## 3 The Proposed Approach
### Proposed Framework  
In just about all instances, conventional clustering approaches would consider all of the data's knowledge during the procedure, which is effective for noiseless data. However, data in the real world are often noised, and the noise may be non-sparse or of a high intensity. The noise may form some pixel blocks in the samples and cause those samples to lose their features. Thus, traditional methods would probably categorize those noised samples into the same cluster. Inspired by our pretests, the HFTC features extracted from face images can maintain valuable morphological information well and greatly weaken the noise, we believe that HFTC extraction can make better use of the key knowledge of low-quality IC images. To be specific, the valuable information of the ICs' images exists in the layout and wiring of the electronic components in the field of IC defect detection (visual detection methods). While the layout and wiring can be effectively retained by HFTC extraction. In the meantime, the impact of environmental noise and pixel loss will be significantly reduced in the extraction process, which is very conducive to the subsequent clustering tasks. In machine learning tasks, each individual sample in the dataset is usually assembled into a column as \(\mathbf{X}=\left\{\mathbf{x}_{i}|i=1,\ldots,N\right\}\). Each individual sample, designated by the letter \(\mathbf{x}_{i}\), must be reshaped during the extraction process to its original height and width, designated by \(\mathbf{X}_{i}\). In order to facilitate the following discussion, the height and width of \(\mathbf{X}_{i}\) are represented by \(H\) and \(W\), respectively. Then, \(\mathbf{\hat{X}}_{i}\) can be made to represent the expression of data \(\mathbf{X}_{i}\) in the frequency domain, which is derived by using a two-dimensional Fourier transform (2D FFT):  
\[\mathbf{\hat{X}}_{i}(s,t)=\sum_{m=1}^{H}\sum_{n=1}^{W}\mathbf{x}_{i}(m,n)e^{-j2\pi( \frac{m}{H}+\frac{m}{W})}, \tag{3}\]  
where \((s,t)\) and\((m,n)\) denote the coordinates. Then, the high-frequency texture components of the images \(\mathbf{\hat{h_{1}}},\cdots,\mathbf{\hat{h_{n}}}\) are obtained by using a high-pass filter \(\mathcal{H}\) to process frequency domain data and apply an inverse two-dimensional FFT to the filtered frequency domain data:  
\[\mathbf{\hat{h_{i}}}(m,n)=\frac{1}{H}\frac{1}{W}\sum_{s=1}^{H}\sum_{t=1}^{W} \mathcal{H}\mathbf{\hat{X}}_{i}(m,n)e^{j2\pi(\frac{m}{H}+\frac{m}{W})}. \tag{4}\]  
Then, \(\mathbf{\hat{h_{i}}}\) is restructured as a column vector \(\mathbf{h}_{i}\) and combined to form the matrix \(H=\left\{\mathbf{h_{1}},\cdots,\mathbf{h_{N}}\right\}\in\mathbb{R}^{d\times n}\). The high-frequency texture components retain most morphological information, which can reveal the essential details for better distinguishing individuals. When the data are noised, the traditional clustering methods may not work as usual, particularly when the noise is not sparse and has a high energy level. Using IC images as examples, when distinct circuit images are obscured by Gaussian noise or flaws, they may be classified as part of the same cluster because of the high energy of the covers. The noise described above is low-frequency information that can be considerably decreased following HFTC extraction, from which it can be concluded that using an HFTC will assist representation learning and increase subsequent clustering performance. Rather than sharing an entire uniform space, the samples of dataset X are typically located in their distinct low-dimensional subspaces. The LRR problem [22] can be reformulated using the self-representation method:  
\[\min_{\mathbf{C}}~{}rank(\mathbf{C})+\lambda\|\mathbf{X}\mathbf{C}-\mathbf{X}\|_{l} \tag{5}\]  
where \(||\mathbf{X}\mathbf{C}-\mathbf{X}\|_{l}\) represents the sparse error measurement.