Quantification of Phase Distribution: Multiple Area Density Maps Using a Bitmap Image

The spatial distribution pattern of constituent phases of rocks has important information for understanding nucleation processes of phases in igneous and metamotphic rocks, timing relationship between partial melting and deformation in migmatites, and effect of one phase on static or dynamic grain growth of another phase in metamorphic rocks. For the quantification of the spatial distribution of a phase, the most widely used is the `distance-to-nearest-neighbor' method. However, the effects of local clustering, size and shape of a phase are not conisdered in this method. Although several methods have been proposed to overcome this problem, they do not provide a simple numerical result or show a complex relationship between the factors affecting the phase distribution. Here, we introduce a new method for much simpler quantification of the phase distiribution using image analysis techinques. In a bitmap image, the `area density map' (ADM) is constructed by calculating pixel density of the objects (grains of a phase) at each pixel position for a unit area and the area of the objects within the unit area. In this single ADM, the density of an object increases continuously toward the center of the object reflecting the object size. By overlaying single ADM's with progessively increasing the size of the unit area, we can produce a synthesized map (or multiple ADM) having an average area density of all ADM's overlaid. In this multiple ADM, the average area density at each pixel position represents a value combining four factors; total fraction, clustering pattern, size and shape of the objects on an image. In a bitmap image containing objects on a null background, a pixel is used as a basic element. Pixels of input images have specific gray scale values to distinguish a grain from the background and individual grains. To construct the multiple ADM, we substitute an average density value calculated from number of pixels for the gray scale value at each position (i.e. attribute of each pixel). To quantify the effect of the size and shape of the objects, pixels only on indivudual objects are used for a statistical calculation. In simple cases having a constant shape, size and fraction of objects, our total area density of input images gives a specific numerical value representing the distribution pattern of objects (e.g. clustered, random and uniform distribution). Also, our multiple ADM can single out a numerical value for the effect of local clustering of the objects and has a potential to numerically analyze the effect of the size and shape of the objects.