This work extends the theory of nonlinear image processing by introducing new filter structures and analysis methods, and demonstrates the utility of these new techniques on a variety of medical images. The filters are based on generalizations of mathematical morphology, which is itself a relatively recent development in image processing. One of these generalizations is the class of linear combinations of morphological operators. This filter class includes the previously defined midrange and pseudomedian filters and leads to the definition of the LOCO filter. The deterministic properties of the linear combinations are similar to those of the constituent morphological filters; however, the linear combinations are statistically unbiased, unlike the conventional morphological operators. This is important in applications like thermography, where the LOCO filter allows shape-based filtering as in standard morphology without introducing a statistical bias.
The other new class of filters introduced here is the value-and-criterion filter structure. This structure is based on morphological opening and closing, but allows the use of both linear and nonlinear operators within the window structure. One useful filter designed with this structure is the Mean of Least Variance (MLV) filter, which takes the mean of the window with the smallest variance within a set of windows defined by the morphological structure. This filter reduces noise and enhances edges in images. Its statistical and deterministic properties resemble those of the averaging filter more than the morphological filters, with several important differences. The MLV filter is valuable in applications like MRI, where noise smoothing and contrast enhancement are the primary goals.
This dissertation also advances the analysis of nonlinear filters by introducing new methods for understanding their behavior. A method of finding the response of nonlinear filters to continuous time periodic signals of various frequencies was described. This technique bears some resemblance to Fourier analysis, but its results are much more limited because of the nonlinear nature of the filters. The breakdown point is another new technique that measures the robustness of filters to outlying signal values. The breakdown point can be used to help design filters with almost any desired level of outlier rejection ability. Corner response analysis is introduced to help quantify the ability of filters to preserve or remove two-dimensional features. Filters that have similar edge responses (such as the median and morphological filters) may have drastically different responses to corners of various angles, and this is illustrated intuitively and quantitatively by polar plots of the corner response of the filters.
To show how the new filter design and analysis techniques introduced here are used in real-world imaging applications, examples from various biomedical imaging fields (thermography, magnetic resonance, and ultrasound) are given. The unbiased linear combinations of morphological operators are good candidates for shape-based filtering of thermograms, since temperature information often must be extracted from the gray levels of these images. By choosing a structuring element shape that matches the shape of the thermal features in the image, the LOCO filter performs well for removing noise and reconstructing smooth thermal contours.
In magnetic resonance and ultrasound imaging, preserving accurate gray levels is not as important as in thermal imaging. Processing goals for these images are more likely to include reducing noise while preserving or enhancing boundaries between tissue regions. In magnetic resonance imaging, contrast enhancement may improve the visual quality of images acquired very quickly, or may be used as a part of algorithms to segment the image into various tissue types automatically. The MLV filter is ideally suited to this task, since it provides noise smoothing and edge enhancement by averaging within homogeneous regions and away from edges. Furthermore, it yields excellent results in a single iteration, unlike other proposed methods such as anisotropic diffusion. In ultrasound imaging, however, the speckle and dropout defects are not amenable to filtering by the LOCO or MLV filters. The standard morphological operators (such as opening and closing) provide better enhancement in ultrasound imaging, thus illustrating a case where the behavior of the conventional filters is preferred.
This dissertation details important new developments in the theory, analysis, and application of nonlinear filtering. Filtering structures are devised to overcome shortcomings in the responses of morphological filters, and new analysis techniques assist in understanding the behavior of both the standard and new nonlinear filters. Appropriate filters for a given application are chosen by using information about the imaging modality and the results of the various analysis methods. Improved options now exist for nonlinear image processing because of the introduction of these new filter design and analysis tools.
© Copyright by Mark A. Schulze, 1994.
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