What Is Image Filtering in the Spatial Domain?

Filtering is a technique for modifying or enhancing an image. For example, you can filter an image to emphasize certain features or remove other features. Image processing operations implemented with filtering include smoothing, sharpening, and edge enhancement.

Filtering is a neighborhood operation, in which the value of any given pixel in the output image is determined by applying some algorithm to the values of the pixels in the neighborhood of the corresponding input pixel. A pixel's neighborhood is some set of pixels, defined by their locations relative to that pixel. (See Sliding Neighborhood Operations for a general discussion of neighborhood operations.) Linear filtering is filtering in which the value of an output pixel is a linear combination of the values of the pixels in the input pixel's neighborhood.

Convolution

Linear filtering of an image is accomplished through an operation called convolution. Convolution is a neighborhood operation in which each output pixel is the weighted sum of neighboring input pixels. The matrix of weights is called the convolution kernel, also known as the filter. A convolution kernel is a correlation kernel that has been rotated 180 degrees.

For example, suppose the image is

A = [17  24   1   8  15
     23   5   7  14  16
      4   6  13  20  22
     10  12  19  21   3
     11  18  25   2   9]

and the convolution kernel is

h = [8   1   6
     3   5   7
     4   9   2]

You would use the following steps to compute the output pixel at position (2, 4):

Rotate the convolution kernel 180 degrees about its center element.
Slide the center element of the rotated convolution kernel so that it lies on top of the (2, 4) element of A.
Multiply each weight in the rotated convolution kernel by the pixel of A underneath.
Sum the individual products from step 3.

Hence the (2, 4) output pixel is

The calculation is shown in the following figure.

Computing the (2, 4) Output of Convolution

A grid of pixels displaying the pixel values. The 3-by-3 pixel neighborhood around the (2, 4) element is highlighted in gray, indicating the position of the convolution kernel. For pixels in the neighborhood, the grid also displays the weights of the convolution kernel, which is equal to the rotated matrix h.

Correlation

The operation called correlation is closely related to convolution. In correlation, the value of an output pixel is also computed as a weighted sum of neighboring pixels. The difference is that the matrix of weights, in this case called the correlation kernel, is not rotated during the computation. The Image Processing Toolbox™ filter design functions return correlation kernels.

The following figure shows how to compute the (2, 4) output pixel of the correlation of A, assuming h is a correlation kernel instead of a convolution kernel, using these steps:

Slide the center element of the correlation kernel so that lies on top of the (2, 4) element of A.
Multiply each weight in the correlation kernel by the pixel of A underneath.
Sum the individual products.

The (2, 4) output pixel from the correlation is

Computing the (2, 4) Output of Correlation