of image regions and embedded image structures such as clusters of points in image
regions of interest. The discovery of such structures is made possible by quantizers.
A quantizer restricts a set of values (usually continuous) to a discrete value. In its
simplest form in computer vision, a quantizer observes a particula r target pixel
intensity and selects the nearest approximating values in the neighbourhood of the
target. The output of a quantizer is called a codebook by A. Gersho and R.M. Gray
[55, §5.1, p. 133] (see, also, S. Ramakrishnan, K. Rose and A. Gersho [164]).
In the context of image mesh overlays, the Gersho–Gray quantizer is replaced by
geometry-based quantizers. A geometry-based quantizer restricts an image region
to its shape contour and observes in an image a particular target object shape
contour, which is compared with other shape contours that have approximately the
same shape as the target. In the foundations of computer vision, geometry-based
quantizers observe and compare image regions with approximately the same
regions such as mesh maximal nucleus clusters (MNCs) compared with other
nucleus clusters. A maximal nucleus cluster (MNCs) is a collection of image mesh
polygons surrounding a mesh polygon called the nucleus (see, e.g., J.F. Peters and
E. İnan on Edelsbrunner nerves in Voronoï tessellations of images [150]). An
image mesh nucleus is a mesh polygon that is the centre of a collection of adjacent
polygons. In effect, every mesh polygon is a nucleus of a cluster of polygons.
However, only one or more mesh nuclei are maximal.
A maximal image mesh nucleus is a mesh nucleus with the highest number of
adjacent polygons. MNCs are important in computer vision, since what we will call
a MNC contour approximates the shape of an underlying image object. A Voronoï
tessellation of an image is a tiling of the image with polygons. A Voronoï tessel-
lation of an image is also called a Voronoï mesh. A sample tiling of a musician
image in Fig. 0.1.1 is shown in Fig. 0.1.2. A sample nucleus of the musician image
tiling is shown in Fig. 0.2.1. The red
dots inside each of the tiling polygons are
examples of Voronoï region (polygon) generating points. For more about this, see
Sect. 1.22.1. This musician mesh nucleus is the centre of a maximal nucleus cluster
shown in Fig. 0.2 .2. This is the only MNC in the musician image mesh in Fig. 0.1.2.
This MNC is also an example of a Voronoï mesh nerve. The study of image MNCs
takes us to the threshold of image geometry and image object shape detection. For
more about this, see Sect. 1.22.2.
Each image tiling polygon is a convex hull of the interior and vertex pixels.
A convex hull of a set of image points is the smallest convex set of the set of points.
A set of image points A is a convex set, provided all of the points on every straight
line segment between any two points in the set A is contained in the set. In other
words, knowledge discovery is at the heart of computer vision. Both knowledge and
understanding of digital images can be used in the desig n of computer vision
systems. In vision system designs, there is a need to understand the composition
and stru cture of digital images as well as the methods used to analyze captured
images.
The focus of this volume is on the study of raster images. The sequel to this
volume will focus on vector images, which are compo sed of points (vectors), lines
and curves. The basic content of every raster image consists of pixels
viii Preface
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