Fast oriented bounding box optimization on the rotation group SO(3, R)


 Constance Campbell
 1 years ago
 Views:
Transcription
1 Fast oriented bounding box optimization on the rotation group SO(3, R) ChiaTche Chang 1, Bastien Gorissen 2,3 and Samuel Melchior 1,2 1 Department of Mathematical Engineering (INMA), Université catholique de Louvain 2 Department of Mechanical Engineering (MEMA), Université catholique de Louvain 3 CENAERO November 30th, 2012 C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 1
2 The problem: minimumvolume OBB Coming up next... The problem: minimumvolume OBB Exact methods in 2D and 3D Classical approaches for the 3D case Our goal Bringing optimization into the game How to solve an optimization problem? Results Conclusion C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 2
3 The problem: minimumvolume OBB The problem Given a set of n points X in 3D, find the minimumvolume arbitrarily oriented bounding box enclosing X. Collision detection, intersection tests, object representation, data approximation... (BV trees... ) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 3
4 The problem: minimumvolume OBB Exact methods in 2D and 3D In 2D: the rotating calipers method A minimumarea rectangle circumscribing a convex polygon has at least one side flush with an edge of the polygon. C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 4
5 The problem: minimumvolume OBB Exact methods in 2D and 3D In 2D: the rotating calipers method A minimumarea rectangle circumscribing a convex polygon has at least one side flush with an edge of the polygon. Compute the convex hull: O(n log n) conv(x ) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 4
6 The problem: minimumvolume OBB Exact methods in 2D and 3D In 2D: the rotating calipers method A minimumarea rectangle circumscribing a convex polygon has at least one side flush with an edge of the polygon. Compute the convex hull: O(n log n) Loop on all edges: O(n) easy and efficient conv(x ) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 4
7 The problem: minimumvolume OBB Exact methods in 2D and 3D In 3D: generalization of the rotating calipers? A minimumvolume box circumscribing a convex polyhedron has at least one face flush with a face of the polyhedron? C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 5
8 The problem: minimumvolume OBB Exact methods in 2D and 3D In 3D: generalization of the rotating calipers? A minimumvolume box circumscribing a convex polyhedron has at least one face two adjacent faces flush with a face edges of the polyhedron. [O Rourke, 1985] C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 5
9 The problem: minimumvolume OBB Exact methods in 2D and 3D In 3D: generalization of the rotating calipers? A minimumvolume box circumscribing a convex polyhedron has at least one face two adjacent faces flush with a face edges of the polyhedron. [O Rourke, 1985] Problem: Loop on all pairs of edges and rotate the box while keeping edges flush O(n 3 ) time complexity... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 5
10 The problem: minimumvolume OBB Classical approaches for the 3D case In practice... O Rourke s algorithm is too slow (cubic time) use faster but inexact methods: C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 6
11 The problem: minimumvolume OBB Classical approaches for the 3D case In practice... O Rourke s algorithm is too slow (cubic time) use faster but inexact methods: PCAbased methods (covariance matrix): very fast and easy to compute but may be very inaccurate C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 6
12 The problem: minimumvolume OBB Classical approaches for the 3D case In practice... O Rourke s algorithm is too slow (cubic time) use faster but inexact methods: PCAbased methods (covariance matrix): very fast and easy to compute but may be very inaccurate Bruteforce all orientations with a small angle increment: large computation time and/or low accuracy C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 6
13 The problem: minimumvolume OBB Classical approaches for the 3D case In practice... O Rourke s algorithm is too slow (cubic time) use faster but inexact methods: PCAbased methods (covariance matrix): very fast and easy to compute but may be very inaccurate Bruteforce all orientations with a small angle increment: large computation time and/or low accuracy Bruteforce a wellchosen set of orientations: may sometimes have (very) good accuracy but still too slow C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 6
14 The problem: minimumvolume OBB Classical approaches for the 3D case In practice... O Rourke s algorithm is too slow (cubic time) use faster but inexact methods: PCAbased methods (covariance matrix): very fast and easy to compute but may be very inaccurate Bruteforce all orientations with a small angle increment: large computation time and/or low accuracy Bruteforce a wellchosen set of orientations: may sometimes have (very) good accuracy but still too slow Guaranteed quality approximation methods: same problem... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 6
15 The problem: minimumvolume OBB Our goal What do we want? Goal: Very good accuracy: find an optimal OBB in (nearly?) all cases If a suboptimal solution is returned, it should be close to the best one Computational cost has to be low C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 7
16 The problem: minimumvolume OBB Our goal What do we want? Goal: Very good accuracy: find an optimal OBB in (nearly?) all cases If a suboptimal solution is returned, it should be close to the best one Computational cost has to be low Our approach: iterative algorithm based on optimization methods C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 7
17 Bringing optimization into the game Coming up next... The problem: minimumvolume OBB Bringing optimization into the game OBB fitting as an optimization problem Requirements Going hybrid How to solve an optimization problem? Results Conclusion C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 8
18 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over size, position, orientation so that the volume of the bounding box all the points are in the box C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
19 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3, position, orientation so that the volume of the bounding box all the points are in the box = ( ξ, η, ζ ) denotes the dimensions of the OBB, C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
20 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3, position, orientation so that ξ η ζ all the points are in the box = ( ξ, η, ζ ) denotes the dimensions of the OBB, C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
21 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3, position, orientation so that ξ η ζ 1 all the rotated and 2 centered points 1 2 = ( ξ, η, ζ ) denotes the dimensions of the OBB, C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
22 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3,Ξ R 3, orientation so that ξ η ζ 1 2 all the rotated points Ξ 1 2 = ( ξ, η, ζ ) denotes the dimensions of the OBB, Ξ is the center of the OBB. C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
23 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3,Ξ R 3,R SO(3,R) so that ξ η ζ 1 2 R(all the points) Ξ 1 2 = ( ξ, η, ζ ) denotes the dimensions of the OBB, Ξ is the center of the OBB. R SO(3, R) is a rotation matrix, C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
24 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3,Ξ R 3,R SO(3,R) so that ξ η ζ 1 2 R(all the points) Ξ 1 2 = ( ξ, η, ζ ) denotes the dimensions of the OBB, Ξ is the center of the OBB. R SO(3, R) is a rotation matrix, SO(3, R) = { R R 3 3 R T R = I = RR T, det(r) = 1 }, C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
25 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min over R 3,Ξ R 3,R SO(3,R) ξ η ζ s.t. 1 2 RX i Ξ 1 2 i = 1,..., N = ( ξ, η, ζ ) denotes the dimensions of the OBB, Ξ is the center of the OBB. R SO(3, R) is a rotation matrix, SO(3, R) = { R R 3 3 R T R = I = RR T, det(r) = 1 }, X = {X i i = 1,..., N } is the considered set of points C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
26 Bringing optimization into the game OBB fitting as an optimization problem A first, direct, formulation min R 3,Ξ R 3,R SO(3,R) ξ η ζ s.t. 1 2 RX i Ξ 1 2 i = 1,..., N = ( ξ, η, ζ ) denotes the dimensions of the OBB, Ξ is the center of the OBB. R SO(3, R) is a rotation matrix, SO(3, R) = { R R 3 3 R T R = I = RR T, det(r) = 1 }, X = {X i i = 1,..., N } is the considered set of points Smooth but constrained optimization problem C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 9
27 Bringing optimization into the game OBB fitting as an optimization problem Unconstrained formulation min R SO(3,R) ( min R 3,Ξ R 3 ξ η ζ } s.t. 1 2 RX i Ξ 1 2 i = 1,..., N {{ } f (R) ) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 10
28 Bringing optimization into the game OBB fitting as an optimization problem Unconstrained formulation min R SO(3,R) ( min R 3,Ξ R 3 ξ η ζ } s.t. 1 2 RX i Ξ 1 2 i = 1,..., N {{ } f (R) ) The objective function f (R) is simply the volume of the AABB of X rotated by R C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 10
29 Bringing optimization into the game OBB fitting as an optimization problem Unconstrained formulation min R SO(3,R) ( min R 3,Ξ R 3 ξ η ζ } s.t. 1 2 RX i Ξ 1 2 i = 1,..., N {{ } f (R) ) The objective function f (R) is simply the volume of the AABB of X rotated by R Unconstrained but nondifferentiable optimization problem C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 10
30 Bringing optimization into the game Requirements Solving this problem requires... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 11
31 Bringing optimization into the game Requirements Solving this problem requires a derivativefree method f (R) is not differentiable everywhere... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 11
32 Bringing optimization into the game Requirements Solving this problem requires a derivativefree method... a global search technique f (R) has many local minima... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 11
33 Bringing optimization into the game Requirements Solving this problem requires a derivativefree method... a global search technique... a fast convergence rate That was the point! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 11
34 Bringing optimization into the game Going hybrid Our idea: using an hybrid method 1. Use a global exploration component: genetic algorithm (GA) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 12
35 Bringing optimization into the game Going hybrid Our idea: using an hybrid method 1. Use a global exploration component: genetic algorithm (GA) 2. Speed up convergence using a local exploitation algorithm NelderMead simplex algorithm (NM) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 12
36 Bringing optimization into the game Going hybrid Our idea: using an hybrid method 1. Use a global exploration component: genetic algorithm (GA) 2. Speed up convergence using a local exploitation algorithm NelderMead simplex algorithm (NM) GA alone would be very slow to converge (GA more suitable for discrete search spaces) NM alone would be stuck in local minima (even with restarts) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 12
37 How to solve an optimization problem? Coming up next... The problem: minimumvolume OBB Bringing optimization into the game How to solve an optimization problem? Genetic algorithms (GA) The NelderMead algorithm (NM) HYBBRID: let s mix GA and NM together! Results Conclusion C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 13
38 How to solve an optimization problem? Genetic algorithms (GA) Global exploration: genetic algorithms Stochastic populationbased evolutionary method (original variant proposed by Holland in the 1970s) Populationbased: keep a large set of candidates at each iteration Evolutionary: generate new candidates by combining current ones depending on their performance C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 14
39 How to solve an optimization problem? Genetic algorithms (GA) The general framework Start with a set of candidates (population) and a performance function (fitness function) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 15
40 How to solve an optimization problem? Genetic algorithms (GA) The general framework Start with a set of candidates (population) and a performance function (fitness function) At each generation: Selection: parents are selected depending on their fitness C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 15
41 How to solve an optimization problem? Genetic algorithms (GA) The general framework Start with a set of candidates (population) and a performance function (fitness function) At each generation: Selection: parents are selected depending on their fitness Crossover: selected parents produce offsprings C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 15
42 How to solve an optimization problem? Genetic algorithms (GA) The general framework Start with a set of candidates (population) and a performance function (fitness function) At each generation: Selection: parents are selected depending on their fitness Crossover: selected parents produce offsprings Mutation: offsprings can be subject to mutations (random modification, gradient step, SA step,...) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 15
43 How to solve an optimization problem? The NelderMead algorithm (NM) The NelderMead simplex algorithm Derivativefree simplicial optimization method (original algorithm proposed by Nelder & Mead in 1965) Simplex (in R n ) = set of n + 1 affinely independent points n = 2 : triangle n = 3 : tetrahedron... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 16
44 How to solve an optimization problem? The NelderMead algorithm (NM) Ideas of the algorithm (details omitted) Reflection Expansion Contraction Reduction Four main ways to move/transform the simplex depending on the performance of its vertices: affine combinations C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 17
45 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = The initial simplex C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
46 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 1: contraction C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
47 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 2: reflection C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
48 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 3: contraction C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
49 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 4: contraction C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
50 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 5: contraction C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
51 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 6: reduction C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
52 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Iteration 7: contraction C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
53 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Some more iterations... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
54 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = Some more iterations... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
55 How to solve an optimization problem? The NelderMead algorithm (NM) NelderMead and the sixhump camel back... 2 Current objective = The final result C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 18
56 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! Mixing NM and GA together (simplified version) Population of M simplices (simplex = set of 4 rotation matrices) Fitness function f (R) (volume of corresponding OBB) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 19
57 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! Mixing NM and GA together (simplified version) Population of M simplices (simplex = set of 4 rotation matrices) Fitness function f (R) (volume of corresponding OBB) Selection: Evaluate fitness of all simplices, keep best 50% C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 19
58 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! Mixing NM and GA together (simplified version) Population of M simplices (simplex = set of 4 rotation matrices) Fitness function f (R) (volume of corresponding OBB) Selection: Evaluate fitness of all simplices, keep best 50% Crossover I: Create M 2 offsprings by mixing vertices: A 1 B 1 C 1 D 1 A 2 B 2 C 2 D 2 A i1 B i2 C i3 D i4, i k {1, 2} C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 19
59 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! Mixing NM and GA together (simplified version) Population of M simplices (simplex = set of 4 rotation matrices) Fitness function f (R) (volume of corresponding OBB) Selection: Evaluate fitness of all simplices, keep best 50% Crossover I: Create M 2 offsprings by mixing vertices: A 1 B 1 C 1 D 1 A 2 B 2 C 2 D 2 A i1 B i2 C i3 D i4, i k {1, 2} Crossover II: Create M 2 offsprings by affinely combine vertices: A 1 B 1 C 1 D 1 A 2 B 2 C 2 D 2 A 3 B 3 C 3 D 3 with A 3 = λa 1 + (1 λ)a 2 C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 19
60 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! Mixing NM and GA together (simplified version) Population of M simplices (simplex = set of 4 rotation matrices) Fitness function f (R) (volume of corresponding OBB) Selection: Evaluate fitness of all simplices, keep best 50% Crossover I: Create M 2 offsprings by mixing vertices: A 1 B 1 C 1 D 1 A 2 B 2 C 2 D 2 A i1 B i2 C i3 D i4, i k {1, 2} Crossover II: Create M 2 offsprings by affinely combine vertices: A 1 B 1 C 1 D 1 A 2 B 2 C 2 D 2 A 3 B 3 C 3 D 3 with A 3 = λa 1 + (1 λ)a 2 Mutation: Apply K NelderMead iterations on each offspring C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 19
61 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! HYBBRID NelderMead algorithm Genetic algorithm on the special orthogonal group SO(3) to solve the optimal OBB problem C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 20
62 How to solve an optimization problem? HYBBRID: let s mix GA and NM together! HYBBRID NelderMead algorithm Genetic algorithm on the special orthogonal group SO(3) to solve the optimal OBB problem = HYbrid Bounding Box Rotation IDentification algorithm C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 20
63 Results Coming up next... The problem: minimumvolume OBB Bringing optimization into the game How to solve an optimization problem? Results Behaviour of HYBBRID Comparison to other algorithms Conclusion C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 21
64 Results Behaviour of HYBBRID Behavior of HYBBRID All algorithms tested on a benchmark set of 300 objects (Gamma db) Implementations done in MATLAB (builtin functions are used) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 22
65 Results Behaviour of HYBBRID Behavior of HYBBRID: yes, it works! All algorithms tested on a benchmark set of 300 objects (Gamma db) Implementations done in MATLAB (builtin functions are used) 100 Tolerance = 1e Performance [%] Population size = Population size = Population size = 40 3 Population size = Computation time [s] Error is less than in 90%+ of the cases! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 22
66 Results Behaviour of HYBBRID How does it scale? 15 Computation time [s] Number of vertices on the convex hull Experimental results show a roughly linear complexity! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 23
67 Results Comparison to other algorithms Comparison to other iterative approaches Failure rate [%] Random search (60000 trials) Relative error tolerance Trying random orientations does not work... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 24
68 Results Comparison to other algorithms Comparison to other iterative approaches Failure rate [%] fmincon, Euler angles (15 restarts) fmincon, Rotation matrix (15 restarts) Random search (60000 trials) Relative error tolerance Constrained smooth opti: success rate 40% but mainly AABBs... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 24
69 Results Comparison to other algorithms Comparison to other iterative approaches Failure rate [%] Nelder Mead (30 restarts) fmincon, Euler angles (15 restarts) fmincon, Rotation matrix (15 restarts) Random search (60000 trials) Relative error tolerance Unconstrained nondiff. opti, random initializations: much better results! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 24
70 Results Comparison to other algorithms Comparison to other iterative approaches HYBBRID (M=50, K=30) Nelder Mead (30 restarts) fmincon, Euler angles (15 restarts) fmincon, Rotation matrix (15 restarts) Random search (60000 trials) Failure rate [%] Relative error tolerance HYBBRID: combining potential solutions does improve the success rate! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 24
71 Results Comparison to other algorithms Comparison to other algorithms First, let s ignore the computational cost and look at the failure rates AABB AABB with post processing Failure rate [%] Relative error tolerance A reference point: the simple AABB C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 25
72 Results Comparison to other algorithms Comparison to other algorithms First, let s ignore the computational cost and look at the failure rates AABB AABB with post processing Min PCA with post processing Continuous PCA with post processing Failure rate [%] Relative error tolerance PCAbased methods: limited accuracy C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 25
73 Results Comparison to other algorithms Comparison to other algorithms First, let s ignore the computational cost and look at the failure rates AABB AABB with post processing Min PCA with post processing Continuous PCA with post processing All Pairs Korsawe (fast variant) Korsawe (slow variant) Failure rate [%] Relative error tolerance Bruteforcing on a set of orientations may be OK... if well chosen! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 25
74 Results Comparison to other algorithms Comparison to other algorithms First, let s ignore the computational cost and look at the failure rates AABB AABB with post processing Min PCA with post processing Continuous PCA with post processing All Pairs Korsawe (fast variant) Korsawe (slow variant) Barequet & Har Peled s GridSearchMinVolBbx (d=60) Failure rate [%] Relative error tolerance Guaranteed approximation algorithms: limited by computational resources C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 25
75 Results Comparison to other algorithms Comparison to other algorithms First, let s ignore the computational cost and look at the failure rates AABB AABB with post processing Min PCA with post processing Continuous PCA with post processing All Pairs Korsawe (fast variant) Korsawe (slow variant) Barequet & Har Peled s GridSearchMinVolBbx (d=60) HYBBRID(M=50, K=30) Failure rate [%] Relative error tolerance HYBBRID: more accurate than these other methods C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 25
76 Results Comparison to other algorithms Comparison to other algorithms What are the computation times? (Tolerance: 10 3 ) Performance [%] Failure rate [%] Computation time [s] Relative error tolerance AABB PCA Continuous PCA Bruteforce on a set of orientations O Rourke s exact algorithm Guaranteed approximation HYBBRID C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 26
77 Results Comparison to other algorithms Comparison to other algorithms What are the computation times? (Tolerance: 10 6 ) Performance [%] Failure rate [%] Computation time [s] Relative error tolerance AABB PCA Continuous PCA Bruteforce on a set of orientations O Rourke s exact algorithm Guaranteed approximation HYBBRID C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 26
78 Conclusion Coming up next... The problem: minimumvolume OBB Bringing optimization into the game How to solve an optimization problem? Results Conclusion C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 27
79 Conclusion Conclusion HYBBRID: NelderMead Genetic algorithm able to approximate optimal OBBs using optimization on SO(3) C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 28
80 Conclusion Conclusion HYBBRID: NelderMead Genetic algorithm able to approximate optimal OBBs using optimization on SO(3) More accurate and/or faster than other algorithms C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 28
81 Conclusion Conclusion HYBBRID: NelderMead Genetic algorithm able to approximate optimal OBBs using optimization on SO(3) More accurate and/or faster than other algorithms Still has room for improvements... C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 28
82 Conclusion Conclusion HYBBRID: NelderMead Genetic algorithm able to approximate optimal OBBs using optimization on SO(3) More accurate and/or faster than other algorithms Still has room for improvements... Thank you for your attention! C.T. Chang et al. Fast oriented bounding box optimization on the rotation group SO(3, R) 28
Fast oriented bounding box optimization on the rotation group
Fast oriented bounding box optimization on the rotation group SO(3, R) CHIATCHE CHANG 1, BASTIEN GORISSEN,3 and SAMUEL MELCHIOR 1, 1 Université catholique de Louvain, Department of Mathematical Engineering
More informationOriented Bounding Box Computation Using Particle Swarm Optimization
Oriented Bounding Box Computation Using Particle Swarm Optimization Pierre B. Borckmans, P.A. Absil Department of Mathematical Engineering, Université catholique de Louvain, B1348 LouvainlaNeuve, Belgium
More informationIntroduction. Optimization
Introduction to Optimization Amy Langville SAMSI Undergraduate Workshop N.C. State University SAMSI 6/1/05 GOAL: minimize f(x 1, x 2, x 3, x 4, x 5 ) = x 2 1.5x 2x 3 + x 4 /x 5 PRIZE: $1 million # of independent
More informationCollision Detection with Bounding Volume Hierarchies
Simulation in Computer Graphics Collision Detection with Bounding Volume Hierarchies Matthias Teschner Computer Science Department University of Freiburg Outline introduction bounding volumes BV hierarchies
More informationLecture 4. Convexity Robust cost functions Optimizing nonconvex functions. 3B1B Optimization Michaelmas 2017 A. Zisserman
Lecture 4 3B1B Optimization Michaelmas 2017 A. Zisserman Convexity Robust cost functions Optimizing nonconvex functions grid search branch and bound simulated annealing evolutionary optimization The Optimization
More informationOptimizing the TracePro Optimization Process
Optimizing the TracePro Optimization Process A TracePro Webinar December 17, 2014 Presenter Presenter Dave Jacobsen Sr. Application Engineer Lambda Research Corporation Moderator Mike Gauvin Vice President
More informationIntroduction to optimization methods and line search
Introduction to optimization methods and line search Jussi Hakanen Postdoctoral researcher jussi.hakanen@jyu.fi How to find optimal solutions? Trial and error widely used in practice, not efficient and
More informationAn Evolutionary Algorithm for Minimizing Multimodal Functions
An Evolutionary Algorithm for Minimizing Multimodal Functions D.G. Sotiropoulos, V.P. Plagianakos and M.N. Vrahatis University of Patras, Department of Mamatics, Division of Computational Mamatics & Informatics,
More informationAn evolutionary annealingsimplex algorithm for global optimisation of water resource systems
FIFTH INTERNATIONAL CONFERENCE ON HYDROINFORMATICS 15 July 2002, Cardiff, UK C05  Evolutionary algorithms in hydroinformatics An evolutionary annealingsimplex algorithm for global optimisation of water
More informationFast Computation of Tight Fitting Oriented Bounding Boxes
1 Fast Computation of Tight Fitting Oriented Bounding Boxes Thomas Larsson Linus Källberg Mälardalen University, Sweden 1.1 Introduction Bounding shapes, or containers, are frequently used to speed up
More informationCollision Detection. Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering
RBE 550 MOTION PLANNING BASED ON DR. DMITRY BERENSON S RBE 550 Collision Detection Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Euler Angle RBE
More informationIntroduction to unconstrained optimization  derivativefree methods
Introduction to unconstrained optimization  derivativefree methods Jussi Hakanen Postdoctoral researcher Office: AgC426.3 jussi.hakanen@jyu.fi Learning outcomes To understand the basic principles of
More informationMathematical Programming and Research Methods (Part II)
Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types
More informationOPTIMIZATION METHODS. For more information visit: or send an to:
OPTIMIZATION METHODS modefrontier is a registered product of ESTECO srl Copyright ESTECO srl 19992007 For more information visit: www.esteco.com or send an email to: modefrontier@esteco.com NEOS Optimization
More informationAlgoritmi di Ottimizzazione: Parte A Aspetti generali e algoritmi classici
Identificazione e Controllo Intelligente Algoritmi di Ottimizzazione: Parte A Aspetti generali e algoritmi classici David Naso A.A. 20062007 Identificazione e Controllo Intelligente 1 Search and optimization
More informationEvolutionary Computation Algorithms for Cryptanalysis: A Study
Evolutionary Computation Algorithms for Cryptanalysis: A Study Poonam Garg Information Technology and Management Dept. Institute of Management Technology Ghaziabad, India pgarg@imt.edu Abstract The cryptanalysis
More informationInitializing the Particle Swarm Optimizer Using the Nonlinear Simplex Method
Initializing the Particle Swarm Optimizer Using the Nonlinear Simplex Method K.E. PARSOPOULOS, M.N. VRAHATIS Department of Mathematics University of Patras University of Patras Artificial Intelligence
More informationModel Parameter Estimation
Model Parameter Estimation Shan He School for Computational Science University of Birmingham Module 0623836: Computational Modelling with MATLAB Outline Outline of Topics Concepts about model parameter
More informationAn Introduction to Evolutionary Algorithms
An Introduction to Evolutionary Algorithms Karthik Sindhya, PhD Postdoctoral Researcher Industrial Optimization Group Department of Mathematical Information Technology Karthik.sindhya@jyu.fi http://users.jyu.fi/~kasindhy/
More informationIntroduction to Genetic Algorithms
Advanced Topics in Image Analysis and Machine Learning Introduction to Genetic Algorithms Week 3 Faculty of Information Science and Engineering Ritsumeikan University Today s class outline Genetic Algorithms
More informationMinimumVolume Box Containing a Set of Points
MinimumVolume Box Containing a Set of Points David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More informationA Genetic Algorithm for Graph Matching using Graph Node Characteristics 1 2
Chapter 5 A Genetic Algorithm for Graph Matching using Graph Node Characteristics 1 2 Graph Matching has attracted the exploration of applying new computing paradigms because of the large number of applications
More informationA Genetic Algorithm Framework
Fast, good, cheap. Pick any two. The Project Triangle 3 A Genetic Algorithm Framework In this chapter, we develop a genetic algorithm based framework to address the problem of designing optimal networks
More informationBounding Volume Hierarchies
Tutorial: RealTime Collision Detection for Dynamic Virtual Environments Bounding Volume Hierarchies Stefan Kimmerle WSI/GRIS University of Tübingen Outline Introduction Bounding Volume Types Hierarchy
More informationEfficiently Approximating the MinimumVolume Bounding Box of a Point Set in Three Dimensions
Efficiently Approximating the MinimumVolume Bounding Box of a Point Set in Three Dimensions Author: Gill Barequet and Sariel HarPeled Presenter: An Nguyen The Problem Input: a set S of points in R 3,
More informationCutLeader Nesting Technology
CutLeader Technology algorithm is the soul of nesting software. For example knapsack algorithm, Pair technology, are able to get a better nesting result. The former is the approximate optimization algorithm;
More informationOffspring Generation Method using Delaunay Triangulation for RealCoded Genetic Algorithms
Offspring Generation Method using Delaunay Triangulation for RealCoded Genetic Algorithms Hisashi Shimosaka 1, Tomoyuki Hiroyasu 2, and Mitsunori Miki 2 1 Graduate School of Engineering, Doshisha University,
More informationIntroduction to Optimization
Introduction to Optimization Second Order Optimization Methods Marc Toussaint U Stuttgart Planned Outline Gradientbased optimization (1st order methods) plain grad., steepest descent, conjugate grad.,
More informationSparse Matrices Reordering using Evolutionary Algorithms: A Seeded Approach
1 Sparse Matrices Reordering using Evolutionary Algorithms: A Seeded Approach David Greiner, Gustavo Montero, Gabriel Winter Institute of Intelligent Systems and Numerical Applications in Engineering (IUSIANI)
More informationCAB: Fast Update of OBB Trees for Collision Detection between Articulated Bodies
CAB: Fast Update of OBB Trees for Collision Detection between Articulated Bodies Harald Schmidl Nolan Walker Ming C. Lin {schmidl walkern lin}@cs.unc.edu Department of Computer Science Sitterson Hall,
More informationHill Climbing. Assume a heuristic value for each assignment of values to all variables. Maintain an assignment of a value to each variable.
Hill Climbing Many search spaces are too big for systematic search. A useful method in practice for some consistency and optimization problems is hill climbing: Assume a heuristic value for each assignment
More informationExperience with the IMMa tyre test bench for the determination of tyre model parameters using genetic techniques
Vehicle System Dynamics Vol. 43, Supplement, 2005, 253 266 Experience with the IMMa tyre test bench for the determination of tyre model parameters using genetic techniques J. A. CABRERA*, A. ORTIZ, E.
More informationConvexity: an introduction
Convexity: an introduction Geir Dahl CMA, Dept. of Mathematics and Dept. of Informatics University of Oslo 1 / 74 1. Introduction 1. Introduction what is convexity where does it arise main concepts and
More informationMULTIOBJECTIVE OPTIMIZATION? DO IT WITH SCILAB
powered by MULTIOBJECTIVE OPTIMIZATION? DO IT WITH SCILAB Authors: Silvia Poles Keywords. Multiobjective optimization; Scilab Abstract: One of the Openeering team goal is to support optimization in companies
More informationPROJECT REPORT. Parallel Optimization in Matlab. Joakim Agnarsson, Mikael Sunde, Inna Ermilova Project in Computational Science: Report January 2013
Parallel Optimization in Matlab Joakim Agnarsson, Mikael Sunde, Inna Ermilova Project in Computational Science: Report January 2013 PROJECT REPORT Department of Information Technology Contents 1 Introduction
More information16.410/413 Principles of Autonomy and Decision Making
16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT)
More informationGA is the most popular population based heuristic algorithm since it was developed by Holland in 1975 [1]. This algorithm runs faster and requires les
Chaotic Crossover Operator on Genetic Algorithm Hüseyin Demirci Computer Engineering, Sakarya University, Sakarya, 54187, Turkey Ahmet Turan Özcerit Computer Engineering, Sakarya University, Sakarya, 54187,
More informationCS4733 Class Notes. 1 2D Robot Motion Planning Algorithm Using Grown Obstacles
CS4733 Class Notes 1 2D Robot Motion Planning Algorithm Using Grown Obstacles Reference: An Algorithm for Planning Collision Free Paths Among Poyhedral Obstacles by T. LozanoPerez and M. Wesley. This
More informationA Direct Search Algorithm for Global Optimization
Article A Direct Search Algorithm for Global Optimization Enrique Baeyens, *, Alberto Herreros and José R. Perán 2 Instituto de las Tecnologías Avanzadas de la Producción (ITAP), Universidad de Valladolid,
More informationCOLLISIONFREE TRAJECTORY PLANNING FOR MANIPULATORS USING GENERALIZED PATTERN SEARCH
ISSN 17264529 Int j simul model 5 (26) 4, 145154 Original scientific paper COLLISIONFREE TRAJECTORY PLANNING FOR MANIPULATORS USING GENERALIZED PATTERN SEARCH Ata, A. A. & Myo, T. R. Mechatronics Engineering
More informationAn Exact Algorithm for Finding Minimum Oriented Bounding Boxes
An Exact Algorithm for Finding Minimum Oriented Bounding Boxes Jukka Jyla nki 2015/06/01 Figure 1: Fast computation of the minimum volume OBB of a convex polyhedron. Left to right: Dragon: 2731149 vertices
More informationCT79 SOFT COMPUTING ALCCSFEB 2014
Q.1 a. Define Union, Intersection and complement operations of Fuzzy sets. For fuzzy sets A and B Figure Fuzzy sets A & B The union of two fuzzy sets A and B is a fuzzy set C, written as C=AUB or C=A OR
More informationLecture 6  Multivariate numerical optimization
Lecture 6  Multivariate numerical optimization Björn Andersson (w/ Jianxin Wei) Department of Statistics, Uppsala University February 13, 2014 1 / 36 Table of Contents 1 Plotting functions of two variables
More informationOriented Bounding Box generation
Oriented Bounding Box generation Overview The ObbUtils library contains all Oriented Bounding Box algorithms. All Oriented Bounding Box Algorithms implement the IObbAlgorithm interface. Additionally the
More informationWE consider the gatesizing problem, that is, the problem
2760 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 55, NO 9, OCTOBER 2008 An Efficient Method for LargeScale Gate Sizing Siddharth Joshi and Stephen Boyd, Fellow, IEEE Abstract We consider
More informationSurfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
More informationTheoretical Concepts of Machine Learning
Theoretical Concepts of Machine Learning Part 2 Institute of Bioinformatics Johannes Kepler University, Linz, Austria Outline 1 Introduction 2 Generalization Error 3 Maximum Likelihood 4 Noise Models 5
More informationUsing a Modified Genetic Algorithm to Find Feasible Regions of a Desirability Function
Using a Modified Genetic Algorithm to Find Feasible Regions of a Desirability Function WEN WAN 1 and JEFFREY B. BIRCH 2 1 Virginia Commonwealth University, Richmond, VA 232980032 2 Virginia Polytechnic
More information336 THE STATISTICAL SOFTWARE NEWSLETTER where z is one (randomly taken) pole of the simplex S, g the centroid of the remaining d poles of the simplex
THE STATISTICAL SOFTWARE NEWSLETTER 335 Simple Evolutionary Heuristics for Global Optimization Josef Tvrdk and Ivan Krivy University of Ostrava, Brafova 7, 701 03 Ostrava, Czech Republic Phone: +420.69.6160
More informationGenetic programming. Lecture Genetic Programming. LISP as a GP language. LISP structure. Sexpressions
Genetic programming Lecture Genetic Programming CIS 412 Artificial Intelligence Umass, Dartmouth One of the central problems in computer science is how to make computers solve problems without being explicitly
More information6th Grade Math. Parent Handbook
6th Grade Math Benchmark 3 Parent Handbook This handbook will help your child review material learned this quarter, and will help them prepare for their third Benchmark Test. Please allow your child to
More informationInformed search algorithms. (Based on slides by Oren Etzioni, Stuart Russell)
Informed search algorithms (Based on slides by Oren Etzioni, Stuart Russell) The problem # Unique board configurations in search space 8puzzle 9! = 362880 15puzzle 16! = 20922789888000 10 13 24puzzle
More informationOPTIMISATION METHODS FOR COMPRESSION, EXTENSION AND TORSION SPRING DESIGN
OPTIMISATION METHODS FOR COMPRESSION, EXTENSION AND TORSION SPRING DESIGN Emmanuel Rodriguez Laboratoire de Génie Mécanique de Toulouse, INSA, F31077 Toulouse, France, phone: 33 5 61 55 97 18, fax: 33
More informationReducing Graphic Conflict In Scale Reduced Maps Using A Genetic Algorithm
Reducing Graphic Conflict In Scale Reduced Maps Using A Genetic Algorithm Dr. Ian D. Wilson School of Technology, University of Glamorgan, Pontypridd CF37 1DL, UK Dr. J. Mark Ware School of Computing,
More informationA Provably Convergent Multifidelity Optimization Algorithm not Requiring HighFidelity Derivatives
A Provably Convergent Multifidelity Optimization Algorithm not Requiring HighFidelity Derivatives Multidisciplinary Design Optimization Specialist Conference April 4, 00 Andrew March & Karen Willcox Motivation
More informationCS 440 Theory of Algorithms /
CS 440 Theory of Algorithms / CS 468 Algorithms in Bioinformaticsi Brute Force. Design and Analysis of Algorithms  Chapter 3 30 Brute Force A straightforward approach usually based on problem statement
More informationModified Order Crossover (OX) Operator
Modified Order Crossover (OX) Operator Ms. Monica Sehrawat 1 N.C. College of Engineering, Israna Panipat, Haryana, INDIA. Mr. Sukhvir Singh 2 N.C. College of Engineering, Israna Panipat, Haryana, INDIA.
More informationCollision processing
Collision processing Different types of collisions Collision of a simulated deformable structure with a kinematic structure (easier) Collision with a rigid moving object, the ground, etc. Collision object
More informationNonDerivative Optimization: Mathematics or Heuristics?
NonDerivative Optimization: Mathematics or Heuristics? Margaret H. Wright Computer Science Department Courant Institute of Mathematical Sciences New York University Kovalevskaya Colloquium Berlin Mathematical
More informationEvolving VariableOrdering Heuristics for Constrained Optimisation
Griffith Research Online https://researchrepository.griffith.edu.au Evolving VariableOrdering Heuristics for Constrained Optimisation Author Bain, Stuart, Thornton, John, Sattar, Abdul Published 2005
More information2006/2007 Intelligent Systems 1. Intelligent Systems. Prof. dr. Paul De Bra Technische Universiteit Eindhoven
test gamma 2006/2007 Intelligent Systems 1 Intelligent Systems Prof. dr. Paul De Bra Technische Universiteit Eindhoven debra@win.tue.nl 2006/2007 Intelligent Systems 2 Informed search and exploration Bestfirst
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationFaculty of Science and Technology MASTER S THESIS
Faculty of Science and Technology MASTER S THESIS Study program/ Specialization: Petroleum Engineering/Reservoir Engineering Writer : Ibnu Hafidz Arief Faculty supervisor: Prof.Dr. Hans Spring semester,
More informationOn the Size of HigherDimensional Triangulations
Combinatorial and Computational Geometry MSRI Publications Volume 52, 2005 On the Size of HigherDimensional Triangulations PETER BRASS Abstract. I show that there are sets of n points in three dimensions,
More informationTracking in image sequences
CENTER FOR MACHINE PERCEPTION CZECH TECHNICAL UNIVERSITY Tracking in image sequences Lecture notes for the course Computer Vision Methods Tomáš Svoboda svobodat@fel.cvut.cz March 23, 2011 Lecture notes
More informationMulti Layer Perceptron trained by Quasi Newton learning rule
Multi Layer Perceptron trained by Quasi Newton learning rule Feedforward neural networks provide a general framework for representing nonlinear functional mappings between a set of input variables and
More informationAlgorithms for convex optimization
Algorithms for convex optimization Michal Kočvara Institute of Information Theory and Automation Academy of Sciences of the Czech Republic and Czech Technical University kocvara@utia.cas.cz http://www.utia.cas.cz/kocvara
More informationGENERATION OF STRUCTURED BLOCK BOUNDARY GRIDS
Congreso de Métodos Numéricos en Ingeniería 2005 Granada, 4 a 7 de julio, 2005 c SEMNI, España, 2005 GENERATION OF STRUCTURED BLOCK BOUNDARY GRIDS Eça L. 1, Hoekstra M. 2 and Windt J. 2 1: Mechanical Engineering
More informationOvercompressing JPEG images with Evolution Algorithms
Author manuscript, published in "EvoIASP2007, Valencia : Spain (2007)" Overcompressing JPEG images with Evolution Algorithms Jacques Lévy Véhel 1, Franklin Mendivil 2 and Evelyne Lutton 1 1 Inria, Complex
More informationOutline of Lecture. Scope of Optimization in Practice. Scope of Optimization (cont.)
Scope of Optimization in Practice and Niche of Evolutionary Methodologies Kalyanmoy Deb* Department of Business Technology Helsinki School of Economics Kalyanmoy.deb@hse.fi http://www.iitk.ac.in/kangal/deb.htm
More informationIntroduction to Design Optimization: Search Methods
Introduction to Design Optimization: Search Methods 1D Optimization The Search We don t know the curve. Given α, we can calculate f(α). By inspecting some points, we try to find the approximated shape
More informationRadio Network Planning with Combinatorial Optimisation Algorithms
Author manuscript, published in "ACTS Mobile Telecommunications Summit 96, Granada : Spain (1996)" Radio Network Planning with Combinatorial Optimisation Algorithms P. Calégari, F. Guidec, P. Kuonen, EPFL,
More informationNeural Network Weight Selection Using Genetic Algorithms
Neural Network Weight Selection Using Genetic Algorithms David Montana presented by: Carl Fink, Hongyi Chen, Jack Cheng, Xinglong Li, Bruce Lin, Chongjie Zhang April 12, 2005 1 Neural Networks Neural networks
More informationComparison of TSP Algorithms
Comparison of TSP Algorithms Project for Models in Facilities Planning and Materials Handling December 1998 Participants: ByungIn Kim JaeIk Shim Min Zhang Executive Summary Our purpose in this term project
More informationBinary Differential Evolution Strategies
Binary Differential Evolution Strategies A.P. Engelbrecht, Member, IEEE G. Pampará Abstract Differential evolution has shown to be a very powerful, yet simple, populationbased optimization approach. The
More informationMATH 890 HOMEWORK 2 DAVID MEREDITH
MATH 890 HOMEWORK 2 DAVID MEREDITH (1) Suppose P and Q are polyhedra. Then P Q is a polyhedron. Moreover if P and Q are polytopes then P Q is a polytope. The facets of P Q are either F Q where F is a facet
More informationUsing an outward selective pressure for improving the search quality of the MOEA/D algorithm
Comput Optim Appl (25) 6:57 67 DOI.7/s589597339 Using an outward selective pressure for improving the search quality of the MOEA/D algorithm Krzysztof Michalak Received: 2 January 24 / Published online:
More informationGenetic Algorithms For Vertex. Splitting in DAGs 1
Genetic Algorithms For Vertex Splitting in DAGs 1 Matthias Mayer 2 and Fikret Ercal 3 CSC9302 Fri Jan 29 1993 Department of Computer Science University of MissouriRolla Rolla, MO 65401, U.S.A. (314)
More informationDirect search methods for an open problem of optimization in systems and control
Direct search methods for an open problem of optimization in systems and control Emile Simon and Vincent Wertz 1 arxiv:1104.5183v3 [math.oc] 4 Jul 2011 Abstract The motivation of this work is to illustrate
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 3: Linear Programming, Continued Prof. John Gunnar Carlsson September 15, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 15, 2010
More informationGradient Methods for Machine Learning
Gradient Methods for Machine Learning Nic Schraudolph Course Overview 1. Mon: Classical Gradient Methods Direct (gradientfree), Steepest Descent, Newton, LevenbergMarquardt, BFGS, Conjugate Gradient
More informationToday. Gradient descent for minimization of functions of real variables. Multidimensional scaling. Selforganizing maps
Today Gradient descent for minimization of functions of real variables. Multidimensional scaling Selforganizing maps Gradient Descent Derivatives Consider function f(x) : R R. The derivative w.r.t. x
More informationParticle Swarm Optimization
Dario Schor, M.Sc., EIT schor@ieee.org Space Systems Department Magellan Aerospace Winnipeg Winnipeg, Manitoba 1 of 34 Optimization Techniques Motivation Optimization: Where, min x F(x), subject to g(x)
More informationWe have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.
Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the
More informationDerivativeFree Optimization
DerivativeFree Optimization Chapter 7 from Jang Outline Simulated Annealing (SA) Downhill simplex search Random search Genetic algorithms (GA) 2 The Big Picture Model space Adaptive networks Neural networks
More informationME964 High Performance Computing for Engineering Applications
ME964 High Performance Computing for Engineering Applications Outlining Midterm Projects Topic 3: GPUbased FEA Topic 4: GPU Direct Solver for Sparse Linear Algebra March 01, 2011 Dan Negrut, 2011 ME964
More informationLecture 2. Topology of Sets in R n. August 27, 2008
Lecture 2 Topology of Sets in R n August 27, 2008 Outline Vectors, Matrices, Norms, Convergence Open and Closed Sets Special Sets: Subspace, Affine Set, Cone, Convex Set Special Convex Sets: Hyperplane,
More informationGenetic Algorithm Performance with Different Selection Methods in Solving MultiObjective Network Design Problem
etic Algorithm Performance with Different Selection Methods in Solving MultiObjective Network Design Problem R. O. Oladele Department of Computer Science University of Ilorin P.M.B. 1515, Ilorin, NIGERIA
More informationRecall: Inside Triangle Test
1/45 Recall: Inside Triangle Test 2D Bounding Boxes rasterize( vert v[3] ) { bbox b; bound3(v, b); line l0, l1, l2; makeline(v[0],v[1],l2); makeline(v[1],v[2],l0); makeline(v[2],v[0],l1); for( y=b.ymin;
More informationConvex Optimization  Chapter 12. Xiangru Lian August 28, 2015
Convex Optimization  Chapter 12 Xiangru Lian August 28, 2015 1 Mathematical optimization minimize f 0 (x) s.t. f j (x) 0, j=1,,m, (1) x S x. (x 1,,x n ). optimization variable. f 0. R n R. objective
More informationMinimumArea Rectangle Containing a Set of Points
MinimumArea Rectangle Containing a Set of Points David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More informationDerating NichePSO. Clive Naicker
Derating NichePSO by Clive Naicker Submitted in partial fulfillment of the requirements for the degree Magister Scientiae (Computer Science) in the Faculty of Engineering, Built Environment and Information
More informationIn this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems.
2 Basics In this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems. 2.1 Notation Let A R m n be a matrix with row index set M = {1,...,m}
More informationIntroduction to ANSYS DesignXplorer
Lecture 5 Goal Driven Optimization 14. 5 Release Introduction to ANSYS DesignXplorer 1 2013 ANSYS, Inc. September 27, 2013 Goal Driven Optimization (GDO) Goal Driven Optimization (GDO) is a multi objective
More informationApplication of Genetic Algorithms to CFD. Cameron McCartney
Application of Genetic Algorithms to CFD Cameron McCartney Introduction define and describe genetic algorithms (GAs) and genetic programming (GP) propose possible applications of GA/GP to CFD Application
More informationLecture Plan. Bestfirst search Greedy search A* search Designing heuristics. Hillclimbing. 1 Informed search strategies. Informed strategies
Lecture Plan 1 Informed search strategies (KA AGH) 1 czerwca 2010 1 / 28 Blind vs. informed search strategies Blind search methods You already know them: BFS, DFS, UCS et al. They don t analyse the nodes
More informationA Different Approach for Continuous Physics. Vincent ROBERT Physics Programmer at Ubisoft
A Different Approach for Continuous Physics Vincent ROBERT vincent.robert@ubisoft.com Physics Programmer at Ubisoft A Different Approach for Continuous Physics Existing approaches Our method Limitations
More informationTriangles and Squares David Eppstein, ICS Theory Group, April 20, 2001
Triangles and Squares David Eppstein, ICS Theory Group, April 20, 2001 Which unitsidelength convex polygons can be formed by packing together unit squares and unit equilateral triangles? For instance
More informationLecture 3: Art Gallery Problems and Polygon Triangulation
EECS 396/496: Computational Geometry Fall 2017 Lecture 3: Art Gallery Problems and Polygon Triangulation Lecturer: Huck Bennett In this lecture, we study the problem of guarding an art gallery (specified
More informationAvailable online at ScienceDirect. Procedia CIRP 44 (2016 )
Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 44 (2016 ) 102 107 6th CIRP Conference on Assembly Technologies and Systems (CATS) Worker skills and equipment optimization in assembly
More information