2️⃣ Convolution (2D) §

$f(x,y)\ast h(x,y)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x^{\prime },y^{\prime })h(x-x^{\prime },y-y^{\prime })d{x}^{\prime }d{y}^{\prime }$

Convolution with a filter §

Image $f$, to which a filter kernel $k$ gets applied by a convolution

$H\{f\}(x,y)=f\ast k={\sum }_{i=-\frac{w_k}{2}}^{\frac{w_k}{2}}{\sum }_{i=-\frac{h_k}{2}}^{\frac{h_k}{2}}f(x-i,y-j)k(i,j)$

Efficient filtering §

Convolution can also be performed like so:

1. Transform $f$ and $k$ to Fourier domain: $F=\mathcal{F}(f)$, $K=\mathcal{F}(k)$
2. Pointwise multiplication: $\mathcal{F}\{f\ast k\}=FK$
3. Transform back to Spacial domaion