Sampling with Halton Points on n-Sphere

Sampling with Halton Points on n-Sphere

Wai-Shing Luk1

1School of MicroelectronicsFudan University

April 6, 2014

Agenda

Abstract

Motivation and Applications

Review of Low Discrepancy SequenceVan der Corput sequence on [0; 1]Halton sequence on [0; 1]Halton sequence on [0; 1]n

Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach

Numerical Experiments

Conclusions

Agenda

Abstract



Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach


Conclusions

Abstract

I Sampling on n-sphere (Sn) has a wide range ofapplications, such as:

I Spherical coding in MIMO wireless communicationI Multivariate empirical mode decompositionI Filter bank design

I We propose a simple yet effective method which:I Utilizes low-discrepancy sequenceI Contains only 10 lines of MATLAB code in our

implementation!I Allow incremental generation.

I Numerical results show that the proposed methodoutperforms the randomly generated sequences and otherproposed methods.

Agenda

Abstract



Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach


Conclusions

Problem Formulation

Desirable properties of samples over Sn

I UniformI DeterministicI Incremental

I The uniformity measures are optimized with every newpoint.

I Reason: in some applications, it is unknown how manypoints are needed to solve the problem in advance

Motivation

I The topic has been well studied for sphere in 3D, i.e. n = 2I Yet it is still unknown how to generate for n > 2.I Potential applications (for n > 2):

I Robotic Motion Planning (S3 and SO(3)) [YJLM10]I Spherical coding in MIMO wireless communication [UL06]:

I Cookbook for Unitary matricesI A code word = a point in Sn

I Multivariate empirical mode decomposition [RM10]I Filter bank design [M+11]

Halton Sequence on Sn

I Halton sequence on S2 has been well studied [CF97] byusing cylindrical coordinates.

I Yet it is still little known for Sn where n > 2.I Note: The generalization of cylindrical coordinates does

NOT work in higher dimensions.

Agenda

Abstract



Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach


Conclusions

Basic: Van der Corput sequence

I Generate a low discrepancy sequence over [0; 1]I Denote vd(k ; b) as a Van der Corput sequence of k points,

where b is the base of a prime number.I MATLAB source code is available at http:

//www.mathworks.com/matlabcentral/fileexchange/15354-generate-a-van-der-corput-sequence

Example

http://www.mathworks.com/matlabcentral/fileexchange/15354-generate-a-van-der-corput-sequence







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Example




Unit Square [0; 1]� [0; 1]

Halton sequence: using 2Van der Corput sequenceswith different bases.

Example[x ; y ] = [vd(k ; 2); vd(k ; 3)]

Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Square [0; 1]� [0; 1]



Unit Hypercube [0; 1]n

I Generally we can generate Halton sequence in a unithypercube [0; 1]n :

[x1; x2; : : : ; xn ] = [vd(k ; b1); vd(k ; b2); : : : ; vd(k ; bn)]

I A wide range of applications on Quasi-Monte CarloMethods (QMC).

Unit Circle S 1

Can be generated by mapping theVan der Corput sequence to [0; 2�]

I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Circle S 1


I � = 2� � vd(k ; b)I [x ; y ] = [cos �; sin �]

Unit Sphere S 2

Has been applied for computergraphic applications [WLH97]

I [z ; x ; y ]= [cos �; sin � cos'; sin � sin']= [z ;

p1� z 2 cos';

p1� z 2 sin']

I ' = 2� � vd(k ; b1) % map to[0; 2�]

I z = 2 � vd(k ; b2)� 1 % map to[�1; 1]

Sphere S 3 and SO(3)

I Deterministic point setsI Optimal grid point sets for S3, SO(3) [Lubotzky, Phillips,

Sarnak 86] [Mitchell 07]I No Halton sequences so far to the best of our knowledge

Agenda

Abstract



Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach


Conclusions

SO(3) or S 3 Hopf Coordinates

I Hopf coordinates (cf. [YJLM10])I x1 = cos(�=2) cos( =2)I x2 = cos(�=2) sin( =2)I x3 = sin(�=2) cos('+ =2)I x4 = sin(�=2) sin('+ =2)

I S3 is a principal circle bundleover the S2

Hopf Coordinates for SO(3) or S 3

Similar to the Halton sequence generation on S2, we performthe mapping:

I ' = 2� � vd(k ; b1) % map to [0; 2�]I = 2� � vd(k ; b2) % map to [0; 2�] for SO(3), orI = 4� � vd(k ; b2) % map to [0; 4�] for S3

I z = 2 � vd(k ; b3)� 1 % map to [�1; 1]I � = cos�1 z

10 Lines of MATLAB Code

1 function[s] = sphere3_hopf(k,b)2 % sphere3_hopf Halton sequence3 varphi = 2*pi*vdcorput(k,b(1)); % map to [0, 2*pi]4 psi = 4*pi*vdcorput(k,b(2)); % map to [0, 4*pi]5 z = 2* vdcorput(k,b(3)) - 1; % map to [-1, 1]6 theta = acos(z);7 cos_eta = cos(theta /2);8 sin_eta = sin(theta /2);9 s = [cos_eta .* cos(psi/2), ...

10 cos_eta .* sin(psi/2), ...11 sin_eta .* cos(varphi + psi/2), ...12 sin_eta .* sin(varphi + psi /2)];

3-sphere

I Polar coordinates:I x0 = cos �3I x1 = sin �3 cos �2I x2 = sin �3 sin �2 cos �1I x3 = sin �3 sin �2 sin �1

n-sphere

I Polar coordinates:I x0 = cos �nI x1 = sin �n cos �n�1I x2 = sin �n sin �n�1 cos �n�2I x3 = sin �n sin �n�1 sin �n�2 cos �n�3I � � �I xn�1 = sin �n sin �n�1 sin �n�2 � � � cos �1I xn = sin �n sin �n�1 sin �n�2 � � � sin �1

How to Generate the Point Set

I p0 = [cos �1; sin �1] where �1 = 2� � vd(k ; b1)I Let fj (�) =

Rsinj �d�, where � 2 (0; �).

Note: fj (�) is a monotonic increasing function in (0; �)I Map vd(k ; bj ) uniformly to fj (�):tj = fj (0) + (fj (�)� fj (0))vd(k ; bj )

I Let �j = f �1j (tj )I Define pn recursively as:pn = [cos �n ; sin �n � pn�1]

S 3 projected on four different spheres

Agenda

Abstract



Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach


Conclusions

Testing the Correctness

I Compare the dispersion with the random point-setI Construct the convex hull for each point-setI Dispersion roughly measured by the difference of the

maximum distance and the minimum distance betweenevery two neighbour points:

maxa2N (b)

fD(a ; b)g � mina2N (b)

fD(a ; b)g

where D(a ; b) =p1� aTb

Random sequences

I To generate random points on Sn , spherical symmetry ofthe multidimensional Gaussian density function can beexploited.

I Then the normalized vector (xi=kxik) is uniformlydistributed over the hypersphere Sn . [Fishman, G. F.(1996)]

Convex Hull with �400 points

Left: our, right: random

Results for S 3

Results for S 4

Agenda

Abstract



Unit Circle S1

Unit Sphere S2

Sphere Sn and SO(3)

Our approach


Conclusions

Conclusions

I Proposed method generates low-discrepancy point-set innearly linear time

I The result outperforms the corresponding randompoint-set, especially when the number of points is small

I The MATLAB source code is available in public (or uponrequest)

References I

[CF97] Jianjun Cui and Willi Freeden, Equidistribution on the sphere, SIAMJournal on Scientific Computing 18 (1997), no. 2, 595–609.

[M+11] DP Mandic et al., Filter bank property of multivariate empirical modedecomposition, Signal Processing, IEEE Transactions on 59 (2011),no. 5, 2421–2426.

[RM10] Naveed Rehman and Danilo P Mandic, Multivariate empirical modedecomposition, Proceedings of the Royal Society A: Mathematical,Physical and Engineering Science 466 (2010), no. 2117, 1291–1302.

[UL06] Zoran Utkovski and Juergen Lindner, On the construction ofnon-coherent space time codes from high-dimensional spherical codes,Spread Spectrum Techniques and Applications, 2006 IEEE NinthInternational Symposium on, IEEE, 2006, pp. 327–331.

[WLH97] Tien-Tsin Wong, Wai-Shing Luk, and Pheng-Ann Heng, Sampling withhammersley and halton points, Journal of graphics tools 2 (1997),no. 2, 9–24.

[YJLM10] Anna Yershova, Swati Jain, Steven M LaValle, and Julie C Mitchell,Generating uniform incremental grids on so (3) using the hopffibration, The International journal of robotics research 29 (2010),no. 7, 801–812.

Sampling with Halton Points on n-Sphere

Technology

corput sequence

corputsequence example

lowdiscrepancy sequence

sn halton sequence

halton points

nsphere sn

matlab source code

prime number