Structure Characterization
Using Mathematical Morphology

Thesis cover

The art my mother created as a cover for my thesis.

This thesis deals with the application of mathematical morphology to images of some kind of structure, with the intention of characterizing (or describing) that structure. The emphasis is placed on measuring properties of the real-world scene, rather than measuring properties of the digital image. That is, we require that the measurement tools are sampling-invariant, or at least produce a sampling-related error that is as small as possible. Filters defined by mathematical morphology can be defined both in the continuous space and the sampled space, but will produce different results in both spaces. We term these differences “discretization errors”. Many of the results presented in this thesis decrease the discretization errors of morphological filters.

The size distribution is the main tool used in this thesis to characterize structures. We estimate it using a granulometry, which is the projection of a morphological scale-space on the scale axis. This morphological scale-space is built with a sieve: an operation that is extensive (or anti-extensive), increasing and absorbing. The volume-weighted, cumulative size distribution of the objects in the image follows by normalization of the granulometry. Two variants of this granulometry receive the most attention: one based on isotropic, structural openings or closings, and one based on Rotation-Invariant Anisotropic (RIA) morphology. RIA openings and closings complement the isotropic ones, in that the latter remove objects based on their smallest diameter, whereas the former remove objects based on any of the other diameters (such as the length).

Isotropic structural openings and closings use a disk (or an n-ball in n-D) as structuring element. The RIA openings and closings we are interested in use line segments as structuring elements. These two shapes are extensively studied in this thesis, and we propose various improvements to the classical algorithms that decrease the discretization errors (that is, they improve the discrete approximation to the continuous operation). For any shape, interpolation directly reduces discretization errors by reducing the relative sampling error of that shape. In addition to that, for disks and balls we propose a small shift with respect to the sampling grid to further reduce discretization errors. For the line segment we propose an algorithm based on skews (with interpolation) of the image. Both these shapes can also be improved by using gray-value structuring elements.

The only way of completely avoiding discretization errors in mathematical morphology is using an alternative image representation. For one-dimensional images we propose to use a piece-wise polynomial representation, based on spline interpolation. Due to the continuous nature of this representation, discretization effects are no longer relevant.

We also study the selection of the morphological operation for the granulometry, and some useful pre-processing steps to prepare the image so that the estimated size distribution is more accurate. Among other things, we look at noise-reduction filters and their effect on the estimated granulometry.

On a somewhat different note, the Radon transform (also known as Hough transform) is studied. It detects parametrized shapes in an image, and can therefore also be used to construct a size distribution. The most important difference between the Radon transform and the granulometry is that the former is linear, whereas the latter is strongly non-linear. Both methods do not require any form of segmentation, although they can benefit from pre-processing.

We show how the Radon transform can be defined such that the resulting parameter response function is band-limited. This makes it possible to define a minimal sampling rate for this function, avoiding aliasing. The parameters can therefore be estimated with sub-pixel accuracy. Secondly, the accuracy and precision of the Radon transform for spheres is examined. In particular, we derive a theoretical approximation for the bias in the estimated radii, and propose a way to modify the transform to reduce this bias. Finally, a memory-efficient algorithm for the Radon transform is proposed.

Resources

Literature

These are the publications produced by this project:

Software

Most of the algorithms implemented for this thesis are included in the DIPimage toolbox, except the sampling-free morphology described in Chapter 4, some versions of the line structuring element described in Chapter 5, and the RIA morphology of Chapter 2 (not yet online). See also a page about usage of the granulometry.

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Last modified June 2, 2010.