What Are Convolution and Correlation?

Convolution and correlation are core image-processing operations. Both involve sliding a small matrix—called a kernel or filter—over an image and computing a weighted sum at each position. These techniques power blurring, edge detection, sharpening and more.

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Correlation vs Convolution

Operation	Description
Correlation	Place the kernel as-is and compute the dot product with the image patch
Convolution	Flip the kernel horizontally and vertically, then compute the dot product

Note: OpenCV’s filter2D implements correlation, even when it’s referred to as convolution.

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Step-by-Step Numerical Example

Given a 3×3 image I and a simple averaging kernel K:

I = [ [10, 20, 30],
      [40, 50, 60],
      [70, 80, 90] ]

K = (1/9) × [ [1, 1, 1],
              [1, 1, 1],
              [1, 1, 1] ]

At the centre pixel (position (1,1)), correlation yields:

Output = (1/9) × (10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90)
       = 450 / 9
       = 50

So the central pixel becomes 50, the average of its 3×3 neighbourhood.

🐍 Python Example (OpenCV)

import cv2
import numpy as np
import matplotlib.pyplot as plt

# Load grayscale image
img_gray = cv2.imread("bird.jpg", cv2.IMREAD_GRAYSCALE)

# Define a 3×3 averaging kernel
kernel = np.ones((3, 3), np.float32) / 9

# Apply filtering (correlation)
filtered = cv2.filter2D(img_gray, -1, kernel)

# Display results
plt.figure(figsize=(10, 5))

plt.subplot(1, 2, 1)
plt.imshow(img_gray, cmap="gray")
plt.title("Original")
plt.axis("off")

plt.subplot(1, 2, 2)
plt.imshow(filtered, cmap="gray")
plt.title("After Convolution (Averaging)")
plt.axis("off")

plt.tight_layout()
plt.show()

🔄 Summary

Term	Description
Kernel	Small matrix used for filtering
Correlation	Slide kernel as-is over the image
Convolution	Slide a flipped kernel over the image
filter2D	OpenCV function that performs correlation

🔄 Correlation in Image Processing

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🧠 What is Correlation?

In image processing, correlation measures similarity between a small filter (kernel) and a region of the image.
Imagine sliding a stencil over an image and checking how well it “matches” each patch.

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🔍 Correlation vs. Convolution — What’s the Difference?

Operation	Flip the Kernel?	Used in Deep Learning?	Intuition
Convolution	✅ Yes (both axes)	✅ Yes	Emphasises pattern matching
Correlation	❌ No	✅ Used in OpenCV & NumPy	Measures similarity directly

📌 In OpenCV, filter2D() actually performs correlation, even though it’s named like convolution.

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🧮 How Correlation Works

Given an image I and a kernel K, for each pixel (x, y) we compute:

Steps to Perform Correlation

1. Slide the kernel over the image
2. Multiply corresponding elements
3. Sum the result → place in the output image

Note: No flipping of the kernel is done here.

🔢 Example: Correlation with a 3×3 Filter

Imagine this 3×3 grayscale image:

I = [ [10, 20, 30],
      [40, 50, 60],
      [70, 80, 90] ]

And a 3×3 kernel (similar to a Sobel X filter):

K = [ [ 1,  0, -1],
      [ 1,  0, -1],
      [ 1,  0, -1] ]

At the centre pixel (1,1), whose original value is 50, the correlation is:

G(1,1) = 10⋅1 + 20⋅0 + 30⋅(-1)
       + 40⋅1 + 50⋅0 + 60⋅(-1)
       + 70⋅1 + 80⋅0 + 90⋅(-1)
       = (10 - 30) + (40 - 60) + (70 - 90)
       = -60

🧪 Python Code: Correlation with OpenCV

import cv2
import numpy as np
import matplotlib.pyplot as plt

# Load image in grayscale
img = cv2.imread('bird.jpg', cv2.IMREAD_GRAYSCALE)

# Define a kernel (like a Sobel X)
kernel = np.array([[1, 0, -1],
                   [1, 0, -1],
                   [1, 0, -1]], dtype=np.float32)

# Apply correlation using filter2D (no kernel flip)
correlated = cv2.filter2D(img, -1, kernel)

# Show result
plt.figure(figsize=(10, 4))
plt.subplot(1, 2, 1)
plt.imshow(img, cmap='gray')
plt.title("Original")
plt.axis("off")

plt.subplot(1, 2, 2)
plt.imshow(correlated, cmap='gray')
plt.title("Correlation Output")
plt.axis("off")
plt.tight_layout()
plt.show()