Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5797654, 12 pages

http://dx.doi.org/10.1155/2016/5797654

## Detection of Surface Defects on Steel Strips Based on Singular Value Decomposition of Digital Image

School of Electronic Information Engineering, Taiyuan University of Science and Technology, 66 Waliu Road, Wanbailin District, Taiyuan, Shanxi Province 030024, China

Received 27 April 2016; Accepted 11 October 2016

Academic Editor: José A. Sanz-Herrera

Copyright © 2016 Qianlai Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Image segmentation technology has been widely used to detect the surface defects in metal industries effectively. In some fields of the manufacturing industry, the determination of defects is more concerned than the accurate location and shape of defects. However, most of current image segmentation algorithms are complex or have difficulty determining the defect. This paper presents a novel method for determining and roughly locating the surface defects of steel strips based on Singular Value Decomposition. The method has no need of image segmentation. The gray level matrix of a digital image is projected on its singular vectors obtained by Singular Value Decomposition. A defect is reflected as a sudden change on the projections. Therefore, the defects can be determined and roughly located according to the sudden changes. The experimental results suggest that this method is valid and convenient for determining the surface defects directly.

#### 1. Introduction

As the raw material of machinery, aerospace, electronics, and other industries, there is a higher requirement for the surface quality of steel strips. But various surface defects occur unavoidably in the production of the steel strip constrained by its production process and physical and chemical properties. Defects ruin not only the appearance of steel strips, but also their important performances, such as the corrosion resistance, wear resistance, and fatigue strength. With defects, some steel strips are regarded as substandard or waste products in the further processing. More seriously, the defects may cause significant economic losses and even safety accidents without effective defect detection. Thus, the effective detection of surface defects is an important part of quality control of steel strips. Especially in China, with the overcapacity in the steel industry, the defect detection is of greater importance for steel strips.

In recent years, there has been a growing interest in detecting surface defects of products based on digital image processing. According to the algorithms of feature extraction and identification of defects, the methods can be roughly divided into three categories: the statistical methods, the frequency spectral methods, and the model-based methods [1]. The statistical methods mainly include Statistical Histogram properties [2, 3], gray level cooccurrence matrix [4, 5], and morphological operations [6–9]. The frequency domain methods range from Fourier analysis [10, 11] and Gabor transform [11–14] to wavelet analysis [15–18]. The model-based methods mainly focus on fractal geometry [19–21].

Many methods are adopted to detect the surface defects of steel strips based on computer vision. The methods cover almost all categories described above. But each method has its own advantages and constraints. In general, the statistical methods determine the existence rather than the location of defects without image segmentation. The frequency spectral methods require complex transformation from spatial domain to frequency domain. The model-based methods lack a standard modeling method so that the model is difficult to be established.

The determination of surface defects rather than their accurate location and shape is more concerned in the further processing of steel strips. Although image segmentation is widely used in the methods available to detect defects on steel surfaces [22], there is a practical need for a simpler and more convenient method.

In order to directly determine and roughly locate the defects on steel strips, this paper presents a novel method based on Singular Value Decomposition (SVD). The method has no need of image segmentation. The gray level matrix is projected on its singular vectors. The defect can be determined and located by the large wave peaks of the projections simultaneously. And the validity of this method is confirmed by experiments.

#### 2. Theories and Method

The Principal Component Analysis (PCA) intends to reveal the internal structure of variables by analyzing the so-called principal components. Since such independent principal components should hold as much information of the initial variables as possible, PCA is usually used to find the correlation between multiple variables. As a multivariate statistical method, PCA has been widely used in social and economic statistics, molecular dynamics simulation, mathematical analysis, and so on. Concerning pattern recognition, PCA is commonly used for filtering, dimensionality reduction of data, and feature extraction. However, it has not been reported that PCA is used to detect and locate the interest objects in an image directly, which deserves more attention and requires more research.

The method presented in this paper is based on SVD which is a classic method of PCA. Therefore, it is necessary to make a brief introduction to SVD before the description of the method.

##### 2.1. Singular Value Decomposition

The theorem of Singular Value Decomposition is introduced as follows [23].

Theorem 1. *Let have rank . There are unitary matrices and , such that where with and is a transpose symbol. The diagonal elements of are called singular values of . Then and are called left and right singular vectors, respectively.**The Singular Value Decomposition can be written in the following form:which is called the singular value factorization of . The columns of the matrix must form an orthonormal basis for the columns of , while the columns of the matrix must form an orthonormal basis for the columns of .*

##### 2.2. Defect Detection Method Based on SVD

Different from the background in pixels, the defect can be regarded as an abnormal area with the change of some elements in the gray level matrix. For the gray level matrix, the singular values and vectors change with its elements in some rows or columns, so that the vectors resulted from projecting the matrix on its singular vectors change too. In an image of the steel strip with defects, the elements of the projection vectors corresponding to the abnormal area change more considerably than those corresponding to the normal area. The indexes of elements with sudden changes correspond to the row or column numbers. With the sudden changes, the defect can be determined and roughly located.

###### 2.2.1. Detection in Ideal Case

In this section, the determination and location are discussed in the ideal case. It is assumed that there is no noise or disturbance in the gray image of the steel strip. In other words, the gray level values of all the pixels are equal for an image without defects. Then the gray level matrix of the image can be described asHere is the gray scale of each pixel in the image.

According to (2), there is where both and are unitary matrices and is a diagonal matrix. Then (4) can be expressed as In the same way, the following equation can be obtained:Thus the left singular vectors are just the eigenvectors of and right singular vectors are just the eigenvectors of .

The left singular vectors can be solved by eigenvalue decomposition. Then is substituted into Let the eigenvalues of be . Obviously, the rank of is 1, and is the only nonzero eigenvalue of . According to eigenvalue decomposition, the eigenvalue should meet the equation , where is defined as in Theorem 1. Then there is

As the solution of the above equation group, the vector must have the form ; here is a real number. So there is Here is called the projection on the first left singular vector of . In the same case, all elements of vector are the same real number too. The vector is called the projection on the first right singular vector of . By using the same method, the projection vectors, both and , are zero vectors.

In the ideal case, the surface defect of the steel strip is regarded as the perturbation of the gray level matrix. According to the matrix perturbation theory, the singular values of a matrix are of good stability [24], so are the singular vectors. In an image of the strip with defects, the projection corresponding to the defect is different from that corresponding to the normal areas. And the rest of the projection will change, but the change is very small.

For example, when an 8-bit pure white image (drawn with Microsoft Paint) is converted into a gray level image shown in Figure 1(a), all the elements of the corresponding gray level matrix will be 255. The singular vectors of the gray level matrix can be evaluated by MATLAB. According to the method in this paper, the gray level matrix is projected on its first left and right singular vector. For convenience, the projections are called left and right projection, respectively. They are shown in Figures 1(b) and 1(c) as left and right projection curves. In Figure 1, both the projection curves are horizontal lines.