Brain Tumor Classification from 3D MRI Using Persistent Homology and Betti Features: A Topological Data Analysis Approach on BraTS2020

Faisal Ahmed

cs.CVMar 14, 2026

Brain Tumor Classification from 3D MRI Using Persistent Homology and Betti Features: A Topological Data Analysis Approach on BraTS2020

Authors: Faisal Ahmed

Organizations: Department of Data Science and Mathematics, Embry-Riddle Aeronautical University, 3700 Willow Creek Rd, Prescott, 86301, Arizona, USA.

Abstract

Accurate and interpretable brain tumor classification from medical imaging remains a challenging problem due to the high dimensionality and complex structural patterns present in magnetic resonance imaging (MRI). In this study, we propose a topology-driven framework for brain tumor classification based on Topological Data Analysis (TDA) applied directly to three-dimensional (3D) MRI volumes. Specifically, we analyze 3D Fluid Attenuated Inversion Recovery (FLAIR) images from the BraTS 2020 dataset and extract interpretable topological descriptors using persistent homology. Persistent homology captures intrinsic geometric and structural characteristics of the data through Betti numbers, which describe connected components (Betti-0), loops (Betti-1), and voids (Betti-2). From the 3D MRI volumes, we derive a compact set of 100 topological features that summarize the underlying topology of brain tumor structures. These descriptors represent complex 3D tumor morphology while significantly reducing data dimensionality. Unlike many deep learning approaches that require large-scale training data or complex architectures, the proposed framework relies on computationally efficient topological features extracted directly from the images. These features are used to train classical machine learning classifiers, including Random Forest and XGBoost, for binary classification of high-grade glioma (HGG) and low-grade glioma (LGG). Experimental results on the BraTS 2020 dataset show that the Random Forest classifier combined with selected Betti features achieves an accuracy of 89.19%. These findings highlight the potential of persistent homology as an effective and interpretable approach for analyzing complex 3D medical images and performing brain tumor classification.

Figures & tables

Panel of Figure 1 — Figure 1: Visualization of topological feature distributions using Principal Component Analysis (PCA). The plots show the first two principal components of Betti-based feature vectors derived from BraTS2020 MRI volumes: (a) Betti-0 features representing connected components, (b) Betti-1 features corresponding to loops or tunnels, and (c) Betti-2 features representing cavities. The separation between low-grade glioma (LGG) and high-grade glioma (HGG) samples demonstrates the discriminative capability of persistent homology-based topological descriptors for brain tumor classification.

Panel of Figure 2 — Figure 2: Representative 2D slices from 3D FLAIR MRI volumes in the BraTS2020 dataset. The first row shows examples of low-grade glioma (LGG) cases, while the second row shows examples of high-grade glioma (HGG) cases. These images illustrate the structural differences in tumor appearance between LGG and HGG samples used for topological feature extraction.

Panel of Figure 3 — Figure 3: Visualization of Betti functions extracted from 3D MRI volumes in the BraTS2020 dataset. Each plot shows the median Betti curve along with $40\%$ confidence bands for the two tumor classes: low-grade glioma (LGG) and high-grade glioma (HGG). The $x$ -axis represents normalized grayscale filtration thresholds, while the $y$ -axis represents the number of topological features at each threshold: connected components for Betti-0, loops for Betti-1, and cavities for Betti-2. These curves illustrate structural differences between LGG and HGG tumors captured by persistent homology.

Figure 4: Cubical sublevel filtration of an MRI slice. A grayscale 3D MRI is progressively thresholded to generate a sequence of binary images. The images $\mathcal{X}_{60}$ , $\mathcal{X}_{80}$ , and $\mathcal{X}_{120}$ correspond to threshold values of $60$ , $80$ , and $120$ , respectively. As the threshold increases, additional voxels become activated, producing a nested sequence of cubical complexes. During this filtration process, topological structures such as connected components ( $k=0$ ), loops or tunnels ( $k=1$ ), and cavities ( $k=2$ ) appear and evolve, which are subsequently captured by persistent homology.

Figure 5: Overview of the proposed TDA-based classification framework. The pipeline begins with a 3D MRI volume composed of multiple 2D slices. Persistent homology is then applied using cubical filtration to compute persistence diagrams for homology dimensions $k=0,1,$ and $2$ , capturing connected components, loops, and cavities within the image structure. These diagrams are subsequently transformed into Betti curves, which describe the evolution of topological features across the filtration scale. The resulting Betti-0, Betti-1, and Betti-2 curves are converted into fixed-length feature vectors and concatenated to form a unified topological representation for each MRI volume. Feature selection is then performed to identify the most informative descriptors, and the selected features are finally used as input to machine learning classifiers to distinguish between low-grade glioma (LGG) and high-grade glioma (HGG) in the BraTS2020 dataset.

Model	Hyperparameter	Value
Random Forest	Number of trees ( $n\_estimators$ )	300
	Split criterion	Entropy
	Minimum samples for split ( $min\_samples\_split$ )	10
	Random seed ( $random\_state$ )	0
XGBoost	Number of boosting trees ( $n\_estimators$ )	1000
	Maximum tree depth ( $max\_depth$ )	25

Table 1: Hyperparameters used for Random Forest and XGBoost classifiers.

X_{1}\subset X_{2}\subset\dots\subset X_{N_{f}}

Algorithm 1 Topological Feature Extraction and Classification for BraTS2020 MRI

Panel of Figure 6 — Figure 6: Visualization of topological feature distributions derived from persistent homology applied to 3D MRI volumes in the BraTS2020 dataset. Each violin plot shows the distribution of aggregated Betti features for the two tumor classes: low-grade glioma (LGG) and high-grade glioma (HGG). (a) Betti-0 features represent connected components, (b) Betti-1 features correspond to loops or tunnels, and (c) Betti-2 features capture cavities within the MRI volume. Differences in the distributions between LGG and HGG indicate that topological descriptors extracted via persistent homology capture meaningful structural variations between tumor grades.

Panel of Figure 7 — Figure 7: Radar plot comparison of classification performance for XGBoost and Random Forest models using topological features derived from persistent homology on the BraTS2020 3D MRI dataset. Each axis represents one evaluation metric: accuracy, precision, recall, and AUC. The curves correspond to different Betti feature configurations, including Betti-0, Betti-1, Betti-2, and their combinations. Results are shown both with and without feature selection, highlighting the impact of selecting the most informative topological descriptors. The radar plots demonstrate that combining Betti features generally improves model performance, while feature selection further enhances classification accuracy and precision.

Feature Set	# Features	Accuracy (%)	Precision (%)	Recall (%)	AUC (%)
Without Feature Selection
B0	100	83.78	83.72	83.78	82.97
B1	100	83.78	83.13	83.78	82.44
B2	100	75.68	71.05	75.68	86.96
B1 + B2	200	78.38	75.72	78.38	88.36
B0 + B1 + B2	300	86.49	88.47	86.49	89.33

Table 2: Performance of XGBoost using Betti features extracted via persistent homology from 3D FLAIR MRI volumes.

Feature Set	# Features	Accuracy (%)	Precision (%)	Recall (%)	AUC (%)
Without Feature Selection
B0	100	81.08	80.04	81.08	83.08
B1	100	83.78	83.72	83.78	89.98
B2	100	81.08	80.04	81.08	86.96
B1 + B2	200	83.78	86.56	83.78	88.36
B0 + B1 + B2	300	83.78	86.56	83.78	89.33

Table 3: Performance of Random Forest using Betti features extracted via persistent homology from 3D FLAIR MRI volumes.

Brain Tumor Classification

Brain tumor classification uses medical images and molecular data to identify tumor types, grades, and subtypes for diagnosis and treatment planning.

See all 51 papers

Bridging Topology and Deep Representation Learning: A TDA-ViT Fusion Model for Four-Class Brain Tumor Classification
May 30, 2026Faisal AhmedVision Transformer Medical Image Classification
Department of Data Science and Mathematics, Embry-Riddle Aeronautical University, 3700 Willow Creek Rd, Prescott, 86301, Arizona, USA.
Enhancing Brain Tumor Classification Using Vision Transformers with Colormap-Based Feature Representation on BRISC2025 Dataset
Mar 22, 2026Faisal AhmedMedical Image Classification Brain Tumor Classification
Department of Data Science and Mathematics, Embry-Riddle Aeronautical University, 3700 Willow Creek Rd, Prescott, Arizona 86301, USA
an interpretable vision transformer framework for automated brain tumor classification
Apr 23, 2026Chinedu Emmanuel Mbonu, Tochukwu Sunday Belonwu, Okwuchukwu Ejike Chukwuogo +1Vision Transformer Medical Image Classification
Department of Computer Science, Nazarbayev University, Astana · Department of Computer Science, Nnamdi Azikiwe University, Nigeria

May 30, 2026J/K move · Enter open · S save

Bridging Topology and Deep Representation Learning: A TDA-ViT Fusion Model for Four-Class Brain Tumor Classification

Faisal Ahmed

Department of Data Science and Mathematics, Embry-Riddle Aeronautical University, 3700 Willow Creek Rd, Prescott, 86301, Arizona, USA.

Vision Transformer Medical Image Classification Brain Tumor Classification Medical Image Representation Learning

Open paper AskPDF arXiv

Accurate brain tumor classification from magnetic resonance imaging (MRI) is a key requirement for early diagnosis and clinical decision-making. Vision Transformers (ViTs) have shown strong performance in medical image analysis by learning global contextual representations, but they often fail to capture intrinsic structural and topological patterns present in tumor regions. To address this limitation, we propose a fusion framework that combines Topological Data Analysis (TDA) features with pretrained Vision Transformer representations for four-class brain tumor classification. In the proposed method, TDA is used to extract complementary topological descriptors that capture geometric structure, connectivity, and shape information from MRI images. In parallel, a pretrained ViT model learns high-level semantic representations from the same images. These two feature spaces are then fused to form a unified and more discriminative representation for classification. The model is evaluated on the BRISC2025 dataset, which contains four brain tumor classes: glioma, meningioma, pituitary tumor, and non-tumor cases. Experimental results show that combining topological and transformer-based features significantly improves performance compared to using either approach alone. The proposed TDA-ViT fusion model achieves an accuracy of 99.10%, precision of 99.27%, recall of 99.15%, F1-score of 99.21%, and an AUC of 99.98%. It also outperforms several state-of-the-art models, including ResNet50, ResNet101, EfficientNetB2, and standalone Vision Transformers. These results demonstrate that topological features provide valuable complementary information that enhances deep representation learning, leading to a robust and highly accurate framework for automated brain tumor classification.

Abstract

Brain Tumor Classification

Explore similar work

Bridging Topology and Deep Representation Learning: A TDA-ViT Fusion Model for Four-Class Brain Tumor Classification

Enhancing Brain Tumor Classification Using Vision Transformers with Colormap-Based Feature Representation on BRISC2025 Dataset

an interpretable vision transformer framework for automated brain tumor classification