Localized Algorithm for Precise Boundary ... - Semantic Scholar
Localized Algorithm for Precise Boundary Detection in 3D Wireless Networks Hongyu Zhou, Su Xia, Miao Jin and Hongyi Wu
integrated Wireless Information Network (iWIN) Lab The Center for Advanced Computer Studies(CACS) University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Outline 1. Introduction 2. Boundary detection 2.1 Unit Ball Fitting (UBF) 2.2 Isolated Fragment Filtering (IFF)
3. Triangular Boundary Surface Construction 4. Simulation 5. Conclusion
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
1. Introduction Motivation Boundary nodes serve as a key attribute that characterizes the network, especially in geographic exploration tasks such as terrain and underwater reconnaissance. Many wireless networks exhibit randomness
Related works All in 2D wireless networks
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Paper Contributions Find a localized method that can precisely detect boundary nodes in 3D wireless networks; Develop an algorithm to construct a 2-manifold planarized triangular mesh surface for 3D boundary
A 3D network
Boundary nodes
Triangular Mesh
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
2. Boundary Node Identification 2.1 Unit Ball Fitting (UBF) Definition 1: arbitrary radio transmission model with a maximum radio transmission range of 1; Definition 2: the nodal density, denoted by ρ, is the average number of nodes in a unit volume; Definition 3: networks are well connected, (1) no nodes are isolated; (2) no degenerated line segment;
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Definition 4: A unit ball is a ball with a radius of r =1+δ, δ is an arbitrarily small constant; Definition 5: An empty unit ball is a unit ball with no nodes inside; Definition 6: A unit ball touches a node if the node is on the surface of the ball; Definition 7: A hole is an empty space that is greater than a unit ball. The space outside the network is treated as a special hole.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Lemma 1: Node A can construct an empty unit ball that touches itself if and only if there exists an empty unit ball touching Node A and its two neighbors. Sufficient condition: If a unit ball touched by Node A and its two neighbors is empty, this empty unit ball always touched by Node A. Necessary condition: If there exits an empty unit ball with Node A on its surface Fix node A and rotate the ball until it touches another node within 2r, denoted by B. If node B does not exist, node A must be an isolated node. Then rotating the ball with Line AB as an axis, until it touches another node, denoted by node C. And node C must exist, otherwise Line AB is degenerated.
r
A
C B A
University of Louisiana at Lafayette
theorem. must be a boundary node. Formally, theorem. we have the following 1: Node can(iWIN) determine heorem. Information Lab@CACS Theorem 1:integrated Node Theorem AWireless can determine ifNetwork itAcan construct an if i ball touches Theorem 1: Node A can determine if it unit can ball construct an unit empty thatempty touches itself bythat testing Θ(ρ2 )itself unit by ballstestin 2 ) unit balls mpty unit ball that touches itself by and testing Θ(ρ Θ(ρ) nodes forand eachΘ(ρ) ball. nodes for each ball. nd Θ(ρ) nodes for each ball. 1, Node Proof: According toProof: LemmaAccording 1, Node A to canLemma exhaustively Theorem 1: Node A can determine if it can construct an empty C to Lemma 1, Node A can C Proof: According test all unit determined by Node A test all unitexhaustively balls determined by balls Node A and its neighbors. 2 unitbyball that itself testing Θ(ρA ) and unit(whose ballstwo and Θ(ρ) to (w est all unit balls determined Node Atouches and its A neighbors. Given Node andby any two neighbors distances Given Node any neighbors nodes forNode each ball. Given Node A and any two neighbors (whose distances to zero A are less than 2r), one than or two unit balls Node A areorless 2r), zero or can onebeor tw A A B than 2r), zero or one formed Node A are less or two unit balls be nodes that can theformed three are on surface(s). 3 the that thethethree nodes are Proof: Accordingsuch to Lemma1, Nodesuch A can exhaustively test all Fig. unit on ormed such that the three nodes are on the surface(s). Fig. 3where illustrates an example two unit balls are determined Ball rotation. balls determined by Node A and its two neighbors. illustrates andistinct example where two unit ba (b) Ball rotation. 4 3 lustrates an example where two unit determined 4 π(2r) ρ,A or Θ(ρ), by balls three are nodes. Since Node Anodes. has about 3 by three Since Node has about for Unit Ball FittingSince (UBF).Node A 3 or Θ(ρ) Θ(ρ), y three nodes. has about Node A neighboring has 34 π(2r)3 ρ, , or neighbor nodes within 2r, it needs Fig. 2. Principles for Unit Ball Fitting (UBF). nodes within the distance of 2r, it needs to test !ρ" neighboring nodes within the distance of 2 2 ! " eighboring !nodes within the distance of 2r, it needs to test to test up up to to Θ(2 × 2 ) = Θ(ρ ) unit unit balls; balls. ρ For each2 unit ball, about " ρ up to Θ(2 × ) = Θ(ρ ) unit balls. For ea 2 4 each 3 2 p to Θ(2 × 2 ) = Θ(ρ ) unit balls. For unit ball, about πr ρ, or Θ(ρ), nodes must be tested to judge if it is empty. 4 3 3 3 ρ, or Θ(ρ), nodes must be Fortested eachtounit ball,if about , or Θ(ρ) nodes must be tested πr ρ, or Θ(ρ), nodes must be tested to ju 3 πr judge it is empty. 3 be employed to test if such an Therefore, the overall computing complexity is Θ(ρ ). Note complexity be employed to test such an3 ).small to judge if itρifisisempty. Therefore, the overall computing complex herefore, thecan overall computing complexity is Θ(ρ Note that usually and bounded. ball that ρ is usually small and3 bounded. hat ρexists. is usually smallunit andball bounded. construct an empty that 3) Algorithm Description: Theorem 1 provides Therefore, the overall computing complexity is Θ(ρ ). a clear 1: Node A cananDescription: construct anball emptyguidance unit ball 3) Algorithm Description: Algorithm Theorem 1 provides clear if 3)there exists empty unit forthat oura algorithm development. It suggestsTheorem a dis- 1 uidance our ifalgorithm development. Itunit suggests a guidance dislf if andfor exists ball forwhere our algorithm development. neighbors ofonly Node Athere (within 2r). an empty tributed and localized algorithm each node tests Θ(ρ2 ) 2) ibuted and localized algorithm where each node tests Θ(ρ ode A and two neighbors of Node A (within 2r). tributed and algorithm where of localized them is empty. To this end, eac the sufficient condition, which is unit balls to judge if any one nit judge if any themweis propose empty. To end, to (UBF) judge algorithm if any oneasofoutlined them is e Weballs firsttoshow the sufficient condition, which theisthis Unitunit Ballballs Fitting ball touched by Node A one and of two we propose the Unit Ball Fitting as outlined mpty, is an empty unit in algorithm Algorithm 1 and elaborated we proposebelow. the Unit Ball Fitting (UBF) alg ard. Ifi.e.,a there unit ball touched by(UBF) Node A and two neighbors below. Aelaborated on i.e., its surface, f Algorithm Node of A Node is1 and empty, there is an empty unit in Algorithm 1 and elaborated below. h an A empty touching ode and unit two ball neighbors ofitself. Node A Algorithm on its surface, 1: Unit Ball Fitting (UBF) Algorithm ntified and Node AanBall isempty a boundary Algorithm 1:such Unit Fitting Algorithm constructed unit(UBF) ball touching itself. //Neighbors of Node i Input: N(i);
Algorithm 1: ofUnit Ball atFitting (UBF) Al University Louisiana Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Unit Ball Fitting (UBF) Algorithm Description Step1: Local coordinates establishment; If all nodes already know their coordinates, this step can be skipped; Step2: Unit Ball Identification; Calculate the center of the unit ball(s); Step3: Empty unit ball checking.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
2.2 Isolated Fragment Filtering (IFF) Observation: A small number of interior nodes may be interpreted by UBF as boundary nodes due to inaccurate nodal coordinates;
Property of IFF: The nodes on a boundary should form a well connected closed surface; Set a threshold γ. Any fragment that consists of less than γ nodes is not considered as a boundary; IFF can also be used to group boundary nodes, e.g, inner boundaries, outer boundary.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
3. Triangular Boundary Surface Construction Step1: Landmark Selection Select a subset of boundary nodes as “landmark”; Any two landmarks must be k-hops apart; Every other nodes will associate with the nearest landmark.
A B
C D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step2: Construction of Combinatorial Delaunay Graph (CDG) Landmarks serve as CDG vertices; If node A and node B are connected in CDG, there must exist a path between landmark A and landmark B and all the nodes on the path are associated with either A or B; CDG is not a planar graph;
A B
C
A B
D
C D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step3: Construction of Combinatorial Delaunay Map (CDM) CDM is a subgraph of CDG and it is a planar graph; If landmarks A and B are connected in CDM, besides all the nodes on the path between them are associated with either landmark A or B, all nodes in the1-hop neighborhood of the path also need to be associated with landmark A or B.
A
A
B
B
C
C
D
D A
A B
C
B C
D
D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step 4: Construction of Triangular Mesh CDM is a planar graph, but not always a triangular mesh; Adding virtual edges in polygons by sending connection packet between landmarks(shortest path based on the identified boundary nodes).
A
A
B
B
C
C
D
D A
A
B
B C
C D
D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step 5: Edge Flip To ensure the triangular mesh is 2-manifold, each virtual edge must be associated with two triangles. After above 4 steps, there still possibly exist edges with three triangular faces.
D
D
B
B E
C A
E
C A
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
A 3D wireless network
Triangular Mesh
UBF
IFF 854 764 658
1862
1951
1085 1480 24
1809
1774
1695
622 1833
Boundary nodes
1386
529
1254
2004 1269
37
1259
162
1286 1614
522 6
CDM
1138
1728 632 505 1696
965
1942
79
2033 69
649
1666
1127
1547
455 570 907
1074 1710
1921
577 1087
348 72
861
854 764 658
1862
1951
1085 1480 24
1809
1774
1695
622 1833
Landmarks
1269
1386
529
1254
2004
37
1259
162
1286 1614
522
632 505 1696
965
1942
79
2033 69
CDG
1138
1728 6
649
1666
1127
1547
455 570 907
1074 1710
1921
577 1087
348
861
72
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
4. Simulation Six 3D wireless networks, over 10,000 nodes, node degree from 5 to 45, average degree 18.5. Random errors from 0 to 100% are introduced in the distance measurement.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Found Correct Mistaken Missing
Boundary Nodes
100%
80%
60%
40%
20%
0%
100%
1 hop 2 hop 3 hop
80%
60%
40%
20%
0%
Distribution of Missing Boundary Nodes
Distribution of Mistaken Boundary Nodes
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Distances Measurement Error
100%
80%
60%
40%
20%
0%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Distances Measurement Error
1 hop 2 hop 3 hop
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Distances Measurement Error
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
5. Conclusion We have proposed distributed and localized algorithms for precise boundary detection in 3D wireless network: 1) Identify the the boundaries nodes of a 3D network; 2) Construct planarized 2-manifold surfaces for inner and outer boundaries. As far as we know, this is the first work for discovering boundary nodes and constructing boundary surface in 3D wireless networks.
University of Louisiana at Lafayette
Thank you!
integrated Wireless Information Network (iWIN) Lab The Center for Advanced Computer Studies(CACS) University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Outline 1. Introduction 2. Boundary detection 2.1 Unit Ball Fitting (UBF) 2.2 Isolated Fragment Filtering (IFF)
3. Triangular Boundary Surface Construction 4. Simulation 5. Conclusion
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
1. Introduction Motivation Boundary nodes serve as a key attribute that characterizes the network, especially in geographic exploration tasks such as terrain and underwater reconnaissance. Many wireless networks exhibit randomness
Related works All in 2D wireless networks
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Paper Contributions Find a localized method that can precisely detect boundary nodes in 3D wireless networks; Develop an algorithm to construct a 2-manifold planarized triangular mesh surface for 3D boundary
A 3D network
Boundary nodes
Triangular Mesh
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
2. Boundary Node Identification 2.1 Unit Ball Fitting (UBF) Definition 1: arbitrary radio transmission model with a maximum radio transmission range of 1; Definition 2: the nodal density, denoted by ρ, is the average number of nodes in a unit volume; Definition 3: networks are well connected, (1) no nodes are isolated; (2) no degenerated line segment;
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Definition 4: A unit ball is a ball with a radius of r =1+δ, δ is an arbitrarily small constant; Definition 5: An empty unit ball is a unit ball with no nodes inside; Definition 6: A unit ball touches a node if the node is on the surface of the ball; Definition 7: A hole is an empty space that is greater than a unit ball. The space outside the network is treated as a special hole.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Lemma 1: Node A can construct an empty unit ball that touches itself if and only if there exists an empty unit ball touching Node A and its two neighbors. Sufficient condition: If a unit ball touched by Node A and its two neighbors is empty, this empty unit ball always touched by Node A. Necessary condition: If there exits an empty unit ball with Node A on its surface Fix node A and rotate the ball until it touches another node within 2r, denoted by B. If node B does not exist, node A must be an isolated node. Then rotating the ball with Line AB as an axis, until it touches another node, denoted by node C. And node C must exist, otherwise Line AB is degenerated.
r
A
C B A
University of Louisiana at Lafayette
theorem. must be a boundary node. Formally, theorem. we have the following 1: Node can(iWIN) determine heorem. Information Lab@CACS Theorem 1:integrated Node Theorem AWireless can determine ifNetwork itAcan construct an if i ball touches Theorem 1: Node A can determine if it unit can ball construct an unit empty thatempty touches itself bythat testing Θ(ρ2 )itself unit by ballstestin 2 ) unit balls mpty unit ball that touches itself by and testing Θ(ρ Θ(ρ) nodes forand eachΘ(ρ) ball. nodes for each ball. nd Θ(ρ) nodes for each ball. 1, Node Proof: According toProof: LemmaAccording 1, Node A to canLemma exhaustively Theorem 1: Node A can determine if it can construct an empty C to Lemma 1, Node A can C Proof: According test all unit determined by Node A test all unitexhaustively balls determined by balls Node A and its neighbors. 2 unitbyball that itself testing Θ(ρA ) and unit(whose ballstwo and Θ(ρ) to (w est all unit balls determined Node Atouches and its A neighbors. Given Node andby any two neighbors distances Given Node any neighbors nodes forNode each ball. Given Node A and any two neighbors (whose distances to zero A are less than 2r), one than or two unit balls Node A areorless 2r), zero or can onebeor tw A A B than 2r), zero or one formed Node A are less or two unit balls be nodes that can theformed three are on surface(s). 3 the that thethethree nodes are Proof: Accordingsuch to Lemma1, Nodesuch A can exhaustively test all Fig. unit on ormed such that the three nodes are on the surface(s). Fig. 3where illustrates an example two unit balls are determined Ball rotation. balls determined by Node A and its two neighbors. illustrates andistinct example where two unit ba (b) Ball rotation. 4 3 lustrates an example where two unit determined 4 π(2r) ρ,A or Θ(ρ), by balls three are nodes. Since Node Anodes. has about 3 by three Since Node has about for Unit Ball FittingSince (UBF).Node A 3 or Θ(ρ) Θ(ρ), y three nodes. has about Node A neighboring has 34 π(2r)3 ρ, , or neighbor nodes within 2r, it needs Fig. 2. Principles for Unit Ball Fitting (UBF). nodes within the distance of 2r, it needs to test !ρ" neighboring nodes within the distance of 2 2 ! " eighboring !nodes within the distance of 2r, it needs to test to test up up to to Θ(2 × 2 ) = Θ(ρ ) unit unit balls; balls. ρ For each2 unit ball, about " ρ up to Θ(2 × ) = Θ(ρ ) unit balls. For ea 2 4 each 3 2 p to Θ(2 × 2 ) = Θ(ρ ) unit balls. For unit ball, about πr ρ, or Θ(ρ), nodes must be tested to judge if it is empty. 4 3 3 3 ρ, or Θ(ρ), nodes must be Fortested eachtounit ball,if about , or Θ(ρ) nodes must be tested πr ρ, or Θ(ρ), nodes must be tested to ju 3 πr judge it is empty. 3 be employed to test if such an Therefore, the overall computing complexity is Θ(ρ ). Note complexity be employed to test such an3 ).small to judge if itρifisisempty. Therefore, the overall computing complex herefore, thecan overall computing complexity is Θ(ρ Note that usually and bounded. ball that ρ is usually small and3 bounded. hat ρexists. is usually smallunit andball bounded. construct an empty that 3) Algorithm Description: Theorem 1 provides Therefore, the overall computing complexity is Θ(ρ ). a clear 1: Node A cananDescription: construct anball emptyguidance unit ball 3) Algorithm Description: Algorithm Theorem 1 provides clear if 3)there exists empty unit forthat oura algorithm development. It suggestsTheorem a dis- 1 uidance our ifalgorithm development. Itunit suggests a guidance dislf if andfor exists ball forwhere our algorithm development. neighbors ofonly Node Athere (within 2r). an empty tributed and localized algorithm each node tests Θ(ρ2 ) 2) ibuted and localized algorithm where each node tests Θ(ρ ode A and two neighbors of Node A (within 2r). tributed and algorithm where of localized them is empty. To this end, eac the sufficient condition, which is unit balls to judge if any one nit judge if any themweis propose empty. To end, to (UBF) judge algorithm if any oneasofoutlined them is e Weballs firsttoshow the sufficient condition, which theisthis Unitunit Ballballs Fitting ball touched by Node A one and of two we propose the Unit Ball Fitting as outlined mpty, is an empty unit in algorithm Algorithm 1 and elaborated we proposebelow. the Unit Ball Fitting (UBF) alg ard. Ifi.e.,a there unit ball touched by(UBF) Node A and two neighbors below. Aelaborated on i.e., its surface, f Algorithm Node of A Node is1 and empty, there is an empty unit in Algorithm 1 and elaborated below. h an A empty touching ode and unit two ball neighbors ofitself. Node A Algorithm on its surface, 1: Unit Ball Fitting (UBF) Algorithm ntified and Node AanBall isempty a boundary Algorithm 1:such Unit Fitting Algorithm constructed unit(UBF) ball touching itself. //Neighbors of Node i Input: N(i);
Algorithm 1: ofUnit Ball atFitting (UBF) Al University Louisiana Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Unit Ball Fitting (UBF) Algorithm Description Step1: Local coordinates establishment; If all nodes already know their coordinates, this step can be skipped; Step2: Unit Ball Identification; Calculate the center of the unit ball(s); Step3: Empty unit ball checking.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
2.2 Isolated Fragment Filtering (IFF) Observation: A small number of interior nodes may be interpreted by UBF as boundary nodes due to inaccurate nodal coordinates;
Property of IFF: The nodes on a boundary should form a well connected closed surface; Set a threshold γ. Any fragment that consists of less than γ nodes is not considered as a boundary; IFF can also be used to group boundary nodes, e.g, inner boundaries, outer boundary.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
3. Triangular Boundary Surface Construction Step1: Landmark Selection Select a subset of boundary nodes as “landmark”; Any two landmarks must be k-hops apart; Every other nodes will associate with the nearest landmark.
A B
C D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step2: Construction of Combinatorial Delaunay Graph (CDG) Landmarks serve as CDG vertices; If node A and node B are connected in CDG, there must exist a path between landmark A and landmark B and all the nodes on the path are associated with either A or B; CDG is not a planar graph;
A B
C
A B
D
C D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step3: Construction of Combinatorial Delaunay Map (CDM) CDM is a subgraph of CDG and it is a planar graph; If landmarks A and B are connected in CDM, besides all the nodes on the path between them are associated with either landmark A or B, all nodes in the1-hop neighborhood of the path also need to be associated with landmark A or B.
A
A
B
B
C
C
D
D A
A B
C
B C
D
D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step 4: Construction of Triangular Mesh CDM is a planar graph, but not always a triangular mesh; Adding virtual edges in polygons by sending connection packet between landmarks(shortest path based on the identified boundary nodes).
A
A
B
B
C
C
D
D A
A
B
B C
C D
D
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Step 5: Edge Flip To ensure the triangular mesh is 2-manifold, each virtual edge must be associated with two triangles. After above 4 steps, there still possibly exist edges with three triangular faces.
D
D
B
B E
C A
E
C A
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
A 3D wireless network
Triangular Mesh
UBF
IFF 854 764 658
1862
1951
1085 1480 24
1809
1774
1695
622 1833
Boundary nodes
1386
529
1254
2004 1269
37
1259
162
1286 1614
522 6
CDM
1138
1728 632 505 1696
965
1942
79
2033 69
649
1666
1127
1547
455 570 907
1074 1710
1921
577 1087
348 72
861
854 764 658
1862
1951
1085 1480 24
1809
1774
1695
622 1833
Landmarks
1269
1386
529
1254
2004
37
1259
162
1286 1614
522
632 505 1696
965
1942
79
2033 69
CDG
1138
1728 6
649
1666
1127
1547
455 570 907
1074 1710
1921
577 1087
348
861
72
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
4. Simulation Six 3D wireless networks, over 10,000 nodes, node degree from 5 to 45, average degree 18.5. Random errors from 0 to 100% are introduced in the distance measurement.
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
Found Correct Mistaken Missing
Boundary Nodes
100%
80%
60%
40%
20%
0%
100%
1 hop 2 hop 3 hop
80%
60%
40%
20%
0%
Distribution of Missing Boundary Nodes
Distribution of Mistaken Boundary Nodes
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Distances Measurement Error
100%
80%
60%
40%
20%
0%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Distances Measurement Error
1 hop 2 hop 3 hop
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Distances Measurement Error
University of Louisiana at Lafayette
integrated Wireless Information Network (iWIN) Lab@CACS
5. Conclusion We have proposed distributed and localized algorithms for precise boundary detection in 3D wireless network: 1) Identify the the boundaries nodes of a 3D network; 2) Construct planarized 2-manifold surfaces for inner and outer boundaries. As far as we know, this is the first work for discovering boundary nodes and constructing boundary surface in 3D wireless networks.
University of Louisiana at Lafayette
Thank you!
