Laplacian Pattern example with Simple Python
Python (NumPy) Example:
Here’s how you might apply a Laplacian filter manually to a small grayscale image:
import numpy as np
from scipy.signal import convolve2d
import matplotlib.pyplot as plt
# Sample grayscale image
image = np.array([
[10, 10, 10, 10, 10],
[10, 50, 50, 50, 10],
[10, 50, 90, 50, 10],
[10, 50, 50, 50, 10],
[10, 10, 10, 10, 10]
])
# Laplacian kernel
laplacian_kernel = np.array([
[0, -1, 0],
[-1, 4, -1],
[0, -1, 0]
])
# Apply the filter
laplacian_output = convolve2d(image, laplacian_kernel, mode='same')
# Show results
plt.subplot(1, 2, 1)
plt.title("Original")
plt.imshow(image, cmap='gray')
plt.subplot(1, 2, 2)
plt.title("Laplacian Edge")
plt.imshow(laplacian_output, cmap='gray')
plt.show()
Summary Table:
| Feature | Description |
|---|---|
| Operator Type | Second derivative (isotropic) |
| Detects | Edges, corners, small intensity changes |
| Sensitive To | Noise (usually combined with Gaussian) |
| Common Use | Edge detection in pre-processing |
| Best With | Cleaned/blurred images |
Step-by-Step Python Example of Laplacian of Gaussian (LoG)
import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import gaussian_filter, laplace
# Create a sample image with sharp edges
image = np.zeros((100, 100))
image[30:70, 30:70] = 255 # A white square in the center
# Step 1: Smooth image using Gaussian filter
blurred = gaussian_filter(image, sigma=2)
# Step 2: Apply Laplacian operator
log_result = laplace(blurred)
# Visualize the results
plt.figure(figsize=(12, 4))
plt.subplot(1, 3, 1)
plt.title("Original")
plt.imshow(image, cmap='gray')
plt.subplot(1, 3, 2)
plt.title("Gaussian Blurred")
plt.imshow(blurred, cmap='gray')
plt.subplot(1, 3, 3)
plt.title("Laplacian of Gaussian (LoG)")
plt.imshow(log_result, cmap='gray')
plt.tight_layout()
plt.show()
Output Description:
| Image Part | Description |
|---|---|
| Original | A sharp-edged white square on black background |
| Gaussian Blurred | Smooth transition from black to white (noise reduced) |
| LoG Output | Edges of the square highlighted with bipolar transitions (black-white borders) |
Intuition:
- Why Gaussian first?
Laplacian is sensitive to noise. The Gaussian blur suppresses noise while preserving overall structure. - Why Laplacian after?
It finds the zero-crossings in second derivative, which signal the edges.