Renée Heilbronner

Image Analysis in Earth Sciences

Part I Looking at Images.- 1 Images and Microstructures.- 2 Acquiring Images.- 3 Digital Image Processing.- 4 Pre-processing.- Part II Segmentation: Finding and Defining the Object.- 5 Segmentation by Point Operations.- 6 Post-processing.- 7 Segmentation by Neighborhood Operations.- 8 Image Analysis.- 9 Test Images.- Part III Measuring Size and Volume.- 10 Volume Determinations.- 11 2-D Grain Size Distributions.- 12 3-D Grain Size.- 13 Fractal Grain Size Distributions.- Part IV Quantifying Shape and Orientation.- 14 Particle Fabrics.- 15 Surface Fabrics.- 16 Strain Fabrics.- 17 Shape Descriptors.- Part V Spatial Relationships.- 18 Spatial Distributions.- 19 Spatial Frequencies.- 20 Autocorrelation Function.- Part VI Orientation Imaging.- 21 Crystal Orientation and Interference Color.- 22 Computer-Integrated Polarization Microscopy.- 23 Orientation and Misorientation Imaging.- Index.

Images and Microstructures

Throughout this book we will be looking at images as we try to extract and analyze in a quantitative way the information contained in them. Most of the images will show microstructures of various types of rocks and the major part of the book is dedicated to methods by which the geometry—the size, shape, number, spatial distribution, etc.—of crystals, grains or surfaces can be quantified. The choice of methods for analysis will depend on the type of microstructure present in an image and on the nature of the image itself. It is therefore useful to ask ourselves at the very beginning what we mean by ‘images’ and what we think ‘microstructures’ are.

Segmentation by Neighborhood Operations

Neighborhood operations are distinct from point operations in that they consider not only the gray value of a given pixel, but also those of the pixels in its neighborhood. In Chap. 5, point operations were used to calculate new gray values from old ones, and point operations were used to achieve segmentations based on gray values only, typically by thresholding or gray level slicing, as discussed in Chap. 1 (Figs. 1.12 and 1.13). Point operations and look-up tables only consider one gray value at the time—they are ‘insensitive’ to the gray values of pixels.

Computer-Integrated Polarization Microscopy

The purpose of the method presented here—called CIP for computer-integrated polarization microscopy—is to derive c-axis orientations from optical micrographs and present the results in the form of pole figures and orientation. For an introduction to the method, see the original publication by Panozzo Heilbronner and Pauli (1993) and the web publication Heilbronner (2000); for application to natural and experimental materials, see, for example, Kilian et al. (2011a, b), Muto et al. (2011), Heilbronner and Tullis (2006), Trullenque et al. (2006) or Stipp et al. (2002). Two implementations of the CIP method exist:

Segmentation by Point Operations

In this chapter we will attempt a first image segmentation; we will try to identify image segments that represent interesting objects such as, for example, mineral grains. To discriminate the pixels that belong to the segments from those that do not belong we consider the gray values of the individual pixels without considering their neighborhoods. In other words, we use point operations (POPs). We will formulate criteria to discriminate segments based on the gray values alone, irrespective of where they occur.

Crystal Orientation and Interference Color

In this chapter we prepare the ground for computer-integrated polarization microscopy (CIP), a method for optical texture analysis and optical orientation imaging (Panozzo Heilbronner and Pauli 1993). The CIP method uses the relation between crystallographic orientations and the interference colors as they appears when thin sections are viewed in the polarization microscope. Optical micrographs taken under specified conditions of polarization serve as input in the calculation of orientation images and pole figures.

Fractal Grain Size Distributions

In the previous chapters, we have been concerned with determining grain size, mean grain size or modal grain size. We looked at grain size distributions, number histograms, and volume weighted histograms. The idea was that the grain size distributions were created by physical processes that select or favor one typical grain size. As a consequence, the analytical effort went into determining that particular grain size as accurately as possible.

2-D Grain Size Distributions

Geomaterials are polycrystalline aggregates—they are composites whose bulk behavior and bulk properties are determined (a) by the physical and chemical properties of the minerals of which they are composed, and (b) by the ‘geometrical properties’ such as shape, size and spatial distribution of the components. As an example, consider a granite and a gneiss: they may be of identical composition, but since their geometry is different, their bulk properties are different too. The aim of image analysis is to quantify the geometry of materials. Our first exercise in the previous chapter (on volume determinations) was aimed at finding a descriptor for a very simple aspect of geometry—the ‘amount’ of a given phase. We used the sum of the cross sectional areas of that phase and determined the volume fraction. We considered the size of the individual cross sections only in so far as we used their statistics to estimate the error, but the focus was on the ‘grand total’ of every phase. In this chapter, we now turn to the size of the individual cross sections.

3-D Grain Size

Converting a size distribution of sectional circles to the corresponding size distribution of spheres is probably one of the oldest problems in stereology. It is not just an academic question but a serious practical problem whenever the grain size of loose sand or particulate matter is to be compared with the grain size of the corresponding rock type or solid material. When a mean grain size has to be determined from thin sections, it is common practice to convert a two-dimensional (2-D) mean to three dimensions by applying a multiplicative factor (of 1.3 or 1.5) to account for the fact that the cross section of a grain can always be smaller but never larger than the grain itself. However, there is no mathematical relationship between the statistical parameters (mean, variance, skewness, etc.) of the size distribution of sections and those of the size distribution of three-dimensional (3-D) grains. In particular, the mean size of the 2-D sections can be smaller than, equal to or larger than the mean size of the 3-D grains. This means that if we compare the means of 2-D grain size distributions, we may draw completely wrong conclusions with regard to the true (3-D) mean grain size.

Orientation and Misorientation Imaging

As mentioned in the previous chapter, the purpose of the CIP method (computer-integrated polarization microscopy) is to perform three basic tasks: