1) Dataset Preprocessing: Since there are different kinds of noises in the images in the sample set, if they are not removed, the recognition results of this experiment will be greatly affected. Each image is denoised by the method of gray value adjustment. It refers to gray value less than a value $a\times 255$ as noise, and changes their values to 0, greater than another value $b\times 255$ as script information of images, and changes them to 255, where $a$ and $b$ a value between 0 and 1. After the images were denoised, we do size normalization of these images. Finally, all of images were saved in size of $n\times n$.

2) Feature Extraction: A very important step of image recognition is how to extract the features of the image in recognition. In computer vision, the gray value of an image is a basic and key feature. Gray value is a way to strip away the color information and only retain and process the brightness information. In a grayscale image, each pixel has a value between 0 and 255. The gray value of the image can describe the contour information of the object in the image, and the gray value around different objects in the image will have a significant difference. Therefore, the gray value of the image can be used as the feature of the image.

As all of images have been denoised and size normalized after preprocessing, the gray values of an image form a matrix. In this paper, the gray value matrix of each image is used as characters for recognition. The gray value of each image is splintered into a column vector, every image can be regarded as a vector with a dimension of $n\times n$.

1) Background of ISOMAP Algorithm: ISOMAP algorithm is one of manifold learning algorithms, which is a nonlinear data dimensionality reduction method that has developed in recent years. Its main idea is to find low-dimensional manifold structures from high-dimensional data, that is to find low-dimensional manifolds in high-dimensional space, in order to achieve dimensionality reduction or data visualization.

ISOMAP was first proposed by Tenenbaum et al. in the famous Science journal [14] , which is a global property preserving manifold learning algorithm, and its low dimensional embedding results can reflect the geodesic distance on the manifold where the high dimensional observation samples are located [15] . So far, ISOMAP algorithm has been applied in many fields. For example, the algorithm has been applied to facial expression recognition in [16] , as well as to face recognition in [17] or classification in [18] or wireless sensor network positioning in [19] , which reflects the practicability of the algorithm.

2) ISOMAP Algorithm: As mentioned earlier, after the features of an image are extracted, we obtain a column vector. The dimension of this column vector is determined by the dimension of the gray value matrix of the image. Assume that we have completed the feature extraction of the data set, and the extracted features are formed into a matrix $\mathcal{X}=\left\{x_1,x_2,\cdots ,x_N\right\}$. The dimensionality of any $x_i$ is $D$. Note that $x_0$ is a point to be identified. The specific steps for recognizing oracle bone script with ISOMAP algorithm are listed here.

$D=\left[d\left(x_i,x_j\right)\right]$(1)

$d_{ij}^2={\left(x_i-x_j\right)}^{\prime }\left(x_i-x_j\right).$(2)

$G=\left[g\left(x_i,x_j\right)\right]$(3)

$g\left(x_i,x_j\right)=\text{min}\left(d_{ij},d_{il}+d_{lj}\right).$(4)

In this paper, in order to solve the problem of edge points which close to other species, we select the result of decision recognition according to the number of points of the nearest category. The label corresponding to $N$ points is classified into $C_1,C_2,\cdots ,C_M$, and find the maximum of them by

$C_j=\text{max}\left\{C_1,C_2,\cdots ,C_M\right\}.$(5)

Finally, we take the label corresponding to the minimum value in $C_j$ as the final identification result. In other words, if the number of points of the some types closest to the test point to be measured is the largest, the point of the type closest to the test point to be measured is selected as the final recognition result. If the closest points are multiple, they are $C_1,C_2,\cdots ,C_p$. Then the minimum Euclidean distance of every category $d_1,d_2,\cdots ,d_p$ correspondence to $C_1,C_2,\cdots ,C_p$ needs to be found, where $p$ is the number of maximum types of points nearest to test point. Finally, the final recognition result is

${\hat{d}}_m=\text{min}\left\{{\hat{d}}_1,{\hat{d}}_2,\cdots ,{\hat{d}}_p\right\}.$(6)

The Corresponding label of ${\hat{d}}_m$ is judged to be the closest image.

From the previous description, the method for recognizing the image of oracle bone script can be written as

${\hat{d}}_i^2={\hat{d}}^2\left(y_i,y_0\right)={\left(y_i-y_0\right)}^{\prime }\left(y_i-y_0\right).$(7)

We take the first $N$ minimum ${\hat{d}}_i$ , and the labels corresponding to $y_i$ in ${\hat{d}}_i$ are divided into M categories. The maximum number of labels $C_1,C_2,\cdots ,C_M$ corresponding to each category is calculated separately. Finally, we calculate $C_j$ by

$C_j=\text{max}\left\{C_1,C_2,\cdots ,C_M\right\},$(8)

where $C_j$ is a type of sample points which closest to $x_0$. We take the label corresponding to the minimum ${\hat{d}}_p$ value in $C_j$ as the final recognition result.

1) Selection of Samples: In this paper, the opening dataset of OBC306 is selected as dataset. There are 6 kinds of images which selected from the dataset, as shown in Figure 4 .

Although only one image from each category is shown here, in fact, each category is made up of 20 identical type images. A sample set composed of 120 images was obtained.

Therefore, the dimension of the gray value matrix of every image is 128 × 128, and finally we concatenate the gray value matrix columns into a column vector, the dimension is 128 × 128.

3) Parameter Setting: The value of $ϵ$-neighborhood is obtained through the Euclidean distance matrix calculated by Equation (2) The distance in the distance matrix of all sample sets ranges from 21,000 to 31,000. To make sure that the adjacency graph is connected, the first small value of $ϵ$-neighbor should be determined through adding $20\text{\%}\delta$ to the minimum distance, where $\delta =31000-21000$. Similarly, the $n$-th small value of $ϵ$-neighbor is determined through adding $20\text{\%}n\delta$ ( $n=2,\cdots ,5$) to the minimum distance. We set 23,000, 25,000, 27,000, 29,000, and 31,000 as the value of different $ϵ$-neighborhood, respectively. To visualize high-dimensional data, the value of embedding dimension $d$ is set to 3. To evaluate the impact of dimensionality, the $d$ is set to 15, 35, 55, 75, and 95 in this study, respectively. The number of the nearest neighbors of the test point is set to $N=1,3,7,13,19$ for the 6 different sample sets, respectively.

The low-embedding of dimensional reduction is obtained by using ISOMAP algorithm on sample set. Then the low-dimension vector is displayed in a 3-dimensional coordinate system, which is shown as follows.

In Figure 6 , the parameter of $ϵ$-neighborhood is 29,000. The different colored points represent different categories, and deepened points represent to be identified.

Through many trials, the average accuracy rate was obtained when the $ϵ$ neighborhoods and the $N$ were set in different values. The experimental results are shown in Table 1 .

From the table, we can observe that when the neighborhood is taken as $ϵ=29000$ and $N=7$, the recognition accuracy reaches 93.3%, which is the best recognition rate under these parameters. In order to observe the impact of changes in $N$ on the average accuracy, a line graph was plotted to show the effect of varying values of $N$ on the recognition rate, as illustrated in Figure 7 .

It can be observed that as $N$ increases, the accuracy shows an increasing trend. When $N$ reaches a certain value, the recognition rate reaches its optimum and gradually begins to decline. As we can intuitively see in the figure, when the value of $N=7$ and $ϵ=29000$, the optimal recognition rate under this parameter can be achieved. This experiment indicates that selecting larger neighbors in the algorithm can achieve a better recognition rate. This is because choosing larger neighboring points can capture the features at sample points and also reduce the introduction of noise.

A large number of experiments are conducted with study, the impact of the embedding dimension and the $ϵ$-neighbor on the average recognition rate, as shown in Figure 8 .

It can be observed that a large value of $ϵ$-neighborhood consistently achieves superior recognition rate compared to the small values. This is because a large value of $ϵ$-neighborhood can capture more discriminative structural features than a small value. It can also be seen that the average accuracy rate tends to decline gradually as the dimension increases. This can be attributed to the introduction of noise components in high-dimensional manifolds, which induces erroneous classifications. Notably, when the embedding dimension exceeds a critical threshold (approximately when d > 55 in this study), the mean recognition accuracy demonstrates asymptotic stabilization. This result reveals that with the increase in dimension, the average recognition rate tends to be stablized and exhibits better robustness under higher embedding dimensions.

To evaluate the parametric robustness of the algorithm through adjusting the value of neighborhood, we formulate a stability metric which refers to the robustness score. It is obtained through formulating the ratio of the average recognition rate to the standard deviation. The results of extensive experiments are presented in Figure 9 .

It can be observed that the curve shows a trend of rising first and then falling with the increase of $ϵ$-neighbor. Notably, the results indicate that when $ϵ=27000$, the experimental results exhibit the best robustness. This is because a small value of $ϵ$-neighbors cannot remain the local structures of the algorithm. While a large value of neighbors leads to the introduction of noise. So it is very important to find a suitable value to maintain the balance between local structure and global structure.

Usually, the efficiency of one algorithm can be reflected by its time cost. To demonstrate the efficiency of this algorithm in recognizing oracle bone inscriptions images, we recorded the running time of tests with different numbers of samples $n$, and $n=100,200,400,600,800$. The recorded results are illustrated in Figure 10 .

It can be observed that as the sample size increases, the time shows an exponential growth trend. The time cost is 5 seconds when we have 100 samples, while it is 86 seconds when we have 800 samples. The results indicate that the algorithm can be directly executed on small-scale datasets. While dealing with large-scale data, it is necessary to introduce approximate computation strategies that reduce the number of shortest paths to reduce time consumption.

In order to compare the advantages of this algorithm, we apply several other algorithms of manifold learning algorithm as a comparison reference in the paper. We also obtain the low dimensional embedding of different algorithms at same sample set, which are shown in Figures 11-13 .

Similarly, we get the average accuracy rate of the responding algorithm in Figure 14 and Figure 15 .

According to the experimental data, we obtain the optimal accuracy rate and time-consuming of all manifold learning algorithms to recognize the images, which are shown as follows.

In Figure 16 , it can be observed that the ISOMAP dimensionality reduction algorithm has a significant advantage in the average recognition rate. This is because the method effectively maintains the geodesic distances of the sampling points in high-dimensional space after dimensionality reduction, thereby preserving the nonlinear characteristics of the data. From Figure 17 , it can be concluded that the time taken by the ISOMAP algorithm is relatively slow, being only faster than the LLE algorithm. This is due to the high computational complexity of ISOMAP algorithm, which requires more time and resources. We can conclude that although the ISOMAP algorithm has certain disadvantages in terms of time consumption, its performance advantages allow it to recognize images more accurately, which is exactly what we expect.

In the experiment, the low-dimensional embeddings of the sample set were first calculated using the ISOMAP algorithm. Subsequently, the results of the low-dimensional embeddings were visualized and analyzed for the effects of dimensionality, indicating that the algorithm demonstrates a better recognition rate at lower dimensions. Next, we investigated the impact of parameter N on recognition accuracy under different neighbors, obtaining the average recognition rates under various parameters after multiple experiments, and compared these results with those of the PCA, MDS, and LLE algorithms. Finally, we analyzed and compared the experimental results, where the ISOMAP algorithm achieved the best recognition rate of 93.3%. Although the ISOMAP algorithm has certain disadvantages in terms of running time, its performance advantage in recognition accuracy allows it to effectively complete the recognition task.