Separable Filter Optimization

Implement efficient separable convolution using the separability property.

A 2D filter K is separable if $K = \mathbf{h} \mathbf{v}^T$ (outer product of two 1D filters).

Computational advantage:

• Direct 2D convolution: $O(N^2 \cdot k^2)$ for k×k kernel
• Separable: $O(N^2 \cdot 2k)$ - much faster!

Algorithm:

1. Check if kernel is separable (rank-1 matrix)
2. If separable, extract h and v vectors
3. Convolve image with h horizontally
4. Convolve result with v vertically

Gaussian, box, and Sobel filters are all separable!