Biomedical Image Processing with Morphology-Based Nonlinear Filters

Mark A. Schulze

Ph.D. Dissertation

The University of Texas at Austin, 1994

Chapter 1: Introduction

1.1. Overview

Nonlinear methods in signal and image processing have become increasingly popular over the past thirty years. There are two general families of nonlinear filters: the homomorphic and polynomial filters, and the order statistic and morphological filters [1] . Homomorphic filters were developed during the 1970's and obey a generalization of the superposition principle [2] . The polynomial filters are based on traditional nonlinear system theory and use Volterra series. Analysis and design of homomorphic and polynomial filters resemble traditional methods used for linear systems and filters in many ways. The order statistic and morphological filters, on the other hand, cannot be analyzed efficiently using generalizations of linear techniques. The median filter is an example of an order statistic filter, and is probably the oldest [3, 4] and most widely used order statistic filter. Morphological filters are based on a form of set algebra known as mathematical morphology. Most morphological filters use extreme order statistics (minimum and maximum values) within a filter window, so they are closely related to order statistic filters [5, 6].

While homomorphic and polynomial filters are designed and analyzed by the techniques used to define them, order statistic filters are often chosen by more heuristic methods. As a result, the behavior of the median filter and other related filters was poorly understood for many years. In the early 1980's, important results on the statistical behavior of the median filter were presented [7] , and a new technique was developed that defined the class of signals invariant to median filtering, the root signals [8, 9] . Morphological filters are derived from a more rigorous mathematical background [10-12] , which provides an excellent basis for design but few tools for analysis. Statistical and deterministic analyses for the basic morphological filters were not published until 1987 [5, 6, 13] . The understanding of the filters' behavior achieved by these analyses is not complete, however, so further study may help determine when morphological filters are best applied.

This dissertation investigates the use of morphology-based nonlinear filters to enhance biomedical images. Specifically, new filters based on mathematical morphology are developed, analyzed, and applied to a variety of medical images. The behavior of the standard morphological filters is undesirable for certain applications, and the new filters are designed to overcome these weaknesses. Some new analysis techniques are introduced, including a method to quantify the response of filters to two-dimensional features. These new analysis methods and the basic statistical and deterministic analyses are used to compare the new filters with the standard filters. Finally, the new nonlinear filters are used to enhance magnetic resonance, thermographic, and ultrasound images and their performance is compared to established filtering techniques for each of the imaging modalities.

The accomplishments of this work include:

1.2. Organization

This dissertation begins with a review of mathematical morphology, including the statistical and deterministic properties of the morphological filters. These properties point out weaknesses (specifically, a bias problem) in the behavior of the standard morphological filters that motivate the development of new filters. Next, new filters that address the bias problem of the standard filters are introduced. Linear combinations of morphological operators are one of the new types of filters. This work develops the deterministic and statistical properties of these filters and illustrates the potential advantages of these filters over the standard morphological filters.

Another new type of filter introduced in this work is the value-and-criterion filter. This filter structure uses the shape-based organization of morphology, but expands the operations used for the filtering beyond just the maximum and minimum operators. Thus, any linear or nonlinear function can be used to determine the output value from values in a window, and to determine which window to use to get the output value. A promising application of this new structure is for designing filters that sharpen edges and smooth noise simultaneously. One of these new filters is the "Mean of Least Variance" filter, or MLV filter, which is a significant improvement over previously defined edge-preserving smoothing filters. The deterministic and statistical properties of the MLV filter are also investigated to contrast its behavior with other morphology-based filters.

Since the usual statistical and deterministic analyses provide only an incomplete understanding of the behavior of nonlinear filters, new analysis methods are introduced here to gain further insight into the response of the filters. A technique to quantify the response of filters to periodic signals of various frequencies is outlined. This method is similar to Fourier analysis for linear filters, but is much more limited in scope because of the nonlinear nature of the filters examined. Nonetheless, this analysis gives valuable clues about the response of nonlinear filters to rapidly fluctuating signals. Another important property of many nonlinear filters is their resistance to outlying values and impulsive noise. The "breakdown point" is a measure of the robustness of filters in the presence of outliers. This method is another way to help explain differences among filters.

The last analysis method developed in this dissertation furthers the understanding of the behavior of filters at two-dimensional structures. This technique, called "corner response analysis," quantifies the percentage of binary corners of various angles that is preserved by a filter. By plotting this information in polar format, the change in the response of a filter to corners of various angles is easily visualized. This method is a major improvement over previous analyses that focused on general characteristics like noise reduction or one-dimensional characteristics like edge preservation. The response of the filter to different rotations of the same feature is also explored using corner response analysis, indicating whether a filter acts similarly to different rotations of 2-D objects.

The final portion of this work illustrates the use of the new nonlinear filters in biomedical image processing applications. The results for the various filters yield important information for selecting the proper filter for a given application. Among the considerations for selecting a filter are the signal and noise characteristics of the specific imaging modality and the type of information that is to be extracted from the data. The imaging modalities considered (thermography, magnetic resonance, and ultrasound) have a variety of different characteristics that call for different filters. The theoretical analyses in the earlier sections provide a solid basis for selecting appropriate filters for each modality.

Thermograms are very noisy, and often accurate temperatures need to be estimated from them. The goal in thermographic imaging, then, is to remove the noise without introducing any statistical bias that would affect the accuracy of temperature readings inferred from the images. The linear combinations of morphological operators match this description and provide more control over shape than previous thermographic filtering techniques. Examples of the new filtering technique are given and compared to established filtering methods for thermography.

In magnetic resonance imaging (MRI), image processing problems of clinical interest include improving the contrast and reducing the noise in images, and segmenting images into regions corresponding to different tissue types. The gray levels in an MR image do not correspond to properties that need to be measured quantitatively, so filters that bias or otherwise alter the gray levels may be used. The MLV filter, one of the new value-and-criterion filters, provides excellent contrast enhancement in MRI by sharpening edges between homogeneous tissue regions and simultaneously smoothing the noise within these regions. The results of a single pass of the MLV filter compare favorably to many iterations of another emerging technique for MRI enhancement, anisotropic diffusion [14] . Examples of both techniques are shown and the noise levels for the filtered images are estimated and compared. The edge enhancing and noise smoothing properties of the MLV filter make it an excellent choice as a pre-processor for segmentation algorithms and contrast improvement schemes in MRI analysis.

In ultrasound images, preserving structures (edges and shapes) in the image is more important than determining accurate gray level values. For example, one might only wish to extract a region of interest from a particular ultrasound image to use for a three-dimensional reconstruction. This is a case where the standard, biased morphological operators are expected to work well. This work shows that the standard morphological filters are more appropriate for ultrasound image processing than any of the newly defined unbiased filters. Examples of filtering by both the biased and unbiased filters are given and compared.

The new nonlinear filter structures and analysis techniques introduced here are useful in biomedical image processing applications because they expand the variety of available filters and tools. The characteristics of the imaging modality can be then used to help select a filter that achieves the desired results. These new filters and analysis tools improve the chances of finding a suitable filter for almost any image processing application. The three biomedical applications investigated here give examples of situations where the new filters are useful and situations where other techniques are preferred.

© Copyright by Mark A. Schulze, 1994.

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Mark A. Schulze
http://www.markschulze.net/
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Last Updated: 17 July 2003