Convolution & Image Filtering

What you’ll learn

• What a kernel is and how it slides over an image
• Why a box blur and a Gaussian blur look different
• How the Sobel kernel turns brightness changes into edges

Intuition

Every output pixel is a weighted sum of its neighbors. The weights form a small grid called a kernel. Change those nine numbers and the same operation can blur, sharpen or highlight edges.

The math

For an input image $I$ and a $(2k+1) \times (2k+1)$ kernel $K$, the output $G$ is

$$G(x, y) = \sum_{i=-k}^{k} \sum_{j=-k}^{k} K(i, j)\, I(x + i,\; y + j)$$

Strictly, this is cross-correlation. True convolution flips the kernel first, $K(-i, -j)$, which makes no difference for symmetric kernels such as a blur.

A blur kernel’s weights sum to $1$, so flat regions keep their brightness. An edge kernel’s weights sum to $0$, so flat regions become $0$ and only changes survive.

Code

From scratch with NumPy:

import numpy as np

def conv2d(img, K):
    """img: (H, W) float32, K: (3, 3). Edge-replicated borders."""
    H, W = img.shape
    p = np.pad(img, 1, mode="edge")
    out = np.zeros_like(img)
    for i in range(3):
        for j in range(3):
            out += K[i, j] * p[i:i + H, j:j + W]
    return out

The same thing with OpenCV:

import cv2
import numpy as np

img = cv2.imread("cameraman.png", cv2.IMREAD_GRAYSCALE).astype(np.float32) / 255
sobel_x = np.array([[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]], np.float32)
edges = cv2.filter2D(img, -1, sobel_x)

Try it

Pick a preset, then edit individual cells. Try making the Sobel kernel detect horizontal edges instead of vertical ones.

Kernel playgroundEdit any cell or pick a preset. The output updates as you type.

Results and pitfalls

Drag the slider to compare the input with a $5 \times 5$ Gaussian blur ($\sigma = 1.4$). The fine stripes in the bottom right disappear first, because blurring is a low-pass filter.

InputGaussian σ=1.4

Common bugs:

• Integer overflow. Filtering a uint8 image with a sharpening kernel clips negative values to 0. Work in float32.
• Border handling. Zero padding darkens the edges of a blurred image. Use mode="edge" or cv2.BORDER_REFLECT.

Exercises

1. Design a kernel that detects only horizontal edges.
2. Why must a blur kernel’s weights sum to 1? What happens if they sum to 2?
3. Show that applying a $3 \times 3$ box blur twice equals one $5 \times 5$ kernel. What does that kernel look like?