Python Hessian based Frangi filter

![Hessian based Frangi filter](

Introduction

The Hessian matrix is a matrix of second-order partial derivatives of a function. In image processing, it can be used to extract meaningful features from an image. The Frangi filter is an edge detection filter that utilizes the Hessian matrix to enhance vessel-like structures in medical images.

The Frangi filter was originally introduced for the detection of blood vessels in medical images, but it can also be applied to other applications such as the enhancement of thin lines in images or the detection of curvilinear structures.

In this article, we will explore how to implement the Frangi filter in Python using the Hessian matrix.

Frangi Filter Algorithm

The Frangi filter algorithm consists of the following steps:

  1. Convert the input image to grayscale.
  2. Compute the Hessian matrix for each pixel.
  3. Compute the eigenvalues of the Hessian matrix at each pixel.
  4. Compute vesselness measure using the eigenvalues.
  5. Threshold the vesselness measure to obtain the final binary image.

Implementation

To implement the Frangi filter in Python, we can use the scipy and numpy libraries. Here's an example implementation:

import numpy as np
import scipy.ndimage.filters as filters

def frangi_filter(image, sigma=1.0, scale_range=(0.5, 2.0)):
    # Convert the image to grayscale
    image = np.mean(image, axis=2)
    
    # Compute the Hessian matrix
    hessian = filters.hessian(image, sigma=sigma)
    
    # Compute the eigenvalues of the Hessian matrix
    eigenvalues = np.linalg.eigvals(hessian)
    
    # Compute vesselness measure
    lambda1 = np.abs(eigenvalues.real)
    lambda2 = np.abs(eigenvalues.imag)
    
    vesselness = np.exp(-(lambda1**2) / (2 * scale_range[0]**2)) * (1 - np.exp(-(lambda2**2) / (2 * scale_range[1]**2)))
    
    # Threshold the vesselness measure
    threshold = 0.5
    binary_image = vesselness > threshold
    
    return binary_image

In the above code, we first convert the input image to grayscale by taking the mean of the RGB channels. Then, we compute the Hessian matrix using the hessian function from the scipy.ndimage.filters module. Next, we compute the eigenvalues of the Hessian matrix using the eigvals function from the numpy.linalg module.

To compute the vesselness measure, we take the absolute values of the eigenvalues and apply a vesselness formula based on the scale range. Finally, we threshold the vesselness measure using a threshold value of 0.5 to obtain the final binary image.

Usage

To use the Frangi filter on an image, you can simply call the frangi_filter function and pass the image as an argument:

import matplotlib.pyplot as plt
from skimage import io

# Load the image
image = io.imread('image.jpg')

# Apply the Frangi filter
binary_image = frangi_filter(image)

# Display the results
plt.subplot(1, 2, 1)
plt.imshow(image)
plt.title('Original Image')

plt.subplot(1, 2, 2)
plt.imshow(binary_image, cmap='gray')
plt.title('Frangi Filter')

plt.show()

In the code above, we first load an image using the skimage.io.imread function. Then, we apply the Frangi filter using the frangi_filter function. Finally, we display the original image and the filtered image using matplotlib.pyplot.imshow.

Conclusion

In this article, we have explored how to implement the Frangi filter in Python using the Hessian matrix. The Frangi filter is a powerful tool for enhancing vessel-like structures in images and can be used in various applications such as medical image analysis.

By understanding the Frangi filter algorithm and implementing it in Python, you can easily apply this technique to your own images and enhance vessel-like structures with ease.