# Linear Filtering

Clifford Watson,
Department of Applied Mathematics,
University of Washington,
Seattle, Washington 98195

The topics discussed here are:

## Low Pass Filters

Low pass filtering, otherwise known as "smoothing", is employed to remove high spatial frequency noise from a digital image. Noise is often introduced during the analog-to-digital conversion process as a side-effect of the physical conversion of patterns of light energy into electrical patterns [Tanimoto].

There are several common approaches to removing this noise:

• If several copies of an image have been obtained from the source, some static image, then it may be possible to sum the values for each pixel from each image and compute an average. This is not possible, however, if the image is from a moving source or there are other time or size restrictions.

• If such averaging is not possible, or if it is insufficient, some form of low pass spatial filtering may be required. There are two main types:

• reconstruction filtering, where an image is restored based on some knowledge of the type of degradation it has undergone. Filters that do this are often called "optimal filters".

• enhancement filtering, which attempts to improve the (subjectively measured) quality of an image for human or machine interpretability. Enhancement filters are generally heuristic and problem oriented [Niblack]; they are the type that are discussed in this tutorial.

## Moving Window Operations

(Select digital convolution for a more detailed derivation of this section.)

The form that low-pass filters usually take is as some sort of moving window operator. The operator usually affects one pixel of the image at a time, changing its value by some function of a "local" region of pixels ("covered" by the window). The operator "moves" over the image to affect all the pixels in the image. Some common types are:

• Neighborhood-averaging filters These replace the value of each pixel, a[i,j] say, by a weighted-average of the pixels in some neighborhood around it, i.e. a weighted sum of a[i+p,j+q], with p = -k to k, q = -k to k for some positive k; the weights are non-negative with the highest weight on the p = q = 0 term. If all the weights are equal then this is a mean filter. "linear"

• Median filters This replaces each pixel value by the median of its neighbors, i.e. the value such that 50% of the values in the neighborhood are above, and 50% are below. This can be difficult and costly to implement due to the need for sorting of the values. However, this method is generally very good at preserving edges.

• Mode filters Each pixel value is replaced by its most common neighbor. This is a particularly useful filter for classification procedures where each pixel corresponds to an object which must be placed into a class; in remote sensing, for example, each class could be some type of terrain, crop type, water, etc..

The above filters are all space invariant in that the same operation is applied to each pixel location. A non-space invariant filtering, using the above filters, can be obtained by changing the type of filter or the weightings used for the pixels for different parts of the image. Non-linear filters also exist which are not space invariant; these attempt to locate edges in the noisy image before applying smoothing, a difficult task at best, in order to reduce the blurring of edges due to smoothing. These filters are not discussed in this tutorial.

## High Pass Filters

This Section is under construction.

Back to the Table of Contents.