of the first and second images respectively, N is the total number of moments (e.g., for order 10, it depends on the number of valid moment indices).

The total energy of an image, based on orthogonal moments, is defined as follows [14,15]:

$$E={\displaystyle {\sum }_{i=1}^N|M_i{|}^2}$$

(11)

where:

E: total energy of the image based on moments

$M_i$: the i-th orthogonal moment (real or complex)

N: total number of moments used (typically depends on the maximum order K)

The MSE measures the average of the squared differences between corresponding pixel intensities of the original and processed images. It is defined as [16,17]:

$$\mathrm{MSE}={\displaystyle \frac{1}{M\times N}}{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N{\left(I_1(i,j)-I_2(i,j)\right)}^2}}$$

(12)

where:

$I_1(i,j)$ and $I_2(i,j)$ are the pixel values at position (i, j) in the reference and the processed image, respectively.

M and N are the dimensions of the image.

The Peak Signal-to-Noise Ratio evaluates the quality of a reconstructed or processed image compared to a reference by expressing the ratio between the maximum possible power of a signal and the power of corrupting noise. It is measured in decibels (dB). Higher PSNR values typically reflect better image quality.

$${\mathrm{PSNR}=10 \log }_{10}( {\displaystyle \frac{MA{X}_I^2}{MSE}})$$

(13)

where:

MAX_I represents the maximum possible pixel value.

MSE is the Mean Squared Error as defined above.

5. Conclusions

The evaluation of three families of orthogonal moments—Zernike, Chebyshev, and Pseudo-Jacobi-Fourier (PJFM)—has highlighted their varying effectiveness in detecting structural defects in images. Zernike moments proved to be reliable from intermediate orders, particularly for capturing global shapes, thanks to their rotational invariance. However, their sensitivity decreases when it comes to detecting small, localized irregularities. Chebyshev moments, known for their simplicity and computational speed, require higher orders to capture fine details and tend to be more sensitive to translations and orientation changes, which may limit their robustness. PJFM, on the other hand, offered the best performance across all evaluations, distinguishing defective images even at low moment orders. Their hybrid construction, which combines radial and angular components, allows them to respond effectively to complex features and deformations. These observations were supported by objective image quality metrics: higher MSE and lower PSNR values consistently indicated visual deterioration in defective images, aligning with the variations observed in moment-based energy and distance. Overall, PJFM appears to be the most promising method for accurate, early, and non-invasive defect detection. Looking forward, future developments could aim to integrate PJFM into real-time monitoring systems, optimize its implementation for embedded platforms, and explore its extension to three-dimensional or multi-modal data. Such advancements would further enhance its applicability in industrial inspection, structural health monitoring, and medical imaging.