Morphological granulometries were conceived by Matheron for the purpose of analyzing images with respect to shape and size of elementary granular components. In particular, granulometric analysis has proven useful in the analysis and identification of textures. Morphological granulometries are filter sequences that measure the residual area after each iteration of the filter. Every component of the filter's structuring element set is dilated for each iteration of the filter and the area resulting from the union of the morphological openings of the original image produces a function of a single variable, the iteration index. If the n components of the structuring element set are allowed to take on values independent of each other, the area after the opening for all combinations of structuring element size provides an n dimensional representation of the image with area after opening a function of all the various sizes of the different structuring elements results. Just as traditional granulometries can be manipulated to provide a signature useful in texture discrimination the n dimensional representation does so as well, although the traditional moment analysis technique is not applicable because the multivariate granulometry does not define a probability density function. Instead an orthonormal projection method is used to represent the transformed image by an arbitrary number of Fourier coefficients. The Fourier coefficients provide a feature space from which a number of features can be selected for the purpose of texture discrimination. For this research, Fourier coefficients (sequency constants) of a Walsh representation are used to characterize the image texture. A feature selector using the Mahalanobis-Like probabilistic distance measure provides a mechanism for reducing the feature set to a mathematically tractable number of features. A Gaussian maximum likelihood classifier is used to identify unknown samples from a finite set of texture patterns. The classifier produces good results when classifying 12 textures in the absence of image noise. Results when the images are corrupted with 10% point noise are poor unless the classifier is trained in the presence of the noise. The classifier also exhibits the ability to distinguish reasonably well the presence or absence of point noise, within a given texture, when the classifer is trained for both conditions.
Library of Congress Subject Headings
Image processing--Digital techniques--Mathematics; Image processing--Mathematics; Visual texture recognition; Walsh functions
Department, Program, or Center
Chester F. Carlson Center for Imaging Science (COS)
Rzadca, Mark, "Multivariate granulometry and its application to texture segementation" (1994). Thesis. Rochester Institute of Technology. Accessed from
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