Image Processing Reference
In-Depth Information
2.5.3
Hyperspheres of m-Neighbor Distance ..................
62
2.5.3.1
Vertices of Hyperspheres ................... 65
2.5.3.2
Errors in Surface and Volume Estimations
65
2.5.3.3
Hyperspheres of Real m-Neighbor Distance
66
2.5.4
Hyperspheres of t-Cost Distance ........................ 68
2.5.5
Hyperspheres of Hyperoctagonal Distances .............
69
2.5.5.1
Vertices of Hyperspheres and
Approximations ............................
70
2.5.5.2
Special Cases of Hyperspheres in 2-D and
3-D .........................................
72
2.5.6
Hyperspheres of Weighted t-Cost Distance .............
75
2.5.6.1
Proximity to Euclidean Hyperspheres .....
78
2.6
Error Estimation and Approximation of Euclidean Distance ...
80
2.6.1
Notions of Error ......................................... 81
2.6.2
Error of m-Neighbor Distance ........................... 81
2.6.3
Error of Real m-Neighbor Distance ..................... 82
2.6.4
Error of t-Cost Distance ................................. 83
2.6.4.1
Error of t-Cost Distance for Real Costs ...
84
2.7
Summary ......................................................... 85
Exercises ......................................................... 85
Image processing in more than two dimensions has attracted a lot of interest
recently. Three-dimensional image processing has several applications, such as
computed tomography. The inclusion of time has increased the three spatial
dimensions to four in studies involving moving objects. Various applications
involving gray-scale pictures and objects with several features pertaining to a
single dimension require representation in higher dimensions.
Studies on the topological properties of quantized spaces are required in im-
age processing applications. This has led to a non-Euclidean geometry known
as digital geometry (or digital topology). A few results in 2-D and 3-D digital
geometry were established early [182, 177, 178] during the evolution of this
area and the need to handle information in higher dimensions necessitated
explorations in arbitrary dimensions (n-D). Many such studies as undertaken
in the past couple of decades are summarized in [118, 114].
Since the measurement of distance between elements is essential in many
applications, a digital analogue of a non-Euclidean distance in n-D forms an
important and interesting part of digital geometry. Such distances find use in
segmentation, merging, thinning, clustering, and the like.
In this chapter, we introduce the generalized notions of neighborhoods,
paths and distances in n-D digital geometry to present a number of classes of
digital distances based on them. We present their shortest path algorithms,
prove the conditions for metricity, study the structures of their hyperspheres
and estimate the errors in geometric (shapes of hyperspheres with respect to
the Euclidean hypersphere) and direct (differences in measures from Euclidean
