Relationship between base vertex and plateau vertex Equation 4. A candidate frame that is a plateau vertex must have one corner coinciding with a base vertex. There are three reasons. For example, the top left base vertex in Figure 4. In fact, a base vertex that does not overlap with any REAL edge of a requested viewing zone does not have a corresponding plateau vertex. Lemma 2 Base Vertex Optimality Condition.
malocbedi.ml At least one optimal frame has one corner coinciding with a base vertex. Using the Base Vertex Optimality Condition BVOC , we can restrict frames to coincide one of its corner with a base vertex, thereby reduce the dimensionality of the problem. The BVOC is true no matter whether the resolution variable is discrete or continuous. However, it is more convenient to use plateau vertices when the resolution variable is discrete.
When the resolution variable is discrete, at least one optimal frame is coinciding with a plateau vertex. Brute force approach Based on the sLemma 3, we can solve the optimization problem by simply checking all combinations of resolution levels and corresponding plateau vertices.
We evaluate the objective function for each of the O n2 plateau vertices and repeat this for each of the m resolution levels. It takes O n time to evaluate a candidate frame c. Therefore, the brute force algorithm runs in O n3 m. The Plateau Vertex Traversal Algorithm is summarized below. It reduces the computation complexity from O n3 m to O n2 m. In step iii of the PVT algorithm, we traverse the vertical facet boundaries of the plateaus one by one. This process only takes O n. Therefore, step iii of the algorithm will take O n2 , proving the following theorem.
Theorem 1. We can solve the Satellite Frame Selection problem in time O n2 m for n users and m resolution levels. Figure a shows how we sweep along x axis to dissect the 2D optimization problem into O n 1D optimization problems.
Instead, we can use base vertex optimality condition to reduce the 3D optimization problem to O n2 1D optimization problems with respect to variable z. Using incremental computation and a diagonal sweep, we show how to improve the running time to O n3. This means that we can reduce the original 3D optimization problem in 4. To study the 1D maximization problem in 4. For simplicity, we assume that the base vertex is at the origin. Moreover, we assume that the base vertex coincides with the lower left corner of the candidate frame.
The base vertex in Figure 4. Placements in which one of the other three corners of the candidate frame coincides with the base vertex are handled in a similar fashion. We may be able to eliminate some of the placements beforehand, but it reduces the computation by only a constant factor.
Now, we gradually increase z and observe the value of s z : Figure 4. Critical z Values and Intersection Topologies. The function s z is a piecewise smooth function see Figure 4. We refer to a maximal z-interval on which s z is smooth as a segment. We consider four questions that form the basis for our algorithms. Can we give a geometric characterization of the endpoints of the segments? How many segments are there?
What is the closed-form description of s z within a single segment, and how complex is the computation of the maximum of s z on that segment? An example of the 1D optimization problem with respect to z.
We start with question 1. Let Zc xv , yv be the set of critical z values for base vertex xv , yv. From 4. It also changes to type 1 once it becomes fully contained. The transitions correspond to critical z values.
We can ignore class o fundamental rectangles because they do not contribute to our objective function. A requested viewing zone that is a fundamental rectangle from class a or b generates at most two critical z values. Many of the requested viewing zones though will not be fundamental rectangles. We resolve this by decomposing those requests. Requested viewing zone decomposition. A requested viewing zone that is not a fundamental rectangle intersects at least one of following: the positive xaxis, the positive y-axis, and the extended diagonal of the expanding candidate 4.
Examples of four requested viewing zone decomposition cases frame.
Transmission : the channel whereby the information is transmitted between the local and the remote zone. Smith AC. Your credit is blocked the forensic issue of diseases. A surgical robot with vision field control for single port endoscopic surgery. For starsOn, for pages as supremely as for pumps, break out our Wiki.
Every fundamental rectangle inherits the zi value of the original request. Since the critical z values partition the z axis into O n segments, on each of which s z is a smooth function, the following lemma is true.
Sharing a Vision. Systems and Algorithms for Collaboratively-Teleoperated Robotic Cameras. Authors: Song, Dezhen. Free Preview. Excellent picture of the . Sharing a Vision: Systems and Algorithms for Collaboratively-Teleoperated Robotic Cameras (Springer Tracts in Advanced Robotics) [Dezhen Song] on.
Lemma 4. For each base vertex, the z-axis can be partitioned into O n segments, on which s z is smooth. Optimization Problem on a Segment. With the knowledge of question 1 and 2 , we are ready to attack question 3 : derive a closed-form representation of s z on a segment and solve the constrained optimization problem.
We have the following lemma. The order of the resulting polynomial depends on the resolution discount factor b , Lemma 5. Therefore, 4. The maximum of 4. Theorem 2. For general CRR metrics that consist of continuous elementary functions, 4. The critical z value belongs to some rectangle. This update only takes constant time.
To exploit this coherence we must sort the elements of Zc xv , yv in the inner loop to be able to consider the segments in order; this takes O n log n time. We replace the inner loop in BV by the following subroutine.
The question is: is it necessary to sort critical z values repeatedly for each base vertex? Each critical point corresponds to the point that the candidate frame start intersecting some requested viewing zone or the point that the intersection between the candidate frame and some requested viewing zone ends.