We prove the convergence of the simulated annealing procedure when the decision to change the current configuration is blind of the cost of the new configuration. In case of filtering binary images, the proof easily generalizes to other procedures, including that of Metropolis. We show that a function Q associated with the algorithm must be chosen as large as possible to provide a fast rate of convergence. The worst case (Q constant) is associated with the "blind" algorithm. On the other hand, an appropriate Q taking sufficiently high values yields a better rate of convergence than that of Metropolis procedure. >

Simulated annealing: a proof of convergence

Let S be a metric space under the distance function d. A metric basis is a subset B ⊆ S such that d(b, x) = d(b, y) for all b ϵ B implies x = y. It is shown that for Euclidean distance, the minimal metric bases for the digital plane are just the sets of three noncollinear points; but for city block or chessboard distance, the digital plane has no finite metric basis. The sizes of minimal metric bases for upright digital rectangles are also derived, and it is shown that there exist rectangles having minimal metric bases of any size ≥ 3.

Metric bases in digital geometry

In this paper, we deal with the problem of sequencing parts and robot moves in a robotic cell where the robot is used to feed machines in the cell. The robotic cell, which produces a set of parts of the same or different types, is a flow-line manufacturing system. Our objective is to maximize the long-run average throughput of the system subject to the constraint that the parts are to be produced in proportion of their demand. The cycle time formulas are developed and analyzed for this purpose for cells producing a single part type using two or three machines. A state space approach is used to address the problem. Both necessary and sufficient conditions are obtained for various cycles to be optimal. Finally, in the case of many part types, the problem of scheduling parts for a specific sequence of robot moves in a two machine cell is formulated as a solvable case of the traveling salesman problem.

Sequencing of Parts and Robot Moves in a Robotic Cell

Digital geometry is the study of geometrical properties of subsets of digital images. If the digitization is sufficiently fine-grained, such properties can be regarded as approximations to the corresponding properties of the "real" sets that gave rise, by digitization, to the digital sets; but it is also important to define how the properties can be computed for the digital sets themselves. Questions of particular interest include how images and image subsets are digitized; how geometric properties are defined for digitized sets; the computational complexity of computing them--in particular, whether they can be computed using simple (e.g., local) operations; characterizing image operations that preserve them; and characterizing digital objects that could be the digitizations of real objects that have given geometric properties. Concepts that have been extensively studied include topological properties (connected components, boundaries); curves and surfaces; straightness, curvature, convexity, and elongatedness; distance, extent, length, area, surface area, volume, and moments; shape description, similarity, symmetry, and relative position; shape simplification and skeletonization.

Digital geometry

Three-dimensional (3D) scan-conversion algorithms, that scan-convert 3D parametric objects into their discrete voxelmap representation within a Cubic Frame Buffer (CFB), are presented. The parametric objects that are studied include Bezier form of cubic parametric curves, bicubic parametric surface patches, and tricubic parametric volumes. The converted objects in discrete 3D space maintain pre-defined application-dependent connectivity and fidelity requirements.The algorithms introduced here emply third-order forward difference techniques. Efficient versions of the algorithms based on first-order decision mechanisms, which employ only integer arithmetic, are also discussed. All algorithms are incremental and use only simple operations inside the inner algorithm loops. They perform scan-conversion with computational complexity which is linear in the number of voxels written to the CFB. All the algorithms have been implemented as part of the CUBE Architecture, which is a voxel-based system for 3D graphics.

Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes

Definitions of 3-D digital surface and plane are introduced. Many geometric properties of these objects are examined. In particular, it is shown that digital convexity is neither a necessary nor a sufficient condition for a digital surface element to be a convex digital plane element, but it is both necessary and sufficient for a digital surface to be a digital plane. Also algorithms are presented to determine whether or not a finite set of digital points is a (convex) digital plane element.

Three-Dimensional Digital Planes

Digital arcs in 3-D digital pictures are defined. The digital image of an arc is also defined. A digital arc is defined to be a digital line segment if it is the digital image of a line segment. It is shown that a digital line segment may be characterized by the chord property holding for its projections onto the coordinate planes. It is also shown that a digital line segment may not be characterized by its own chord property. A linear time algorithm is presented that determines whether or not a digital arc is a digital line segment.

Three-Dimensional Digital Line Segments

Geometric properties of digital disks are discussed. An algorithm is presented that determines whether or not a given digital region is a digital disk.

https://ieeexplore.ieee.org/iel5/34/4767514/04767531.pdf

Digital Disks

A period on digital line segments is defined. It is shown that each of the two nearest supports contains a preimage of the digital line segment and at least one of them contains two points of the digital line segment. Linear time algorithms are presented which find the nearest support of a 2-D digital line segment, four and eight parameters that represent 2-D and 3-D digital line segments, respectively, and construct the complete set of preimages of 2-D and 3-D digital line segments.

Chul E. Kim

Papers

Three-Dimensional Digital Planes

Three-Dimensional Digital Line Segments

Digital Disks

Representation of digital line segments and their preimages

On the recognition of digital planes in three-dimensional space