Topological insights and hybrid feature extraction for breast cancer detection: a persistent homology classification approach

Related works

Persistent Homology (PH), a core method in TDA, has gained increasing attention in medical imaging due to its ability to capture intrinsic structural features of complex data. By encoding both local and global topological properties, PH has been applied across diverse medical domains. For example, Ren et al. (2023) applied PH to ECG analysis, Yadav, Nisha & Coskunuzer (2023) to musculoskeletal imaging, Tan et al. (2023) to brain signals, and Chen et al. (2023) to retinal image analysis. In oncology, Hahn et al. (2021) studied DNA repair in breast cancer cells, while Malek et al. (2023) applied PH to mammography microcalcifications.

In breast cancer classification, Anjum et al. (2021) combined histogram of orientedgradients (HOG) and Canny edge detection with support vector machine (SVM) achieving 94% accuracy. Similarly, Mu’jizah & Novitasari (2021) integrated HOG, principal component analysis (PCA), and SVM, reporting 100% sensitivity and 97.5% specificity. These works demonstrate that hybrid methods leveraging gradient-based and topological features can enhance diagnostic performance.

Despite these advances, state-of-the-art approaches still rely heavily on convolutional neural network (CNN)-based deep learning models (El Houby & Yassin, 2021; Karthiga, Narasimhan & Amirtharajan, 2022; Jabeen et al., 2023; Yadav & Kumar, 2025), with some exploring ensemble-based learning (Shah et al., 2024; Haq et al., 2022; Batool & Byun, 2024) which, although effective, require extensive preprocessing and computational resources, limiting their scalability in real-world settings.

Research gap

To date, limited studies have systematically explored the integration of PH with hybrid feature extraction and dimensionality reduction methods for mammography classification. Existing works either focus exclusively on deep architectures (Jabeen et al., 2023; Yadav & Kumar, 2025) or apply PH in isolation without leveraging its synergy with descriptors like HOG. This leaves a gap for computationally efficient yet accurate frameworks that combine PH with conventional feature extraction to provide reliable alternatives to deep networks. Addressing this gap, our study introduces a persistent homology classification algorithm (PHCA)-based classification framework with HOG and PCA, aiming to deliver strong diagnostic performance while maintaining efficiency and scalability.

In summary, the combination of PH with hybrid feature extraction methods such as HOG and dimensionality reduction techniques such as PCA is a promising approach for breast cancer classification. These methods enable the extraction of rich topological features from high-dimensional image data, contributing to more accurate and efficient diagnostic tools. The major contribution of this study is the development of a novel breast cancer classification framework that integrates HOG for feature extraction, PCA for dimensionality reduction, and PHCA for classification. Unlike deep learning approaches, the proposed method provides competitive performance while being computationally efficient, making it suitable for large-scale screening applications. This work further demonstrates the potential of topological methods in medical image analysis, effectively bridging the gap between traditional feature-based and deep learning approaches.

Dataset

The dataset used in this article is extracted from the INbreast database, the summarized description of which is presented in Table 1. It consists of 7,632 mammogram images categorized into two classes: 2,520 benign and 5,112 malignant images from Huang & Lin (2020). The mammography images in the INbreast database were originally collected from the Centro Hospitalar de S. Joao (CHSJ) Breast Center in Porto. The database contains data collected from August 2008 to July 2010 and includes 115 cases with a total of 410 images (Moreira et al., 2012). Of these, 90 cases concern women with abnormalities in both breasts. Four different types of breast disease are recorded in the database: Mass, calcification, asymmetries and distortions. The mammograms are recorded from two standard perspectives: Craniocaudal (CC) and Mediolateral Oblique (MLO). In addition, breast density is classified into four categories based on the BI-RADS standards: Fully Fat (Density 1), Scattered Fibrous-Landular Density (Density 2), Heterogeneously Dense (Density 3) and Extremely Dense (Density 4). The images are stored in two resolutions: 3,328 $\times$ 4,084 pixels or 2,560 $\times$ 3,328 pixels, in DICOM format. A total of 106 mammograms depicting breast masses were selected from the INbreast database.

To enhance the dataset for model training, data augmentation techniques were applied. Image preprocessing method using contrast limited adaptive histogram equalization (CLAHE) was used to the 106 mammograph scans (Huang & Lin, 2020), making the lesions more detectable but may cause feature distortion. Then, multiple-angle rotation ( $\theta ={30}^{\circ }-{360}^{\circ }$ with ${30}^{\circ }$ increments) and image flipping were then implemented, imitating variations in breast orientation which helps the model to train on generalized acquisition angles. From these steps, the dataset is increased from 106 to 7,632 mammography images.

Each image is preprocessed to ensure a uniform size and format. In particular, all images are resized to 224 $\times$ 224 pixels to ensure uniformity across the dataset and converted to grayscale to reduce the computational complexity associated with multi-channel (RGB) data.

Grayscale conversion is particularly important in medical imaging. In this way, the intensity values of the images are preserved while the color information, which is not relevant for this classification task, is discarded. By focusing exclusively on the pixel intensity, the computational effort is significantly reduced without losing important information. In addition, data augmentation techniques, including image rotation, have been applied to increase the variability of the dataset. The expansion helps to prevent overfitting during model training by exposing the model to a wider range of image variations.

Feature extraction using histogram of oriented gradients

HOG is a feature descriptor used to define an image by the pixel intensity and intensities of gradients of pixels (Madhuri & Negi, 2023). It does so by computing for edges and shapes on a dense grid of uniformly spaced cells and improves its performance by using overlapping contrast normalization. Particularly, the method divides the image into a small spatial region called cell. The combination of magnitude and angle derived from each cell will be accumulated to develop the $1$-dimensional histogram of gradients. Then, the contrast normalization is performed on the local histogram by accumulating the derived 1-dimensional histogram of gradients of each cell performed by relatively larger spatial regions called blocks. In total, three parameters are usually set for HOG, namely: dimension of a cell, dimension of a block, and number of orientations.

One of HOG’s main advantages is its robust nature in capturing local edge and shape information, effective for distinguishing structural patterns from subtle texture variations. This makes it useful specific to identifying tumors where boundary and margin details are crucial diagnostic features. However, despite HOG being easy to implement and more computationally efficient than deep learning approaches, its performance is sensitive to parameter selection. It also results with high-dimensional descriptors, requiring the need for dimensionality-reduction methods.

Dimension reduction using principal component analysis

Principal component analysis (PCA) aims to take high-dimensional dataset to a lower-dimensional form while minimizing loss of information. It aims to find uncorrelated variables of features derived from the original dataset that successively maximize the variance. New variables or principal components are derived similar to the eigenvalue/eigenvector problem. Its theoretical root is elaborated by Jolliffe & Cadima (2016). Its essence lies in the fact that reducing the number of dimensions which provides the best separation between classes will improve the classification process. By retaining the principal components that capture most of the variance in the data, PCA ensures that the model remains both robust and accurate even with complex, high-dimensional inputs (Bakar et al., 2023).

In breast cancer recognition, PCA can help improve HOG-extracted descriptors by reducing it to lower-dimensional features. In this process, classification becomes less prone to overfitting while still preserving most of the explained variance of the original data. However, due to its process, PCA may remove clinically relevant low-variance descriptors which may affect classification performance, requiring a caution in the parameter (% of explained variance) used.

Feature scaling

Feature scaling is a technique under data normalization. Its relevance is already established and its impact have been validated to improve classification performance in various fields, such as medical data classification, multimodal biometrics systems, vehicle classification, faulty motor detection, predicting stock market, leaf classification, credit approval data classification, genomics, and some other application areas (Singh & Singh, 2020). In this article, the Standard, Minmax, and Robust scalers were compared as feature scaling methods.

The standard scaler follows from the standard normal distribution which makes the mean across instances of each feature to be 0 and scales the data to unit variance. That is, for each feature $i$, the updated $i$th feature values of all instances are calculated as

(6) $$X_i^{\mathrm{\prime }}=\frac{X_i-X_{\mu }}{X_{\sigma }}$$where $X_i$ are the original feature values, and $X_{\mu }$ and $X_{\sigma }$ are the mean and standard deviation of $X_i$, respectively. Meanwhile, the minmax scaler translates each individual feature in such a way that the range of the translated values falls within 0 and 1. This is computed as

(7) $$X_i^{\mathrm{\prime }}=\frac{X_i-X_{min}}{X_{max}-X_{min}}$$where $X_i$ are the original feature values, and $X_{min}$ and $X_{max}$ are the minimum and maximum of $X_i$, respectively. Finally, the robust scaler was used to scale the features using statistics that are robust to outliers. This does so by removing the median of each feature, scaling the features according to the quartile ranges, and centering independently on each feature.

Persistent homology classification algorithm

PHCA, a supervised classification algorithm that utilizes algebraic topology in analyzing the shape or topological properties of data, was used to classify the breast cancer tumor images to either benign or malignant. De Lara (2023) presented a comprehensive discussion of the theoretical basis behind PHCA. Detailed here is a brief breakdown of how the algorithm works.

• a.

  Input: Training data

  • •

    The training data X consists of points in $(n+1)$-dimensional space, where the first $n$ dimensions represent attributes and the $(n+1)$-th dimension gives the class label.

  • •

    The data is divided into $k$-classes $X_1,X_2,\dots ,X_k$, with each class being a point cloud. For this study, there are only two classes or values for k, benign or malignant.

• b.

  Computing persistent homology

  • •

    For each class $X_i$, the persistent homology is computed which captures the topological features, particularly the connected components, of the point cloud at different resolutions.

  • •

    The persistent homology of a point cloud is represented by a matrix $\mathcal{P}(X_i)$ where each row records a topological feature, including:

    • –

      The dimension of the feature (0 for connected components, 1 for loops, 2 for voids, etc.).

    • –

      The birth and death times, indicating when the feature appears and disappears during the filtration process.

• c.

  Filtration and parameter setting

  • •

    A filtration is a process that analyzes the point cloud at various scales, where smaller scales capture finer details, and larger scales capture broader structure.

  • •

    The maximum scale $maxsc$ is chosen based on the data, often as half the maximum distance between any two points in the cloud.

• d.

  Classifying a new point

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    Given a new point $p$ that needs to be classified, the algorithm computes the persistent homology of the augmented point clouds $Y_i=X_i\cup \{p\}$ for each class $i$.

  • •

    The algorithm then measures the change in topological features between $\mathcal{P}(X_i)$ and $\mathcal{P}(Y_i)$.

• e.

  Score function

  • •

    For each class $i$, a score is computed based on the difference in the lifespan of the topological features before and after introducing $p$ to the point cloud.

  • •

    The score is calculated as (8) $$Score(X_i)=\left|\sum _{a\in Y_i}l(a,1)-\sum _{b\in X_i}l(b,1)\right|$$where $l$ is the map that sends a basis element with birth index of $1$ to its death index.

• f.

  Classification decision

  • •

    The class that is least impacted by the addition of the new point (i.e., the class with the smallest change in lifespan of the topological features) is selected as the predicted class for $p$.

  • •

    Mathematically, $p$ is classified under class $j$ if $Score(X_j)\le Score(X_i)$ for all $i\in \{1\dots ,k\}$.

In summary, PHCA uses persistent homology to analyze the topological structure of different classes in the training data. By evaluating how much the persistent homology changes when a new point is added to each class, the algorithm determines which class the point most likely belongs to. The class with the least topological disturbance due to the addition of the point is the predicted class for the new observation. Algorithm 1 shows the pseudocode for PHCA.

Performance evaluation

To evaluate the performance of PHCA, five metrics are obtained. The values of these metrics rely on the confusion matrix corresponding to the predicted classes.

Overall framework

The framework of the implementation is presented in Fig. 1. All 7,632 grayscale images (Fig. 1A) are split into 80:20 training/validation and test sets (Fig. 1B). Using the training/validation set, the goal is to determine which combination of HOG parameters and scaler produces the best result (Fig. 1C). In this way, the experiment guarantees the best features extracted from the images. To do this, a grid search cross-validation with five folds is employed. This allows the comparison of different combinations of parameters and scaler shown in Table 3.

As an example, the first combination of parameters and scaler is 4 × 4 pixels per cell (PPC), 2 × 2 cells per block (CPB), and Standard scaler. Using 5-fold cross-validation, the training and validation folds undergoes feature extraction using HOG with 4 × 4 PPC, 2 × 2 CPB, and nine orientations. Then, the extracted HOG descriptors is reduced using PCA with a standard 95% parameter. Finally, the reduced features are scaled using Standard scaler and the output of which is used to train the PHCA model. For each fold, the F1-score is calculated and the average F1-score across the five folds is obtained. This serves as the “score” of the HOG parameter and scaler combination in the tuning process.

The parameter and scaler combination having the best average F1-score is then used to predict the 20% training set. The corresponding HOG parameters of this combination are used to extract the HOG descriptors (Fig. 1D). Then, using PCA with 95% parameter and the corresponding scaler in the best combination, the final features are obtained (Fig. 1E). Finally, classification is implemented using PHCA on these features (Fig. 1F). The output of PHCA is a vector of binary values (0 or 1) representing the predicted class of each test data. The performance of PHCA is then quantified using five evaluation metrics: precision, recall, F1-score, specificity, and overall accuracy.

The specified combinations in Table 3 are declared based on studies that used HOG for similar application. The PCA is used with a standard 95% parameter to preserve at least 95% of total variance of the dataset. This is set based on common practice on use of the algorithm. In this way, dimension reduction capability of PCA is maximized while minimizing the information loss. Similar strategy is set for the orientations parameter of HOG, using a standard of nine in the experiment.

The implementation in this article is done on a computer with the following configurations: 3,401 MHz AMD Ryzen 9 5950X 16-core processor, NVIDIA GeForce RTX 3080 graphics cards, 64 GB 3,200 MHz RAM, and a 1 TB solid-state drive. For reproducibility, the repository code is available at https://doi.org/10.5281/zenodo.14862036 (Jetomo, 2025).

HOG parameter and scaler tuning

The 80% training/validation set is used to identify the best combination of HOG parameters (PPC and CPB) and scaler in terms of average F1-score. Table 4 shows the implementation result on this tuning stage. The table shows that the average F1-score ranges from 90.58% to 95.74% which indicates good overall performance of the data preprocessing algorithms used. On average, the combinations having $8\times 8$ PPC with variations in CPB and scaler resulted in the least average F1-scores (90.82%). Meanwhile, the combinations with $4\times 4$ and $16\times 16$ PPC obtained at least 95% average F1-scores, considering the margin of errors. For CPB, the $3\times 3$ parameter obtained a better overall average F1-score of 93.94% than $2\times 2$ having 93.64%. Additionally, the standard, minmax, and robust scalers obtained an average F1-score of 93.75%, 93.93%, and 93.70%, respectively, indicating no significant differences in their contribution to the result.

Comparing all combinations altogether, both $4\times 4$, $3\times 3$, standard and $16\times 16$, $3\times 3$, minmax combinations obtained the highest average F1-score of 95.74%, with the former being more robust due to its smaller margin of error. Due to this result, we use both combinations in classifying the test set and compare their performance in terms of the various classification metrics and computational efficiency.

Classifying the test set

In the final classification stage, the PHCA model is trained using the 80% or 6,105 training images used for tuning the HOG and scaler. Then, the 20% or 1,527 test images were predicted using the trained PHCA model. In the test set, 504 images are benign while 1,023 images are malignant, capturing the distribution of the classes in the entire dataset. The results of this classification is shown in Table 5. The table highlights the final classification result of the two best HOG parameter and scaler combinations in the tuning stage: combination A represents the $4\times 4$ PPC and standard scaler and combination B represents the $16\times 16$ PPC and minmax scaler, both of which has $3\times 3$ CPB.

For combination A, the number of HOG descriptors obtained are 236,196. Reducing this using PCA, the final feature vector size is 2,432. On the other hand for combination B, the HOG descriptors has dimension 11,664 while the final feature vector size is 739, significantly smaller than the first (see Table 6). Table 5 shows that the classification scores of the two combinations are consistently high, both for class 0 (benign) and class 1 (malignant) images.

Initial analysis on the scores shows that combination A and B have variations in performance per metric. For precision, combination A obtained a higher value for class 0 (97.03%) than class 1 (95.64%) whereas combination B obtained higher for class 1 (98.23%) than class 0 (95.48%). Recall scores reflect a different observation. Here, both combinations A and B obtained higher values for class 1. However, the difference in recall score of combination A for benign and malignant images is $\sim 8\mathrm{\%}$ whereas the same quantity for combination B is $2\mathrm{\%}$. This indicates robustness of combination B’s precision against data imbalance as noted that there are twice as much malignant than benign test images. The same observation is true for F1-score. Combination A still resulted with a higher difference in F1-scores for class 0 and 1 than combination B. This additionally indicates that robustness is more observed in combination B in terms of F1-score, a metric that incorporates both precision and recall.

Specificity tells a different story. For combination A, a higher specificity score is obtained for class 1 while combination B scored higher for class 0. However, in class 0, combination B still obtained higher specificity scores than combination A. The overall accuracy value further highlights how combination B outperforms combination A by scoring 97.31%, 1.24% better than the other. From these analyses, we’ll dig further our discussion on the result of combination B. The confusion matrix of which is provided in Fig. 2.

Computational complexity analysis

The computational times of the PHCA model together with two HOG parameters and scaler combinations are monitored and summarized in Table 6. The corresponding feature dimensions for each stage of the data preprocessing are also provided. From the result, it is clear that the choice of data preprocessing technique, either of the two combinations, had a major impact on the overall computational cost.

Combination A required 1,078.03 s or approx. 18 min to extract the features while combination B did the same task in only 128.52 s or approx. 2.14 min. From these values, it shows that data preparation takes at least 8 times longer for combination A than B. This reflects the added computational complexity of a more granular PPC parameter of the HOG algorithm— $4\times 4$ for combination A compared to $16\times 16$ for B. Using smaller cell size implies that gradient orientations are computed at a finer spatial resolution, producing a larger number of descriptors—almost 240,000 for combination A while only 12,000 for combination B.

On the other hand, the training time of both combinations required only $\sim 3.4$ s or 0.05 min. This indicates that the preprocessing approach, at least between combinations A and B, does not significantly affect model training complexity even with the significant difference in size of the final feature vectors of the two combinations. This suggests that the information to distinguish between benign and malignant scans captured by the 2,432-dimensional feature vectors of combination A is also captured by the more memory-efficient 739-dimensional feature vectors of B.

Meanwhile, the predicting time of combination A took 4,837.67 s or 80.63 min to predict 1,527 test images while combination B took 4,623.51 s or 77.06 min for the same task, slightly faster than that of combination A. The $\sim 200$ s difference is also largely due to the size difference of the final feature vector of the two combinations. The PHCA model required fewer computations for a smaller-sized feature vector from combination B, resulting with a smaller prediction time per image. This suggests that the lighter data preprocessing of combination B may lead to marginally more efficient inference.

From the previous section, we highlighted how combination B outperformed combination A in terms of overall classification performance. Despite the more detailed representation of the images using combination A with the $4\times 4$ PPC, the added preprocessing time does not justify its results being similar or worse in some cases to that of combination B. For practical applications, a breast cancer diagnostic tool requires efficiency and trustworthy results. The faster overall classification runtime of combination B and its demonstrated robustness and strong performance with the PHCA model further justifies its reliability for the task.

Comparison with existing literature

In Table 7, we present the performance comparison of our proposed method with existing literature that used the same INbreast dataset. Papers cited here have implemented varying preparation techniques while some used the same 7,632 images as in this article. Majority of the existing literature utilized deep networks, specifically CNN-inspired models, to distinguish benign from malignant mammography scans.

El Houby & Yassin (2021) used 387 out of the 410 raw images where the INbreast dataset originated. Noise removal, image enhancement using CLAHE, and region of interest (ROI) extraction has been used to obtain the features from the images. A fine-tuned CNN model was trained which resulted with a recall, specificity, and accuracy of 96.55%, 96.49%, and 96.52%, respectively. A similar approach is proposed by Karthiga, Narasimhan & Amirtharajan (2022) but using Polynomial Curve Fitting as their segmentation technique to partition the input image into multiple regions that are easy to isolate and analyze, and thus construct the ROI. Classifying their extracted features using CNN with RMS Prop optimizer, the authors were able to obtain an accuracy of 96.53%. The proposed work offers improvements to these results as reflected in the obtained metrics shown in Table 5. Additionally, the proposed work did not require any noise removal, ROI extraction, or a deep network like CNN to achieve a better result, highlighting the capability of the HOG descriptors and PH-based model for the task.

More recent literature offers a more complex approach to the problem. Jabeen et al. (2023) proposes a hybrid framework to extract features from the INbreast images. The authors used their haze-reduced local-global method for contrast enhancement, extracted deep features using a pre-trained EfficientNet-b0 model, and applied feature selection technique using their Equilibrium-Jaya controlled Regula Falsi. A feature fusion strategy is applied by the authors which resulted with an accuracy of 99.60% using the Medium Gaussian SVM (MGSVM) model. With the feature selection strategy, they obtained a slightly lower accuracy (99.40%) using Cubic SVM but with a significant reduction in computation time (2.20 s from 20.01 s of MGSVM). These results clearly outperformed PHCA but the marginal performance gap of 3–4% can be clinically accepted. Also, Jabeen et al. (2023) uses a 50:50 train-test ratio with a more balanced set of images (1,216 benign and 1,120 malignant) whereas our proposed work still performed strongly on a 80:20 train-test ratio with a significantly imbalanced dataset (2,520 benign and 5,112 malignant). The proposed feature selection strategy, however, clearly outperforms the proposed work since the PHCA model predicts an image for about 3.03 s whereas that of Jabeen et al. (2023) takes 0.018 s, indicating an area of improvement for PHCA.

In Yadav & Kumar (2025), the authors proposes an advanced image-filtering method through adaptive mean filtering which enhances the quality of the mammography scans. They used the Particle Swarm Optimization algorithm to optimize a Fractional CNN (Frac-CNN) and compared this with state-of-the-art models like ResNet50, DenseNet, and CNN-SVM using the same data preparation framework. With a 70:30 train-test on the INbreast dataset identical to this article, the authors achieved 99.00% recall, 98.20% specificity, and 99.35% accuracy using the fine-tuned Frac-CNN model, outperforming the state-of-the-art models and the proposed work. The authors highlighted that the training time of Frac-CNN is 7 h on 7,632 images (or approx. 3.30 s per image), about 5–10% increase compared to standard CNNs. Comparing this with the training time of our proposed work which is 3.41 s on 6,105 images (or approx. 0.0006 s per image), indicating less complexity of the PHCA model in terms of training capability. Moreover, from the result of Yadav & Kumar (2025) on the state-of-the-art models, we can also see how the proposed work performed at par, if not better, in some of the metrics. PHCA is shown to outperform DenseNet and CNN-SVM in recall and specificity which are priority metrics for breast cancer diagnosis. The model also exceeded ResNet50 in terms of accuracy, a good indicator for a model’s classification power. This further highlights how PHCA can offer strong results with fewer parameter configurations and less complexity than deep networks.

Limitations and future directions

It is noted that the proposed work is constrained in terms of the preprocessing algorithms used, limited comparison with state-of-the art models, and application to only one dataset. The following future steps could be done to address these limitations.

• Further refinements of the approach can be done by exploring other parameters for the preprocessing techniques. A more exhaustive search algorithm can be implemented, e.g., random search cross validation, Bayesian optimization, or metaheuristic approaches like particle swarm optimization and Genetic Algorithm to identify best parameters. Other preprocessing algorithms can also be explored to identify which works best with the topology-based classifier.

• While PHCA is compared with existing literature, benchmarking PHCA with baseline methods (e.g., SVM, ensemble-based models, neural networks, etc.) using the same preprocessing approach can be implemented to have a one-to-one comparison. This comparison may focus on classification performance, computation time, and resources consumption.

• Use PHCA and the proposed approach to other datasets such as the CBIS-DDSM (Sawyer-Lee et al., 2016) and MIAS (Suckling et al., 2015) database. This can offer a more comprehensive performance evaluation and help identify which breast cancer data structures is more applicable for the model.

• The authors also aim to coordinate with radiologists or medical institutions to deploy and evaluate the proposed approach in a clinical setting. This will provide additional insights on the model’s performance in real-world diagnostic systems.