## Separable convolutions

Thursday, August 19th, 2010

The convolution is an important tool in image processing. All *linear* filters are convolutions (like, for example, the Gaussian filter). One of the reasons linear filters are so prevalent is that they imitate physical systems. For example, an analog electric circuit (containing resistors, capacitors and inductors) can be described by a convolution. The projection of a cell on a slide, through the microscope’s optical systems, onto the CCD sensor, can be described by a convolution. Even the sampling and averaging that occur in the CCD can be described by a convolution.

Two properties of the convolution are quite interesting when looking for an efficient implementation:

- The convolution is a multiplication in the Fourier domain:
*f*(*x*)⊗*h*(*x*) ⇒ F(*ω*)⋅H(*ω*) . This means that you can compute the convolution by applying the Fourier transform to both the image and the convolution kernel, multiplying the two results, then inverse transforming the result. - The convolution is associative: (
*f*⊗*h*_{1})⊗*h*_{2}=*f*⊗(*h*_{1}⊗*h*_{2}) . This post is about the repercussions of this property.