## 2 Materials and Methods
### The Proposed Method  
In this section, the detailed design and implementation of FMDR are introduced, including its mathematical expressions, theoretical derivations and its application in semi-supervised classification problems. In practice, images are usually assembled as a data matrix, where each column (or row) represents a vectorized sample. At first, a transformation will be adopted for extracting the discriminative information and reducing the interference in frequency domain. The transformation consists of several steps. Firstly, the vectorized samples are reshaped as images according to their height and width. Then, each image in the original spatial domain is converted into the frequency domain using 2D DFT. Based on the distribution of crucial  
Figure 2: The effectiveness of extracting high-frequency texture components from images of different objects. (**a**) Images of various objects, including a mountain, human face, aerial photo and wood. (**b**) Extraction results of high-frequency texture components. morphological features, a designed filter is employed to extract the high-frequency texture components. Lastly, in order to maintain consistency in the data structure, the extracted features are then vectorized and assembled into a new matrix. For the \(i\)th sample in the data matrix \(\mathbf{X}\), the high-frequency texture components \(\mathbf{HF}_{i}\) of pixel matrix \(\mathbf{Pix}_{i}\) can be written in the matrix multiplication form according to (2):  
\[\mathbf{HF}_{i}=\mathbf{B}_{1}\mathbf{KB}_{2}\odot\mathbf{B}_{1}\mathbf{Pix}_ {i}\mathbf{B}_{2}. \tag{3}\]  
Base matrices \(\mathbf{B}_{1}\) and \(\mathbf{B}_{2}\) are introduced in (2), which are the irrelevant bases. \(\mathbf{K}\) refers to the convolution kernel of the filter in the original domain. \(\odot\) is a Hadamard product operator. For each pixel point, the transformation can be rewritten in a more detailed form:  
\[\mathbf{HF}_{i}(u,v)=\sum_{m=0}^{H-1}\sum_{n=0}^{W-1}\mathcal{F}\mathbf{Pix}_{ i}(m,n)e^{-j2\pi(\frac{mm}{H}+\frac{m}{W})}, \tag{4}\]  
where \(\mathcal{F}\) represents the designed filter. In practical applications, the distribution of valuable information within images is different across every dataset. There is no universally applicable filter design that can suit all scenarios. In this study, we employed the Butterworth filter, which is a high-pass filter and primarily involves two adjustable parameters: the order and the cut-off frequency. Based on experimental findings, the characteristics of high-frequency texture components are correlated with the filter's parameters. Specifically, the quantity of features is linked to the cut-off frequency, while the texture thickness depends on the order of the filter. With lower parameters, a significant amount of irrelevant information and interference is retained. As the parameters increase, more low-frequency interference is filtered out, making the texture components more pronounced. However, excessively large parameters result in the abandonment of a substantial portion of the information. Samples from diverse datasets contain noise with varying distributions and structures. To achieve optimal performance, comprehensive consideration of filter parameter tuning is essential in practical applications. After extracting the high-frequency texture components corresponding to \(n\) samples, we can finally reassemble the frequency information into a data matrix by vectorization:  
\[\mathbf{HF}=[vec(\mathbf{HF}_{1}),vec(\mathbf{HF}_{2}),\cdots,vec(\mathbf{HF}_ {n})]. \tag{5}\]  
Subsequently, a novel kernel function is designed for measuring the relationship between samples based on the frequency domain information. The key to manifold learning algorithms is to determine the distance measurement of samples in the high-dimensional manifold space. A more precise distance measure leads to better preservation of sample relationships after dimensionality reduction. With conventional graph-based methods, the distance between samples is calculated by the absolute distance.