Computer Vision, Graphics, and Image Processing
Digital images of geometric pictures
Abstract
Digital geometric pictures, which are the digital images of real geometric pictures, are introduced. Especially, rectangles, equilateral triangles, and isosceles right triangles are treated here. Characterizations of sets of lattice points constructing parallel lines and right angles are investigated first. A necessary and sufficient condition for a given finite set of lattice points to be the digital image of a rectangle is given by making use of these characterizations. Then, characterizations of angles of 60° and 45° are investigated from a point of view of digital geometry. Finally, necessary and sufficient conditions for digitized equilateral triangles and isosceles right triangles are shown.
References (8)
- C.E Kim
On cellular straight line segments
Comput. Graphics Image Process
(1982) - A Nakamura et al.
Digital circles
Comp. Vision Graphics Image Process
(1984) - A Rosenfeld
Digital straight line segments
IEEE Trans. Computer
(1974) - A Rosenfeld et al.
How a digital computer can tell us whether a line is straight
Amer. Math. Monthly
(1982)
Cited by (10)
Reconstruction of a digital circle
1994, Pattern RecognitionSince digitization always causes some loss of information, reconstruction of the original figure from a given digitization is a challenging task. Reconstruction of digital circles has already been addressed in the literature. However, an in-depth analysis of an OBQ image of a continuous circle as well as a solution to its domain construction problem is still lacking. In this paper a detailed analysis of digital circles has been carried out. A modified I_R method is formulated to numerically compute the domain of each digital quarter circle for a given radius. Several properties of the OBQ image of a circle reveal that in many cases it is possible to split a digital circle into four digital quarter circles, such that the domains of the individual quarter circles can be combined to obtain the domain of the full circle. Moreover, the domain of a quarter circle is geometrically characterized.
Multi-orientation scanning: A necessity in the identification of pore necks in porous media by 3-D computer reconstruction from serial section data
1994, Journal of Colloid And Interface ScienceThe ability to correctly partition the three-dimensional pore space network of a porous medium, e.g., a petroleum-bearing reservoir rock, into the physically significant bulges (pores) and constrictions (necks) depends on our ability to identify and locate the necks in a 3-D reconstructed data set. P;Simulated model media giving necks of known position and orientation in 3-D were used to validate the multiorientational scanning approach used in our work and to demonstrate that scanning from a single orientation cannot detect all necks and probably misses more than half of them. The required resolution of the data set for correct partitioning of the pore space as a function of the number of scanning orientations is established, which allows us to draw conclusions on the optimal number of scanning orientations to use. The multiorientational scanning approach does not mean that physical scanning of an actual sample from several orientations is required. Generally all but one of the scans are scans of the 3-D array created in computer memory from data input from a single orientation physical scan.
We define the critical departure angle for a given scanning scheme as the angle at which the scans from all orientations used completely cover the whole orientational space. This is a constant in the continuous domain but is a weakly increasing function of resolution in the discrete domain of the data set. We define the maximum detection angle for a given scanning scheme as the largest angle between the unit normal of a neck and a scanning orientation at which the neck will be detected from the scan from that orientation. It is an increasing function of the resolution of the data set. The minimum resolution required for a scheme to have the ability to correctly identify necks of any orientation in 3-D is that at which the maximum detection angle exceeds the critical departure angle.
Possible multiorientational scanning schemes which may be formed and applied for neck detection in 3-D were outlined. Four of the schemes were investigated and compared: (a) the three-orientation scheme (scans from the three orthogonal directions), (b) the seven-orientation scheme (scans as in (a) plus from four of the six diagonal directions), (c) the nine-orientation scheme (scans as in (a) plus from the six diagonal directions), and (d) the thirteen-orientation scheme (scans as in (c) plus from the four corner-to-corner directions). The quantitative relative efficiency analysis of the four schemes indicates that the nine-orientation scheme is much more efficient than the seven-orientation and three-orientation schemes, and probably slightly less efficient than the thirteen-orientation scheme. However, incorporating the four corner-to-corner scans necessary for the thirteen-orientation scheme would require a very substantial software development effort. On balance, considering both processing and development costs, we conclude that the nine-orientation scheme is the optimal approach.
Parameter estimation and reconstruction of digital conics in normal positions
1992, CVGIP: Graphical Models and Image ProcessingReconstruction of the original curve (and the estimation of its parameters) from its digitization is a challenging problem as quantization always causes some loss of information. So we often estimate at least one (or all) continuous curve(s) which is (are) isomorphic to the original one under discretization. Some work has already been done in this respect on straight lines, circles, squares, etc. In this paper, we have attempted this problem for a specialized class of conics which are said to be in normal positions. In normal position the center of the conic is situated at a grid point and its axes are parallel to the coordinate axes. For circles and parabolas, we can directly formulate the domain, i.e., the entire set of continuous curves which produces the same digitization. For ellipses (and this can be extended to hyperbolas too), we first compute the smallest rectangle containing the domain of the given digitization and then estimate the domain itself. The major contribution of this paper lies in the development of a new method of analysis (via the iterative refinement of parameter bounds) which can be easily extended to other 1- or 2-parameter piecewise monotonic shapes such as straight lines or circles with known radius.
Efficient measurement procedures for compound part profile by computer vision
1991, Computers and Industrial EngineeringIt is apparent that automated inspection for manufacturing is on the threshold of broad industrial utilization. One key problem in manufacturing applications of automated inspection is how to find fast and efficient methods using economical computers that industry can afford. Moreover, most mechanical designers set only overall tolerances for part geometric features. In computer vision inspection, as well as in the use of other automated inspection devices and coordinate measuring machines (CMM) the errors of representing each geometric features should be identified separately. The proposed statistical inference method provides a scientific basis for setting inspection tolerances in original geometric space which are compatible with engineering specifications.
The results of this presentation should supply industrial practitioners with an accurate and fast approach for on-line part profile inspection.
Digital squares
1990, Computer Vision, Graphics and Image ProcessingAs one of our serial works on digital geometry, digital squares are discussed. The purpose of this paper is to give an algorithm to decide whether or not an arbitrary set of lattice points is the digitization of a real square.
Picture processing: 1985.
1986, Computer Vision, Graphics, & Image ProcessingThis paper presents a bibliography of nearly 1100 references related to the computer processing of pictorial information, arranged by subject matter. Coverage is restricted, for the most part, to a selected set of U.S. journals and proceedings of specialized meetings. The topics covered include digitization, approximation, and compression; transforms, filtering, enhancement, restoration, and reconstruction; architectures, systems, software, and techniques; pictorial pattern recognition; feature detection, segmentation, and image analysis; matching and time-varying imagery; shape and pattern; geometry; texture; and three-dimensional scene analysis. No attempt is made to evaluate or summarize the items cited; the purpose is simply to provide a convenient compendium of references.