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Energy-aware stage illumination.

Energy-aware stage illumination. Written by: Friedrich Eisenbrand Stefan Funke Andreas Karrenbauer Domagoj Matijevic. Presented By: Yossi Maimon. Illumination Problem. Input: Given a stage and a set of light sources. Target: To illuminate the stage such that each point

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Energy-aware stage illumination.

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  1. Energy-aware stage illumination. Written by: Friedrich Eisenbrand Stefan Funke Andreas Karrenbauer Domagoj Matijevic Presented By: Yossi Maimon.

  2. Illumination Problem Input: Given a stage and a set of light sources. Target: To illuminate the stage such that each point on the stage receives a sufficient amount of light (one unit) while minimizing the overall Power assignment.

  3. Illuminate Vs Guarding • Illuminate is a different version of guarding problem. • Energy from a light source is decreasing quadratically with the distance . • Add up: A point can accumulate energy from different light sources.

  4. Light Energy is homogenously distributed over the surface.

  5. Example

  6. Article Contribution • A polynomial-time solution based on convex programming. • A approximate solution based on a discretization and linear programming. • A purely combinatorial O(1) approximate solution with running time

  7. Convex programming S – light source. X - The energy of light source. P – Point on the stage.

  8. Convex programming (cont) K-Convex body All point in distance from K All point in distance from K contained in the convex Target: To determine if a point is in K. The constraints are not bounded so we will look for lighting LP and combinatorial

  9. Pruning light sources • Under the assumption that each light source can be assigned arbitrary high power. • Only light sources whose Voronoi cells are intersect the stage can be part of optimal solution. Let s be a light source whose Voronoi cell does not intersect the stage, be the first neighbors to the left and right whose Voronoi cells intersect the stage.

  10. A approximation scheme • Goal: • Discrete the problem by using a finite number of guard. • Solve the linear programming only for the guards • power up all light sources • In the end: each point on the stage that isn’t a guard will Receive a enough light. Guard: a point on the stage that receive a sufficient amount of energy Definition:

  11. approximation (cont) Set of guards Construct: Assume |S|=1. Let p0 be the closest point to s. Add p0 to G. Build p-1 and p+1 in

  12. approximation (cont 1) Numbers of guards: • Several light sources: • For each light source |Gs| will be computed. • Union all the sets. D denote the length of L. The constraints is depend on the length of the stage.

  13. approximation (cont 2) Powering: Powering every light source in Ensures that every point receive enough light. Summery: The light source energy can be found by solving LP with constraint and n variables.

  14. A simple O(1) approximationAlgorithm Restricting the problem to O(n) guards. Transfer back to the original problem in O(1) in terms of quality. Lemma: 4*Xv is power assignment to all point on the stage. A 4 approximation can be solve by LP with n+1 constraint. Independent to the length of the stage.

  15. simple O(1)-Pruning guards • Compute for each guard p the ens(p). • Sort the guards in decreasing order. • For i=1…n if has not been remove yet, remove all guards at distance • Return the guards as Gp.

  16. simple O(1) • Compute the set of guards Gv (via the Voronoi diagram of S). • Prune the set of guards Gv with pruning constant to obtain Gp, |Gp|=m. • Let Gp be ordered such that • For all i=1..m Running time: No guard gets more then a constant amount of energy.

  17. simple O(1) – (cont) Definitions: will be the amount of light in Energy from light sources where j<i Energy from light source Energy from light sources where j>i

  18. Open problems Art gallery illumination with fix number of light source. Given polygon with n vertices and k fix number of light sources, determine the position and power to each light source such that each point (on edge or interior) has at least 1 unit of energy. Another variant is to restrict the position of the light source only on vertices or edges.

  19. Open problems (cont) Stage illuminations with obstacles: The same problem only this time with obstacle. The pruning light sources nor the disretization can be applied immediately.

  20. Results Performance according to the Analysis. D-The length of the stage. |Gv|-Number of light sources.

  21. Actual result Adaptive power up. Using a refined power up strategy the algorithms achieve result closest to The optimal.

  22. Conclusions • Solving stage illuminate. • The model take in consideration decreasing energy over distance. • Using a fix set of light source. • Minimize the overall energy.

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