MATHEMATICAL PROPERTIES OF THE PSEUDOMEDIAN FILTER by MARK ALLEN SCHULZE, B.A., B.S.E.E.

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

Morphological image processing filters preserve shapes related to the structuring element shape of the operator. The basic morphological operators are minimum (erosion) and maximum (dilation) operations performed on the pixels within a structuring element. Although these operators (and the compound operators formed from them) are able to smooth noise, they also introduce a statistical and deterministic bias, which is unacceptable in some applications. However, since every morphological operator has a complementary operator that is equally and oppositely biased, we propose averaging the complementary operators to alleviate the bias. Of the three filters formed by averaging the standard morphological operators, two are the previously-defined midrange filter and pseudomedian filter, while one is a new filter, which we call the LOCO filter. Under most conditions, the LOCO filter is the best of these at reducing impulses and noise.

method for mobile robot to avoid obstacles with a real time system in an uncertain environment. The method deals with the problem of robot in driving from an initial position to a target position based on elastic force and potential field force. To avoid the obstacles, the robot has to modify the trajectory based on signal received from the sensor system in the sampling times. It was evident that with the combination of Modification Elastic strip and Pseudomedian filter to process the nonlinear data from sensor uncertainties in the data received from the sensor system can be reduced. The simulations and experiments of these methods were carried out. Keywords—Collision avoidance, Avoidance obstacle, Elastic Strip, Real time collision avoidance.