We introduce a continuous time method to analyze the response of median, pseudomedian, average (mean), and midrange filters to certain periodic signals. The filter definitions are generalized to continuous time, and these definitions are applied to periodic signals such as triangle, square, and sinusoidal waves of varying frequencies. These operations yield "amplitude response" measures which are analytic functions of the frequency of the input signal. In addition, a "correlation" measure is defined to indicate the level of distortion introduced by each filter. Examples of this analysis for the median, pseudomedian, average, and midrange filters show similarities and differences among them.

Nonlinear filtering techniques are becoming increasingly important in image processing applications, and are often better than linear filters at removing noise without distorting image features. However, design and analysis of nonlinear filters are much more difficult than for linear filters. One structure for designing nonlinear filters is mathematical morphology, which creates filters based on shape and size characteristics. Morphological filters are limited to minimum and maximum operations that introduce bias into images. This precludes the use of morphological filters in applications where accurate estimation of the true gray level is necessary. This work develops two new filtering structures based on mathematical morphology that overcome the limitations of morphological filters while retaining their emphasis on shape. The linear combinations of morphological filters eliminate the bias of the standard filters, while the value-and-criterion filters allow a variety of linear and nonlinear operations to be used in the geometric structure of morphology. One important value-and-criterion filter is the Mean of Least Variance (MLV) filter, which sharpens edges and provides noise smoothing equivalent to linear filtering. To help understand the behavior of the new filters, the deterministic and statistical properties of the filters are derived and compared to the properties of the standard morphological filters. In addition, new analysis techniques for nonlinear filters are introduced that describe the behavior of filters in the presence of rapidly fluctuating signals, impulsive noise, and corners. The corner response analysis is especially informative because it quantifies the degree to which a filter preserves corners of all angles. Examples of the new nonlinear filtering techniques are given for a variety of medical images, including thermographic, magnetic resonance, and ultrasound images. The results of the filter analyses are important in