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Energy-aware stage illumination.

Written by:

Friedrich Eisenbrand

Stefan Funke

Andreas Karrenbauer

Domagoj Matijevic

Presented By: Yossi Maimon.

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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.

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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.

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LightEnergy is homogenously distributed over the surface.

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Example

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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 2( )O n

(1 )

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Convex programming

2

min

. .

: 1( , )

0

ss S

s

s

Objective

x

s txp L

d p sx

S – light source.

X - The energy of light source.

P – Point on the stage.

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Convex programming (cont)

( , )S K ( , )S K All point in distance from K

All point in distance from K contained in the convex

K-Convex body

The constraints are not bounded so we will look for lighting LP and combinatorial

Target:To determine if a point is in K.

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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.

,l rs s

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A approximation scheme(1 )

Guard: a point on the stage that receive a sufficient amount of energy

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.

, ( ) min ( , )s Sp L ens p d p s Definition:

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approximation (cont)(1 )

good

, : ( , ) * ( )G Lp L g G d p g ens p

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

02 ( )ens p

1( , ) 2 ( )i i id p p ens p

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approximation (cont 1)(1 )

log| | ( )

DGs O

D denote the length of L.

The constraints is depend on the length of the stage.

Numbers of guards:

Several light sources:

•For each light source |Gs| will be computed.

•Union all the sets.

• , : ( , ) ( )p L g G d p g ens p

| | ( log[ ])n D

Gs On

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approximation (cont 2)(1 )

Powering:

Powering every light source in

Ensures that every point receive enough light.

(1 6 )

Summery:

(1 ) opts s

s S s Sx x

The light source energy can be found by solving LP with

constraint and n variables.( log )n DOn

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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.

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1. Compute for each guard p the ens(p).

2. Sort the guards in decreasing order.

3. For i=1…n

if has not been remove yet, remove all guards at distance

4. Return the guards as Gp.

simple O(1)-Pruning guards

ip

| * | *max( ( )* ( )), 0i j i jg g C ens g ens g C

jp * ( )iens p

,

| | ( )

i i j

i j j

p Gv p Gp p Gp

p p ens p

2 2(1 ) * 4(1 )Xv Xp

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simple O(1)

1. Compute the set of guards Gv (via the Voronoi diagram of S).

2. Prune the set of guards Gv with pruning constant to obtain Gp, |Gp|=m.

3. Let Gp be ordered such that

4. For all i=1..m

1 2( ) ( ) .... ( )mens p ens p ens p

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max{0,| * | *(1 )}| * |

i ji i i j

i j

xx p s

p s

Running time: 2( )O n

No guard gets more then a constant amount of energy.

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simple O(1) – (cont)Definitions:

will be the amount of light in ipiP

4 1 6 11i i i iP P P P

i

i

i

P

P

P

Energy from light sources where j<i

Energy from light source

Energy from light sources where j>i is

is

is

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Open problemsArt 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.

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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.

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ResultsPerformance according to the Analysis.

D-The length of the stage.

|Gv|-Number of light sources.

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Actual resultAdaptive power up.

Using a refined power up strategy the algorithms achieve result closest to

The optimal.

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Conclusions

•Solving stage illuminate.

•The model take in consideration decreasing energy over distance.

•Using a fix set of light source.

•Minimize the overall energy.