The convolution integral is expressed in one dimension by the relationship

This represents the convolution of two time functions, and ; commonly is a time varying signal, e.g. speech, and is the impulse (time) response of a particular filter. is a dummy variable which represents the shift of one function with respect to the other, as illustrated in Figure 7: Convolution in One Dimension

The importance of the convolution integral is based to a large extent on the convolution theorem which relates multiplication in the time domain to convolution in the frequency domain and vice versa. Representing the convolution of two functions by the symbol , then

and

In 2-dimensional image processing terms, the continuous convolution integral may be expressed

In a manner analagous to one-dimensional convolution, the function is simply the image function rotated by 180 degrees about the origin. The function is the function further translated to move the origin of the image function g to the point in the plane. The functions are then pointwise multiplied and the product function is integrated over 2 dimensions. Convolution of digital sampled images is analagous to that for continuous images, except that the integral is transformed to a discrete summations over the image dimensions, m and n.

Since both and are non-zero over a finite domain, i.e. the field of view of the imaging system, the summation is necessary only over the area of non-zero overlap. The number of required multiply and add operations is equal to the number of pixels in times the number of pixels in

Hence an algorithm for digital convolution of an image of pixel dimensions p by q with a mask, of dimensions 2m+1 by 2n+1, is expressed,

For each row, i =1 to p

		 For each column, j =1 to p

		 	 (* Process pixel  *)

	 		 Set output image pixel, =0

	 		 For each mask row, k=-m to m

		 		 For each  mask column, l=-n to n

		 		  =