X-ray Coronary Artery Image Segmentation

A lightweight approach for automatic segmentation of 2D X-ray coronary artery images using shallow multi-layer perceptrons and morphological filtering.

Theory

Hessian Matrix

The Hessian matrix stores second-order spatial derivatives of pixel intensity values for detecting tubular structures:

$$H = \begin{bmatrix} I_{xx} & I_{xy} \\\ I_{yx} & I_{yy} \end{bmatrix}$$

Where I_xx and I_yy are second-order partial derivatives along x and y axes, and I_xy = I_yx is the mixed partial derivative.

Frangi Vesselness Filter

The Frangi filter enhances narrow, elongated structures while suppressing blobs and background:

$$V(\sigma) = \begin{cases} 0, & \text{if } \lambda_2 > 0 \\\ f(\gamma) \exp\left(-\frac{R^2}{2\alpha^2}\right) \left[1 - \exp\left(-\frac{S^2}{2\beta^2}\right)\right], & \text{otherwise} \end{cases}$$

Where:

• math R = \frac{|\lambda_1|}{|\lambda_2|} differentiates between blobs and tubes
• math S = \sqrt{\lambda_1^2 + \lambda_2^2} measures combined eigenvalue magnitude
• α controls blob structure penalization
• β controls plate-like structure penalization
• γ modulates sensitivity to background texture

Multi-Layer Perceptron

A shallow neural network with:

• Single input neuron (pixel intensity)
• 2 hidden layers with 9 neurons each
• ReLU activation for hidden layers
• Softmax output activation
• Weighted binary cross-entropy loss to handle class imbalance

Otsu's Adaptive Thresholding

Finds optimal threshold by maximizing inter-class variance:

$$\sigma^2(\tau) = w_1(\tau)w_2(\tau)[\mu_1(\tau) - \mu_2(\tau)]^2$$

Where w_1, w_2 are class probabilities and μ_1, μ_2 are class means.

Methodology

1. Data Augmentation: Horizontal flipping for robustness
2. Preprocessing:
   • Frangi multiscale filtering (σ ∈ {1.8, 1.9, ..., 4.0})
   • Small-object removal (5000 pixel threshold) to eliminate artifacts
3. Training: 5-fold cross-validation with 212 training samples
4. Segmentation: Otsu's method applied to predicted foreground probabilities

Dataset

DCA1 Database: 130 grayscale X-ray coronary angiograms (300×300 pixels) with expert-labeled ground truth from the Mexican Social Security Institute Cardiology Department.

• Training: 212 samples
• Validation: 56 samples
• Class imbalance: Vessel pixels comprise 15-19% of total pixels

Benchmarking

Metric	Filtered	Unfiltered
AUROC	0.948	0.760
Dice	0.61	0.15
Sensitivity	0.79	0.21
Specificity	0.95	0.91
Precision	0.51	0.15
IoU	0.44	0.10

Results

• Preprocessing improved Dice score by 4x and sensitivity by 3.8x
• Otsu thresholds: 0.10-0.14 (filtered) vs 10⁻³-10⁻² (unfiltered)
• Performance comparable to Cervantes et al. with much simpler architecture

Tools

• Python 3.10.12 on Google Colab
• Intel Core i5-1135G7 (2.40GHz, 16GB RAM)
• NumPy, scikit-image, Matplotlib, PyTorch

References

1. Young, I. (1983). Image analysis and mathematical morphology. Cytometry, 4, 184-185.
2. Frangi, A. et al. (2000). Multiscale Vessel Enhancement Filtering. Medical Image Computing and Computer-Assisted Intervention, 1496.
3. Luo, Y. & Sun, L. (2023). Digital subtraction angiography image segmentation based on multiscale Hessian matrix. Journal of Radiation Research and Applied Sciences, 16(3).
4. Ma, G., Yang, J., & Zhao, H. (2020). A coronary artery segmentation method based on region growing. Technology and Health Care, 28, S463-S472.
5. Cervantes-Sanchez, F. et al. (2019). Automatic Segmentation of Coronary Arteries using Multiscale Analysis and Artificial Neural Networks. Applied Sciences, 9(24), 5507.
6. Park, T. et al. (2022). Deep Learning Segmentation in 2D X-ray Images. Diagnostics, 12(4), 778.
7. Iyer, K. et al. (2021). AngioNet: a convolutional neural network for vessel segmentation. Scientific Reports, 11, 18066.

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Code Repository: https://github.com/eigenchip/xca_image_segmentation
License: MIT