Applications of Multiscale Geometric Analysis in Image Texture Recognition and Classification

Compared to traditional wavelet transform methods, although wavelet transforms can perform multiscale decomposition, their ability for directional decomposition is limited and often insufficient to capture complex texture directions in images. Therefore, this study introduces the Contourlet transform to optimize the image texture feature extraction process. Compared to wavelet transforms, the Contourlet transform not only retains the characteristics of multiscale decomposition but also provides richer directional information, which is crucial for revealing the subtle texture structures in images. This high directional sensitivity is especially important in complex image texture analysis, such as vegetation cover in natural scenes or pathological tissues in medical images, significantly enhancing the discriminability and efficiency of texture features.

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Figure 1. Principle of the discrete Contourlet transform filter bank decomposition

The discrete Contourlet transform, also known as the pyramidal directional filter bank (DFB), is a composite filter structure consisting of a Laplacian pyramid (LP) and a DFB. This transform is particularly important in the application of image texture feature extraction because it can effectively capture the texture information and local details in images. In the implementation of the Contourlet transform filter bank decomposition, the LP is used to decompose the original image into approximate low-frequency components and high-frequency detail components. This decomposition helps reveal the basic structures and singularities in the image, such as texture edges and breakpoints. By recursively decomposing the approximate component further with LP, the system can extract the texture and structural information of the image at multiple scales, which is crucial for complex texture analysis. Figure 1 shows the principle of the discrete Contourlet transform filter bank decomposition.

The implementation process of the Contourlet transform filter bank decomposition involves using the DFB to perform multi-directional decomposition on the high-frequency detail components obtained after LP decomposition. At this stage, singularities located in the same direction are merged into single coefficients, forming the Contourlet transform coefficients. As the scale increases, the number of directions in the Contourlet transform also increases, allowing the transform to capture finer texture details at various scales and directions. The coefficients in each direction have a gear-shaped support base, which is very suitable for capturing lines and curves in the image, such as texture stripes, thus effectively depicting the texture contours of the image. This method is particularly suitable for dealing with complex texture features in images, such as natural landscape tree textures or fabric detail textures.

Once the Contourlet transform is completed in the discrete domain, this transform can be extended to the square-integrable space of continuous functions, allowing for more detailed image analysis in the continuous domain. This process, through iterative filter bank operations, decomposes the image space into a sequence of multiscale and multidirectional subspaces, allowing for exhaustive extraction of texture features across various directions and scales. This decomposition method is particularly suitable for texture recognition and classification, as it captures the directionality and scale variations of textures, which is crucial for analyzing complex texture patterns in natural scenes such as trees, water ripple textures, or artificial environments such as fabrics and materials. Moreover, the multiscale and multidirectional processing capability of the Contourlet transform ensures that texture details from coarse to fine are effectively captured and described, greatly enhancing the effectiveness and application scope of image texture feature extraction. The orthogonal operation is denoted by ⨁, and the approximation component at the lowest level is denoted by N₀, while the detail component at scale 2^mk-1 is denoted by Q^mk_k,j, then there is:

${{L}^{2}}\left( {{R}^{2}} \right)={{N}_{0}}\oplus \left( \underset{k\le 0}{\mathop{\oplus }}\,\left( \underset{k\le 0}{\overset{{{2}^{{{m}_{k}}}}}{\mathop{\oplus }}}\,q_{k,j}^{{{m}_{k}}} \right) \right)$     (1)

If the scale, direction, and position parameters are represented by k, j, and v respectively, the continuous domain Contourlet function is denoted by {ϑ^mk_k,j,v(s)}, which gives the expression:

$\vartheta _{k,j,v}^{{{m}_{k}}}\left( s \right)=\sum\nolimits_{l\in {{c}^{c}}}{h_{j}^{{{m}_{k}}}}\left( l-a_{j}^{{{m}_{k}}}v \right){{\omega }_{k,l}}\left( s \right)$     (2)

Assuming the low-pass analysis filter is represented by h^mk_j, the framework defined in the low-dimensional space R² is denoted by ω_k,l(s), and the oversampling matrix is represented by t^mk_j, the following definitions apply:

$t_{j}^{{{m}_{k}}}=\left\{ \begin{align}  & DIAG\left( {{2}^{{{m}_{k}}}},2 \right),0\le j\le {{2}^{{{m}_{k}}-1}} \\ & DIAG\left( 2,{{2}^{{{m}_{k}}}} \right),{{2}^{{{m}_{k}}}}\le j\le {{2}^{^{{{m}_{k}}}-1}} \\\end{align} \right.$     (3)

In the application of image texture feature extraction, the high dimensionality of features typically leads to a high model complexity, thereby affecting learning efficiency. To effectively address this challenge, this paper employs the Contourlet-KSR method, which is a technique that combines the Contourlet transform with the KSR. The Contourlet transform, with its capability for multiscale and multidirectional decomposition, can extract rich image texture information; while KSR, as a nonlinear dimensionality reduction technique, not only reduces redundant features but also preserves the local geometric structure of the data, i.e., the inherent manifold structure of image texture features. This method is particularly suitable for image texture feature extraction because it not only optimizes the representativeness and classification performance of the features but also simplifies the computational process through dimensionality reduction, enhancing processing speed. Compared to other application scenarios, image texture feature extraction places a greater emphasis on capturing and describing subtle texture differences within images, which is crucial for enhancing the accuracy and efficiency of image analysis.

In the process of image texture feature extraction, the first step of applying the KSR algorithm involves constructing a neighborhood graph H, which models the low-dimensional manifold structure of image data in high-dimensional space. In this process, the graph H contains v vertices, each representing an image sample a_u. If two samples a_u and a_k belong to the same texture category, they are connected by an edge in the neighborhood graph. Such a connection strategy is based on the assumption that samples of the same category have similar texture properties and therefore should remain close in the low-dimensional space. The construction of this neighborhood graph ensures that samples which are close to each other in the original high-dimensional space remain close when mapped to the low-dimensional space, effectively reflecting the intrinsic structure and texture patterns of the original data.

The second step involves determining the weight Q of the edges in the neighborhood graph. In graph H, if there is an edge connection between vertices u and k, the weight q_uk is set to 1/ m_j, where m_j is the number of samples belonging to the same category j, with z representing the total number of categories. This weighting design aims to balance the influence of each category's samples in the graph, preventing bias in the overall dimensionality reduction results due to a larger number of samples in certain categories. If there is no edge connection between vertices u and k, the corresponding weight q_uk is 0. This weighting helps maintain the cohesion of samples within the same category during the dimensionality reduction process while allowing samples from different categories to be appropriately separated in the low-dimensional space, thus enhancing the classifier's performance and accuracy in handling image texture features.

The third step of the KSR algorithm involves calculating the response vector b, which is central to the dimensionality reduction process. Initially, imagine projecting the neighborhood graph H onto a one-dimensional space, resulting in the response vector y=[y₁,y₂....y_N]^T.

$\begin{align}  & MIN\sum\nolimits_{u,k=1,u\ne k}^{v}{{{q}_{uk}}{{\left( {{b}_{u}}-{{b}_{k}} \right)}^{2}}} \\ & s.t.\ {{b}^{S}}Fb=1 \\\end{align}$      (4)

To achieve this goal, we first construct a diagonal matrix F, where the diagonal element F_uu equals the weighted degree f_u of vertex u, with f_u being the sum of the weights q_uk between vertex u and all other vertices. Next, we define the Laplacian matrix M as M=F-Q, where Q is the weight matrix. This equation is obtained through a linear transformation:

$\begin{align}  & {{b}^{*}}=\underset{{{b}^{S}}Fb=1}{\mathop{ARGMIN}}\,{{b}^{S}}Mb=ARGMIN\frac{{{b}^{S}}}{{{b}^{S}}Fb} \\ & \text{   }=ARGMIN\frac{{{n}^{S}}Qn}{{{n}^{S}}Fn} \\\end{align}$      (5)

The Laplacian matrix crucially reflects the topological structure of the graph and is used to find a low-dimensional representation that preserves the local structure of the original data as much as possible. Solving the generalized eigenvalue problem of this matrix yields the first z largest generalized eigenvectors b₁, b₂, ..., b_z, which define the new coordinate axes mapped to the low-dimensional space. In the application of image texture feature extraction, these vectors help us capture the most representative texture features and maintain the relative positions and distances between samples in the original high-dimensional space, effectively revealing the internal texture structure and changes in the image. The corresponding generalized eigenvalue problem is:

${{Q}_{b}}=\eta Fb$      (6)

The fourth step involves the implementation of regularized kernel least squares, which is used to optimize and precisely calculate the features expressed after dimensionality reduction. In this step, we construct a linear function b_u=d(a_u)=β^Sa_u, where β is a coefficient vector obtained by solving a regularized least squares problem. x_j is the solution to the following regularized least squares problem:

${{x}_{j}}=\underset{x}{\mathop{ARGMIN}}\,\left( \sum\nolimits_{u=1}^{v}{\left( {{x}^{S}}{{a}_{u}}-b_{u}^{j} \right)+\alpha {{\left\| x \right\|}^{2}}} \right)$      (7)

In practice, this optimization problem is converted into solving a set of linear equations, which are expressed in a kernelized form, transforming the original problem into a regularized kernel least squares problem:

$\left( a{{a}^{T}}-\alpha U \right){{x}_{j}}=a{{b}_{j}}$     (8)

The kernel method, by introducing a kernel function J(a,a_u), allows us to perform linear learning in a high-dimensional feature space while the actual calculations are still performed in the original dimensions.

$\left( J+\alpha U \right){{x}_{j}}={{b}_{j}}$     (9)

Thus, the final function d(a) is expressed as d(a)=Σ^v_u=1x^j_uJ(a,a_u), where a^k_i is the solution to the regularized kernel least squares problem.

$\underset{d\in {{G}_{j}}}{\mathop{ARGMIN}}\,\left( \sum\nolimits_{u=1}^{v}{{{\left( d\left( {{a}_{u}} \right)-b_{u}^{j} \right)}^{2}}}+\alpha {{\left\| d \right\|}^{2}}_{{{G}_{J}}} \right)$     (10)

The fifth step involves using the eigenvectors obtained in previous steps to construct the projection matrix Φ=[x₁,x₂....,x_z-1]∈R^v×z-1, which is obtained by solving the least squares problem. This step is critical in the entire dimensionality reduction process, mapping the high-dimensional image data to a z-1 dimensional subspace:

a→c=Φ^SJ(:,a)     (11)

This projection matrix Φ not only simplifies the dimensionality of the data but also preserves the important structural features of the original data, ensuring that the dimensionality-reduced data still retains key information about image texture. In this way, the sample data is embedded into a lower-dimensional space, effectively extracting the most critical features for classification recognition, while reducing computational complexity and enhancing processing efficiency.

In the application of the Contourlet-KSR feature extraction method to image texture feature extraction, parameter settings should be carefully considered to ensure the method can effectively process and analyze various texture types. First, for the setting of Contourlet transform parameters, adjustments need to be made based on the complexity and diversity of different image textures to optimize the performance of multiscale and multidirectional decomposition, thereby capturing more detailed and distinctive texture features. Second, in the implementation of the KSR algorithm, the construction of the neighborhood graph uses Euclidean distance to measure the distance between sample points, as Euclidean distance provides an intuitive and practical measure of similarity in most texture analysis scenarios. The weights of the edges are set using the heat kernel, with the kernel parameter set to 3, which helps to strengthen the connections within local neighborhoods, ensuring local continuity of texture structure and overall class cohesion in the image.

Finally, a support vector machine (SVM) is used for the final classification, where the SVM's kernel parameter is selected within a range from 0 to 4 in steps of 0.01. Such fine-tuning helps find the optimal model complexity, balancing the risks of overfitting and underfitting, while the penalty factor C remains default, providing a standard level of regularization. These parameter settings work together, aiming to maximize the recognition capability and classification accuracy of texture features extracted from complex image data while maintaining manageable computation.

In the experimental steps of applying the Contourlet-KSR feature extraction method for image texture feature recognition, the process can proceed as follows, as detailed in Figure 2:

(1) Perform a Contourlet transform on each image, decomposing it using multiscale and multidirectional methods to obtain several directional subbands and a low-pass subband.

(2) Calculate the mean and variance of each subband image and integrate these statistics into a one-dimensional long vector, see Figure 3 for the process.

(3) Vertically concatenate the statistical feature row vectors of all images to form a Contourlet feature matrix.

(4) Normalize the feature matrix and divide the samples into a training set and a test set, typically using even-numbered samples for training and odd-numbered samples for testing.

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Figure 2. Flowchart of image texture feature recognition process

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Figure 3. Flowchart of the process for acquiring a one-dimensional long vector

(5) Apply the KSR algorithm to perform dimensionality reduction on the feature matrix of the training set, and use the obtained projection matrix to perform the same reduction on the feature matrix of the test set.

(6) Normalize the feature matrices of the dimensionally reduced training and test sets to standardize the data input format, enhancing the performance of the classification algorithm.

(7) Finally, input the normalized features and their corresponding category labels into a SVM for final classification, with its efficient boundary decision function, SVM can accurately distinguish between different texture categories.

In comparing the experimental results of different image texture feature extraction methods, we can observe that each technique performs variably in terms of feature dimension, time consumed, recognition rate, and recognition rate after KSR dimension reduction (see Table 1). Wavelet transformation, with a feature dimension of 31 and a time consumption of 28, achieved a recognition rate of 92.31%, which increased to 93.45% after KSR dimension reduction. Local Binary Patterns (LBP) had a higher feature dimension (245) and a time consumption of 31, with an original recognition rate of 81.2%, which improved to 87.56% after KSR reduction. Gabor filters showed the best time efficiency (21) and reached a recognition rate of 91.25% with a feature dimension of 241, slightly improving to 92.31% after KSR reduction. Dual-Tree Complex Wavelet Transform (DT-CWT), with the lowest feature dimension (25) and a slightly higher time consumption (33), achieved a recognition rate of 93.54%, which increased further to 95% after KSR reduction. The method discussed in this paper, although slightly higher in time consumption and feature dimension (51), approached the optimal recognition rate at 93.5%, reaching 95.69% after KSR reduction, showing excellent performance. From the data analysis, it is evident that the Contourlet-KSR image texture feature extraction technique effectively combines the multiscale and multidirectional properties of the Contourlet transform with the spectral response characteristics of Krawtchouk polynomials, significantly enhancing the feature's descriptive ability and interference resistance. Despite higher feature dimension and time consumption, the use of KSR dimension reduction technology maintained a high recognition rate and even improved it to 95.69%, surpassing other traditional methods in performance.

Table 2 displays the experimental results of the Contourlet-KSR image texture feature extraction method under different configurations of decomposition levels and direction numbers. The data shows that with the increase in decomposition levels and changes in direction numbers, feature dimension, feature extraction time, and classification time all increase, but the corresponding recognition rates also improve. With three decomposition levels and a direction number configuration of (8,8,8), the feature dimension was 51, feature extraction time was 62 milliseconds, classification time without dimension reduction was 152 milliseconds, and the recognition rate was 92.56%, which significantly increased to 96.82% after using KSR dimension reduction. In four decomposition levels, using a direction number configuration of (4,8,8,16), the feature extraction time reached 81 milliseconds, classification time without dimension reduction was 178 milliseconds, and the recognition rate was 93.68%, stabilizing at 96.87% after KSR reduction. These data suggest that by increasing decomposition levels and adjusting direction numbers, classification accuracy can be significantly improved at the expense of some computational efficiency. From these experimental results, it can be concluded that the application of Contourlet-KSR technology is very effective in image texture feature extraction. Although feature extraction and classification time slightly increase with the addition of decomposition levels and direction numbers, the significant improvement in recognition rates demonstrates the advantage of this method in enhancing image texture recognition accuracy. Particularly, the use of KSR dimension reduction technology not only substantially reduces the feature dimension but also maintains a lower classification time while achieving a very high recognition rate.

Table 1. Experimental results of different image texture feature extraction methods

Table 2. Experimental results of the contourlet-KSR image texture feature extraction method

Figure 5 shows the result distribution after processing the test set data using the Contourlet-KSR feature extraction method, which is a visualization representing features projected onto a two-dimensional plane. In this figure, different colors represent different image texture categories. Observations show that the data points of smooth textures are highly concentrated, exhibiting the least intra-class dispersion, indicating that images of smooth textures have high similarity and consistency in feature space. Granular textures are slightly more dispersed but still concentrated. In contrast, line textures show the greatest intra-class dispersion, indicating significant variability in feature expression for this category. Despite this, the inter-class dispersion among the three texture categories is similar, indicating that although each texture type varies in intra-class similarity, they are distinctly distinguishable from each other in feature space. These observations validate the effectiveness of the Contourlet-KSR feature extraction technique in the task of image texture classification. This method not only clearly distinguishes between different categories of textures but also reveals the continuous variations within similar textures. Therefore, the multiscale and multidirectional characteristics of the Contourlet-KSR technique, combined with the spectral response characteristics of Krawtchouk polynomials, successfully enhance the descriptive power of texture features and strengthen the robustness of the classification model.

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Figure 5. Data distribution after KSR dimension reduction

Table 3 details the image texture classification results using the texture image classification method based on domain MCM proposed in this paper. These data display the specific recognition effects of different texture types at an overall recognition rate of up to 97.8%. From the table, it is evident that the recognition rates for line and irregular textures reached 100%, demonstrating that the method is highly effective for these distinctly characterized texture types. The recognition rate for wave textures is also very high, at 98.58%. Although the recognition rates for smooth and regular textures are slightly lower, at 92.31%, they still perform well overall. Other texture types such as rough and mesh textures also showed high recognition rates exceeding 96%. These results indicate that the method maintains high accuracy even in complex texture classification scenarios. Analysis of the data in Table 3 allows us to conclude that the texture image classification method based on domain MCM performs with high precision and robustness in practical applications. This method effectively utilizes the multiresolution co-occurrence characteristics of images to improve classification accuracy, especially when dealing with images with complex texture features. Additionally, the high recognition rates and low misclassification rates demonstrated when processing various texture types confirm the practicality and reliability of this method in texture classification tasks.

Table 4 provides the correlation matrix data between non-subsampled wavelet subband energy and MCM features (entropy, contrast, and correlation), reflecting the degree of association between these features across different wavelet subbands. The data show that wavelet energy maintains high correlation with entropy across all subbands, especially in the HH1 and HH2 subbands, where the correlation coefficients are 0.9654 and 0.9574, respectively, indicating a very strong positive correlation. Although the overall correlation with contrast features is slightly lower, it also reaches 0.9321 and 0.9236 in the HH1 and HH2 subbands, respectively, indicating a strong association with energy features. In contrast, the correlation coefficients between wavelet energy and correlation are lower across all subbands, particularly lowest in the HH1 subband at only 0.3569. These data reveal the close relationship between wavelet energy and entropy and contrast in texture analysis, while the low correlation with correlation may point to the diversity and complementarity of feature extraction.

Table 3. Image texture classification results when recognition rate is 97.8%

Table 4. Correlation matrix between non-subsampled wavelet subband energy and multiresolution co-occurrence features

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Figure 6. Image texture classification results before and after KSR dimension reduction

These experimental results emphasize the effectiveness of the non-subsampled wavelet transform in extracting image texture features, particularly its high correlation with entropy and contrast, providing a powerful tool for texture analysis. The high correlation of wavelet energy with entropy and contrast supports their capability to capture the complexity and clarity of image textures, which are key factors for effective texture classification. The texture image classification method based on domain MCM, by combining these strongly correlated features, can more accurately distinguish different texture types, enhancing classification precision and robustness.

Figure 6 details the recognition rates on the training and test sets in image texture classification tasks under different kernel parameter settings, before and after optimization. Before optimization, the recognition rate on the training set continually increased with the kernel parameter, eventually stabilizing at a high level of 99%. The test set also showed a similar trend, initially at 82% and gradually increasing to a peak of 100%. This indicates that as the kernel parameter is adjusted, the model becomes more precisely adapted to and predictive of different texture types. However, post-optimization data showed a different trend, with a slight decrease in the training set's recognition rate, mostly stabilizing around 96.8%, while the test set's recognition rate significantly dropped from an initial 93.8% to a low of 88.6%. This suggests that the optimization process may have led to overfitting, where the model is overly tuned to the training data, reducing its generalization ability on unknown data. These experimental results suggest that while the model's performance on the training set may slightly improve under some parameter settings after optimization, its performance on the test set indicates that the optimization strategies may not have effectively improved the model's generalization capability. This phenomenon emphasizes the need for careful handling of overfitting issues in model training and optimization in image texture classification tasks, especially when dealing with multiple parameters and high-dimensional feature spaces. The Contourlet-KSR technique and the method based on domain MCM, intended to enhance the descriptive power of features and robustness of classification, suggest that more detailed parameter adjustments and model validation processes might be necessary to ensure the practicality and effectiveness of the developed models.

Table 5. Comparison of texture image classification accuracy rates for various features

Table 5 compares the accuracy rates of various features in texture image classification tasks, including individual spatial gray-level co-occurrence features, wavelet energy features, their combinations, and MCM with uniform and non-uniform quantization. The data reveal that using spatial gray-level co-occurrence features alone resulted in the lowest accuracy rate of 74.23% with a standard deviation of 2.14%, indicating that while this method has a baseline effectiveness, it does not perform particularly well with complex texture images. Using wavelet energy features alone significantly improved the accuracy to 91.25%, showing a higher effectiveness. Combining spatial gray-level co-occurrence features with wavelet energy yielded an accuracy rate almost the same as using wavelet energy alone, indicating that adding spatial gray-level co-occurrence features did not bring additional improvements. The accuracy rates for MCM uniform quantization and non-uniform quantization were 85.34% and 92.34%, respectively, with non-uniform quantization performing better, showcasing the potential of this technology. The highest accuracy came from the combination of wavelet energy and MCM features, reaching 93.27%, also with a low standard deviation of 1.08%, indicating optimal stability and efficiency.

These experimental results highlight the effectiveness and superiority of the texture image classification method based on domain MCM proposed in this paper. In particular, the combination of wavelet energy and MCM features not only increased the classification accuracy but also ensured the stability of results, suggesting that integrating these two technologies can more effectively capture and utilize texture information for precise classification. Moreover, the performance of non-uniform quantization of MCM features being better than uniform quantization further proves the importance of considering non-uniform strategies in texture feature extraction and classification.