Linear Filtering
Clifford Watson,
Department of Applied Mathematics,
University of Washington,
Seattle, Washington 98195
The topics discussed here are:
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.
(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.
This Section is under construction.
Back to the Table of Contents.
watson@amath.washington.edu or tanimoto@cs.washington.edu
Last modified: Mon Oct 10 15:03:58 PST 1994