Convolution

A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image:

It is written:

Suppose H is a Gaussian or mean kernel. How does convolution differ from cross-correlation?

Continuous filtering

We can also apply continuous filters to continuous images.

In the case of cross correlation:

In the case of convolution:

Note that the image and filter are infinite.

Image gradient

The gradient of an image:

The gradient points in the direction of most rapid change in intensity

Effects of noise

Consider a single row or column of the image

Plotting intensity as a function of position gives a signal

Solution: smooth first

Derivative theorem of convolution

This saves us one operation:

Laplacian of Gaussian

Consider

2D edge detection filters

Edge detection by subtraction

Gaussian - image filter


	A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image:



	It is written:


	Suppose H is a Gaussian or mean kernel. How does convolution differ from cross-correlation?


	We can also apply continuous filters to continuous images.
	In the case of cross correlation:



	In the case of convolution:




	Note that the image and filter are infinite.


	The gradient of an image:


	The gradient points in the direction of most rapid change in intensity


	Consider a single row or column of the image
		Plotting intensity as a function of position gives a signal