Self-Supervised Texture Image Anomaly Detection By Fusing Normalizing Flow and Dictionary Learning

A backbone network will be utilized by deep learning-based anomaly detection metheds to extract features. The backbone network has significant feature extraction capabilities after pre-training on enormous labeled datasets. Real anomaly detection tasks lack a sizable amount of labeled data for backbone network training, and the cost of labeling huge datasets is quite expensive. We try to solve this problem by the model itself, so self-supervised learning is very suitable. By establish auxiliary tasks, self-supervised learning will extract supervised information from unsupervised data. The backbone network is trained using the extracted supervised information to increase the feature extraction capability of the network. For better feature extraction by self-supervised learning, a suitable auxiliary task is essential. A frequent additional task in semantic single classification research is rotation prediction. During the research, texture images with periodic backgrounds lacked subjects, resulting in the rotation prediction task not producing good performance. CutPaste^[20] is utilized as a auxiliary task in this essay. CutPaste crop a random rectangular patch from the training image and pastes it at any location on the image. With this method, irregularity are generated in the training samples to more realistically simulate realistic anomalies.

Anomalies in the texture are typically tiny. By using normalizing flow^[33], one may learn the distribution of data characteristics and estimate their densities with accuracy. Therefore normalizing flow is sensitive to minute irregularities. In order to improve the model’s ability to detect minute anomalies, we attempt to add normalizing flow into feature learning.

Fit unknown complicated feature distributions by normalizing flow. To simulate the unkown distribution, the approach applies a number of bijective transformations to a basic distribution. The normalizing flow is defined as follow:

For $p$ is an initial simple distribution and $x$ obeys $p$ in Equation 1, $F$ stands for the invertible transformation, and both $x$ and $t$ have the same dimension, being D-dimensional. The Jacobian matrix in Equation 1 is:

A Jacobi matrix is produced by the bijective transformation process on the simple distribution as shown in Equation 2.

We make the assumption that the features in latent space follow some sort of normal distribution during the bijective transformation procedure on feature space and latent space. Figure 3 illustrates the addition of the affine coupling layer structure in Real-NVP to the model.

In figure 3, $l$ and $n$ are arbitrary differentiable functions. The partial residual vector of $k_{1}$ couples additively and multiplicatively with vector $k_{2}$ to obtain $m^{(1)}_{2}$:

The vector $m^{(1)}_{1}$ is the same as vector $k_{1}$:

New vectors $m^{(1)}_{1}$ and $m^{(1)}_{2}$ are spliced again to form a whole. Random ordering and repartitioning of the spliced out new vectors. This method fully mixes information extracted from image. Above operations are repeated several times to form a affine coupling layer. During the multiplicative coupling operation, the exponential function is used to ensure that coefficients are non-zero.

Following, the model works various data interpolation processes. On the data, the feature extractor extracts features. A perfect parameter is discovered by the normalizing flow during training. This variable maximize the similarity between the known and unknown distributions:

The probability function in Equation 5 is applied for determining the parameters. The parameter values are derived by discretizing the continuous function into a discrete function and deriving the derivative, taking the logarithm of likelihood function. Normalizing flow, in contrast to other generative model training methods, trains generative networks just in mapping space, without making distinctions or making approximations about the accuracy of the data.

We design a loss function for normalizing flow training as shown in formula six. A typical distribution in the potential space is $t$. mapping the initial distribution to the prospective space. The distribution created after bijective transformation is more similar to the genuine distribution in the potential space, according to a smaller value of the second paradigm based on potential space. The Jacobi matrix’s second term in Equation six is a logarithmic determinant, and it represents the scale coefficients’ total before the exponential operation. The proper scale of feature transformation must be chosen in order to produce the best outcomes for the model. Part IV of the article will discuss the implementation’s specifics.

Another name for dictionary learning is sparse coding. Its major goal is to extract the crucial characteristics of the data and omit the rest. The repeating arrangement of some important information is ultimately what gives texture image backgrounds their periodicity. This property matches well with the idea of dictionary learning. To improve detection outcomes in the stage of anomaly detection, we attempt to join dictionary learning in the detecting phase.

Dictionary learning solves the problem of representation about signal $y$ in N-dimensional space. As shown in Figure 4, The signal $y$ can be expressed as $y\approx Dx$. $y$ is a signal to be reconstructed, $D$ is an overcomplete dictionary, and $x$ denotes the sparse coefficient. The mathematical expression for dictionary learning (sparse coding) can be described as:

$||x||_{0}$ indicates the number of non-zero vectors in the dictionary. $x$ is a coefficient for constructing the sparse coding and $x\in R^{M}$, D is an overcomplete dictionary and $D\in R^{N\times M}$, $y$ is an input signal to be reconstructed, $\xi$ is an arbitrarily small positive number.

There are two steps for anomaly detection stage. Firstly, a feature dictionary is used to reconstruct a new image. The error between the reconstructed image and the target image is compared. Whether an image contains anomalies depends on the size of the inaccuracy. In accordance with the aforementioned ideas, selecting a suitable error threshold is essential for anomaly detection, and thresholding the reconstruction error further enhances the model’s detection speed.

The best sparse coefficients are produced during the optimization iteration process of building a feature dictionary:

In the equation eight, dictionary matrix $D={d_{1},d_{2},...,d_{N}}$, where $d_{n}$ is a feature vector in the dictionary matrix. $x_{i}\in R_{N}$ is the sparsity coefficient. The $K$ patches of features $y$ are used as training data for dictionary learning. The second paradigm of the feature vectors in the dictionary must be 1, in order to prevent nonsense values from being produced, due to the uncertainty of multiplying eigenvectors with sparse coefficients.

Finding anomalous locations is necessary for an anomaly detection model. To identify image anomaly regions, we employ the method of calculating anomaly scores. The abnormal portions of the anomalous images focus more on high-value anomaly scores. Our model will choose a few regions with the highest anomalous scores to combine, creating a new region that will be the anomaly region located by the model, in accordance with the requirements of the detection task. The five locations with the highest anomalous ratings are combined for the actual detection process. The localized locations have been manually verified to be genuine anomalous regions.

The MVTec AD dataset, a popular dataset for assessing the performance of unsupervised anomaly detection systems, is used to evaluate the effectiveness of our method. It uses actual industrial anomaly images for its images. 5354 color high-resolution photos totaling more than 70 different anomaly categories are included. In this study, we emphasize the identification of texture anomalies and confirm the model’s accuracy in identifying certain texture types, such carpet.The rationale for focusing on texture anomalies in this research is given below

In industrial inspection, texture-based industrial anomaly detection is an important detection method. Taking carpet as an example, surface anomalies of these products are often generated by cuts and scrapes, which are less intense and cannot be detected on the product using intensity attributes, and require texture analysis of the sample.Texture is a visual representation of the repetitive patterns of features on the surface of an object. The textures contained in texture images are often complex, and it is difficult for humans to analyze and recognize texture features by eye only. Traditional anomaly detection methods rely on manually selected feature points, which cannot be applied to such texture anomaly data. The use of CNN for image texture learning can avoid the interference generated by textures.

Texture images are very common in the field of anomaly detection, and many existing detection models for texture images are not robust enough to be widely applicable to different types of texture images. In the following sections, we hope to build a more robust anomaly detection model for texture images, and we describe the implementation process.

The method, which draws inspiration from the two-stage image classification model^[30], also separates feature learning and anomaly detection.Based on the baseline model, we optimized the baseline model. There are so two training processes in the improved approach. An image patch is randomly cropped to the training sample and pasted to a randomly chosen area of the sample prior to training. The aforementioned tasks attempt to the greatest extent to imitate abnormalities in actual anomalies. For extracting picture features, our model uses a pre-trained version of AlexNet. Five convolutional layers and three fully connected layers make up AlexNet^[35], which uses ReLU as its activation function. The AlexNet network structure is less complex than VGG and ResNet’s. It can make a model less complex. AlexNet does not perform worse than VGG and ResNet in terms of feature extraction.

Normalizing flow helps better learning of minute variations, as was already indicated. Figure 5 illustrates the normalizing flow structure we employ in this paper. The input data are separated into two pieces and sorted at random. On both sections of the input, fully connected activities are carried out concurrently. The internal operations of the fully linked neural network are shown in figure 5 as n and l. Normalizing flow has eight coupling components.

The t-SNE plots produced by the model using the Carpet data are displayed in Figures 6(a) and 6(b). Figure 6(b) makes it obvious that the data processed by the auxiliary task and the normal and abnormal data no longer overlap. This shows that normalizing flow enhances the model’s capacity to detect minute irregularities. The model is more sensitive to minute variations in the data.

The input size for the training phase’s photos is 128 and the learning rate is 2. During the experiment, it was discovered that the loss value of the model stabilized after 270 epochs of training. Input images have a batch size of 2. A model created by self-supervised representation learning extracts features from training samples during the anomaly detection phase. There are 32 inputs in the batch. Extracted features will form feature dictionaries. For image reconstruction and sparse coding, feature dictionaries are crucial.  A judgment on the performance of the anomaly detection model is also crucial for anomaly detection. ROC curves are used to assess model performance. The ROC curve is a series of values obtained under particular circumstances using various evaluation criteria. AUC value is the area of the ROC curve with the coordinate axis. Higher AUC value means better detection accuracy of the model.

As indicated in Table 1, during the trial we compared the outcomes of the upgraded approach with the baseline method. The detection accuracy of the modified approach in this work is enhanced by 31$\%$ on Carpet data. Additionally, the revised approach slightly enhances the Tile and Wood data. As demonstrated in Table 2, in addition to comparisons with the baseline approach, the approach also included comparisons with Sparse coding^[22], 1-NN^[40], GANomaly^[37], and other approaches.

In contrast, it is discovered that the enhanced method has significant robustness and does well in each of the five texture datasets. It may be used to identify anomalies in a variety of texture types. The table in bold is the best performance.

By computing anomaly scores, abnormal locations are localized. To distinguish between normal and anomalous regions, an error threshold is set. Figure seven shows a line graph of anomaly scores for partial Carpet images. Red dots in the figure represent normal images and blue dots represent anomalous images. From figure seven, we can find that the scores of normal images are significantly lower than those of abnormal images.

In summary, the article describe a method for detecting textural anomalies that combines self-supervised learning and dictionary learning. The proposed anomaly detection method’s performance is clearly described in Tables 1 and 2. The research backs up the idea that a normalizing flow can aid the model in capturing tiny anomalies more effectively. It improves  the detection accuracy of the model and makes the model more sensitive to minute anomalies. Additionally, neural networks can improve the efficiency of conventional methods for feature extraction. Combining convolutional neural networks with traditional methods will significantly improve the performance of detection methods.