Convolution
The engine behind blur, sharpening, edge detection, and neural networks.
Convolution is a mathematical operation that combines two functions by computing the integral of their pointwise product as one slides over the other. In image processing, it applies a small kernel matrix across every pixel. In neural networks, it learns these kernels automatically. The same operation runs audio filters, fluid simulations, and probability distributions.
Mathematics
(f * g)(t) = ∫ f(τ) g(t − τ) dτ
Discrete form:
(f * g)[n] = Σ f[m] g[n − m]
Key Facts
- Separable convolutions reduce 2D operations to two 1D passes
- Frequency domain: convolution becomes multiplication (Fourier)
- CNNs learn optimal kernels through backpropagation
- GPU implementations use shared memory tiling for efficiency
Where It Appears
- Image blur, sharpen, edge detection
- Convolutional neural networks (CNNs)
- Audio filtering and reverb
- Fluid simulation velocity smoothing
- Probability — sum of independent distributions