Convolution

Convolution is the core operation of spatial image filtering: a small matrix called a kernel slides across the image, computing a weighted sum of surrounding pixels at each position. Nearly all spatial filters - blur, sharpen, edge detection, emboss - are implemented as convolutions.

The convolution procedure:

• Slide the kernel (e.g. 3×3) across the image from top-left to bottom-right, one pixel at a time
• At each position, multiply each kernel element by the corresponding pixel value
• Sum all products to produce the output pixel value
• Repeat for every pixel in the image

Representative kernels and their effects:

• Box filter: All elements equal 1/9 in a 3×3 kernel. Produces simple averaging blur
• Gaussian filter: Weights follow a Gaussian distribution. Creates natural blur and is optimal for noise reduction
• Sobel filter: Detects horizontal and vertical edges by approximating first-order derivatives
• Laplacian filter: Detects edges in all directions via second-order derivatives. Zero-crossings indicate edge locations
• Unsharp mask: Subtracts a blurred version from the original to enhance sharpness. A staple in print and photography

Computationally, applying an N×N kernel to an M×M image requires O(M²N²) operations. For large kernels, converting to frequency domain (FFT) multiplication is faster. Separable kernels like Gaussian can be decomposed into two 1D passes, reducing complexity to O(M²N).

Deep learning CNNs (Convolutional Neural Networks) use the same convolution operation, but learn kernel values automatically through training rather than manual design.