Induced well-distributed sets in Riemannian spaces

Induced Well-Distributed Sets inRiemannian Spaces

LOTHAR WENZEL, RAM RAJAGOPAL, and DINESH NAIRNational Instruments

The concept of Riemannian geometries is used to construct induced homogeneous point sets onmanifolds that are based on well-distributed point sets in unit cubes of an appropriately chosen Eu-clidean space. These well-distributed point sets in unit cubes are based on standard low-discrepancysequences. The approach is algorithmic, that is, the methods developed in this article have beenimplemented and tested. Applications in image processing, graph theory and measurement-basedexploration are presented.

Categories and Subject Descriptors: J.2 [Computer Applications, Physical Science andEngineering]

General Terms: Algorithms, Theory, Measurement

Additional Key Words and Phrases: Riemannian geometry, well-distributed point sets, low-discrepancy sequences, image processing

1. INTRODUCTION

This article is organized as follows. Section 2 briefly introduces the conceptsof Riemannian geometries and low-discrepancy sets in an n-dimensional unitcube. Section 3 deals with (abstract) surfaces and develops an algorithm thatallows the construction of well-distributed sets of points lying on arbitrary sur-faces. In Section 4, simple examples of well-distributed points on arbitrarysurfaces are shown. Section 5 generalizes the results to the case of higher di-mensional spaces. The algorithms developed in this article are used in Section 6to solve real-world problems. It is shown that an image can be regarded asa Riemannian space and based on this a generalized edge detector is de-rived. Efficient exploration of higher dimensional spaces is another applica-tion field. A specific fiber optics alignment problem was the original reason todeal with these questions. The embedding of large graphs in Euclidean spacescombined with detection of dimensionality and determination of homogeneous

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ACM Transactions on Mathematical Software, Vol. 29, No. 1, March 2003, Pages 82–94.

Induced Well-Distributed Sets in Riemannian Spaces • 83

subgraphs is an application where discrete objects are replaced with continuousones.

In almost all of the potential applications the use of mappings of low-discrepancy sequences offers advantages. Low-discrepancy sets are always welldistributed. An additional criterion can be applied to stop a computation whena certain objective is achieved.

2. RIEMANNIAN SPACES AND WELL-DISTRIBUTED SETS OF POINTS

We begin with some well-known facts regarding Riemannian metrics (e.g., Gray[1998]) and low-discrepancy sequences. Given an abstract surface S with aRiemannian metric defined for (u, v) in [0, 1]2,

ds2 = E(u, v) du2 + 2F (u, v) du dv+ G(u, v) dv2 (1)

E(u, v), F (u, v), and G(u, v) are differentiable functions in u and v, and EG-F 2

is nonnegative. The area element dA is defined by:

dA =√

E(u, v)G(u, v)− F (u, v)2 du ∧ dv (2)

The function

9(u, v) =√

E(u, v)G(u, v)− F (u, v)2 (3)

is nonnegative in [0, 1]2 and 92 is differentiable. An abstract n-dimensionalmetric is defined by

ds2 =n∑

i, j=1

gij (u1, . . . , un) du2i du2

j (4)

where the matrix consisting of gij : [0, 1]n→R is always symmetric, differen-tiable, and positive semi-definite. An embedding of an abstract space (4) in anm-dimensional Euclidean space is a diffeomorphism (patches) g of the cube[0, 1]2 with

g1 (u1, . . . , un) , g2 (u1, . . . , un) , . . . , gm (u1, . . . , un) : Rn→ R,

where the Riemannian metric of this embedding is described by (4). Usually,this definition is too restrictive. Instead, local diffeomorphisms (patches) shouldbe used where these patches cover the whole space under consideration.

To quantify the homogeneity of a finite 1-dimensional set of points, the defi-nition of discrepancy of a set X was introduced (Corput [1935]):

D (X ) = supR|m (R)− p (R)| (5)

In (5), R runs over all rectangles [0.r] with 0 ≤ r ≤ 1, m(R) stands for thelength r of the closed interval R, and p(R)is the ratio of the number of points ofX in R and the number of all points of X . The definition given in Eq. (5) can begeneralized to the case of d dimensions (d = 2, 3, . . . ), where the term intervalmust be interpreted as a d -dimensional rectangle. The lower the discrepancythe better or more homogeneous is the distribution of the set. The discrepancy

ACM Transactions on Mathematical Software, Vol. 29, No. 1, March 2003.

84 • L. Wenzel et al.

of an infinite sequence X = {x1,x2, . . . , xn, , . . . } is a new sequence of positivereal numbers D(X n) , where X n stands for the first n elements of X .

There exists a set of points of given length, that realizes the lowest discrep-ancy. It is well known (e.g., Kocis and Whiten [1997]) that inequality (6) holdstrue for all finite sequences X of length n in the d -dimensional unit cube

D (X ) ≥ Bd

(log n

)(d−1)/2

n. (6)

Bd depends only on d . Except for the trivial case d = 1, it is unknownwhether the theoretical lower bound is attainable. Many schemes to build finitesequences X of length n do exist that deliver a slightly worse limit

D (X ) ≤ Bd

(log n

)d

n. (7)

There are also infinite sequences X with (7) for all subsequences consist-ing of the first n elements. The latter result gave rise to the definition oflow-discrepancy (infinitse) sequences X . The inequality in Eq. (7) must bevalid for all subsequences, where Bd is an appropriately chosen constant. Low-discrepancy sequences are also known as quasi-random sequences.

Many of the well-studied low-discrepancy sequences in the d -dimensionalunit cube can be constructed as combinations of 1-dimensional low-discrepancysequences. The most popular low-discrepancy sequences are based on schemesintroduced by Richtmyer [1951, 1958], Halton [1960], Sobol [1967], andNiederreiter [1992]. We discuss the topic of low-discrepancy (space-) curvesand efficient scanning strategies in a separate article, Wenzel et. al. [2001].

3. INDUCED POINT DISTRIBUTIONS

The question arises whether an area-preserving map up to a constant factorfrom [0, 1]2onto an abstract surface S can be found. For sufficiently smooth bi-jective functions f (u, v) and g (u, v) the determinant of the Jacobian representsthe ratio between the two area elements (see also Lemma 15.12 in Gray [1998]).Based on such mappings, a set of points defined in [0, 1]2 could be transformedonto the abstract surface, preserving low discrepancy in the sense of the metricdefined for the surface. Therefore, given a nonnegative function 9(u, v) definedon [0, 1]2 are there functions f (u, v), g (u, v) and a constant c with

∂ f (u, v)∂u

∂ g (u, v)∂v

− ∂ f (u, v)∂v

∂ g (u, v)∂u

= c9(u, v), (8)

where ( f , g ) is a diffeomorphism of [0, 1]2. We show that under certain condi-tions this choice is always possible.

THEOREM 1. Let 9(u, v) be a nonnegative function on [0, 1]2 where 92 iscontinuously differentiable. Let 9(u, v) be positive with exception of a set L of



points (u, v) of Lebesgue-measure 0. For (u, v) /∈ L, let

f (u, v) = 1∫ 10 du9(u, v)

∫ u

0du9(u, v)

g (u, v) = g (v) = 1∫ 10

∫ 10 du dv9(u, v)

∫ 1

0

∫ v

0du dv9(u, v)

(9)

Furthermore, let the functions f and g be extendable to continuously differen-tiable mappings defined on [0, 1]2. Then the extension functions ( f, g) define adiffeomorphism of [0, 1]2

/L where (8) is always valid.

PROOF. 9 is a continuous function. L is a closed subset with Lebesque-measure 0. L can be excluded from the aforementioned integrations withoutchanging the results. The denominators in (9) are positive. The extensions off and g map [0, 1]2 onto [0, 1]2 and are continuously differentiable. g (v)isstrictly monotone in v (the Lebesgue-measure of L is 0). For (u, v) /∈ L, it is

∂ f (u, v)∂u

∂ g (u, v)∂v

− ∂ f (u, v)∂v

∂ g (u, v)∂u

= ∂ f (u, v)∂u

g ′(v)= 1∫ 10

∫ 10 du dv9(u, v)

9(u, v)

g (u, v) = g (u′, v′) implies v = v′ and from f (u, v) = f (u′, v) it follows u = u′.

Theorem 1 allows the construction of well-distributed sets D of points onRn that are embeddings of abstract surfaces. The procedure is based on givendistributions of sets in [0, 1]2. An Rn embedding of an abstract surface (1) is amapping x(u, v) = (x1(u, v), x2(u, v), . . . , xn(u, v)) with E = xuxu, , F = xuxv andG = xvxv. All functions xk are differentiable.

The algorithm is straightforward and works as follows:

Algorithm I:

(I.1) Given an abstract surface S defined on [0, 1]2 where 9(u, v) satisfies the proposi-tions of Theorem 1. In addition, let x(u, v) be a known embedding of S in Rm.

(I.2) According to Theorem 1 a diffeomorphism ( f (u, v), g (u, v)) of [0, 1]2/L can be con-

structed.(I.3) Compute the inverse transform ( f −1(u, v), g−1(u, v)).(I.4) Let D be a well-distributed set in [0, 1]2. The image of D under the transform

x( f −1(u, v), g−1(u, v)) forms a well-distributed set lying on the aforementionedembedding in Rm.

Because of (8) the latter transform preserves areas (up to a constant factor). Inparticular, (I.1)–(I.4) provides a method to preserve low-discrepancy properties.

Algorithm (I.1)–(I.4) also produces a new abstract surface:

ds2 = E( f −1(u, v), g−1(u, v)) du2 + 2F ( f −1(u, v), g−1(u, v)) du dv+G( f −1(u, v), g−1(u, v)) dv2

In this surface, the resulting area element dA2 is independent of u and v.



Fig. 1. Original low-discrepancy distribution (left) and induced version (right).

4. SOME SURFACE EXAMPLES

One of the simplest examples is the abstract surface defined by:

ds2 = du2 + u2 dv2

There is an embedding in R2, x(u, v)= (u cos(2πv), u sin(2πv)). This embed-ding is simply a unit circle in polar coordinates and gives 9(u, v) = u.Steps (I.1)–(I.4) deliver the diffeomorphism ( f (u, v), g (u, v)) = (u2, v), that is,( f −1(u, v), g−1(u, v)) = (

√u, v) which, of course, produces a new parameteri-

zation (√

u cos(2πv),√

u sin(2πv)) of the given circle. Figure 1 compares a low-discrepancy set derived from such a sequence in [0, 1]2 (i.e., discard those pointsin [0, 1]2 that are outside the circle) with the image of such a sequence underan induced transform. In both cases, a rectangular region in proximity to thecenter of a circle is depicted. The distributions look alike; however, Figure 2reveals that the Fourier transforms are remarkably different. Pronounced ori-entation dependency in the first case and isotropic frequency behavior in thelatter case are characteristic properties of these approaches. In almost all cases,the latter approach offers significant advantages. The reason for this is simplythe fact that an axis-dependent sampling strategy neglects the real structureof the underlying problem (e.g., image processing grid, shape of the domainof integration). In many machine vision applications, rotation invariance isessential. For example, numerous pattern matching projects allow situationswhere template and image can be rotated. An x − y oriented sampling schemeprefers these two directions. This can result in mismatches or undetectedmatches. Additionally, the first approach based on derived low-discrepancysets in [0, 1]2 suffers from serious drawbacks in higher dimensional spaces(see Section 5).



Fig. 2. The Fourier transforms reveal remarkable differences.

A more sophisticated example is a part of a sphere given as an abstractsurface by

ds2 = cos2(v) du2 + dv2,

where (u, v) is located in [0, 1]2. An R3 embedding is x(u, v) = (cos(u) cos(v),sin(u) cos(v), sin(v)). Algorithm (I.1)–(I.4) produces the diffeomorphism ( f (u, v),g (u, v)) = (u, sin(v)/sin(1)), that is, ( f −1(u, v), g−1(u, v)) = (u, arcsin(sin(1)v)).

The third example deals with the surface of a torus. The R3 embedding andthe abstract surface have the descriptions (b < a)

x(u, v) = ((a + bcos(2πv)) cos(2πu), (a + bcos(2πv)) sin(2πu), bsin(2πv))ds2 = 4π2(a + bcos(2πv))2 du2 + 4π2b2 dv2

Algorithm (I.1)–(I.4) delivers

( f (u, v), g (u, v)) =(

u, v+ b2aπ

sin(2πv)).

After computing the inverse transform homogeneous point sets lying on thesurface of a torus can be generated. Figure 3 shows such a distribution andprojections onto different planes.

5. HIGHER DIMENSIONAL RIEMANNIAN GEOMETRIES

In 3- or n-dimensional Riemannian geometries, the term area preserving mustbe replaced with volume preserving or content preserving, respectively. Volumeor content preserving mappings (up to a constant factor) are characterized by

9(u1, . . . , un) =√

det(gij(u1, . . . , un)) = const. (10)

It is natural to generalize Theorem 1 in the following manner.

THEOREM 2. Let 9(u1, u2, . . . , un) be a nonnegative function on [0, 1]n where92 is continuously differentiable. Let9(u1, u2, . . . , un) be positive with exception



Fig. 3. Well-distributed point sets on the surface of a torus.

of a set L of points (u1, u2, . . . , un) of Lebesgue-measure 0. For (u1, u2, . . . , un) /∈L, let

f1(u1, u2, . . . , un) =∫ u1

0 du19(u1, . . . , un)∫ 10 du19(u1, . . . , un)

· · ·fn−1(un−1, un) =

∫ un−1

0

∫ 10 · · ·

∫ 10 du1 · · ·dun−19(u1, . . . , un)∫ 1

0

∫ 10 · · ·

∫ 10 du1 · · ·dun−19(u1, . . . , un)

fn(un) =∫ un

0

∫ 10 · · ·

∫ 10 du1 · · ·dun9(u1, . . . , un)∫ 1

0

∫ 10 . . .

∫ 10 du1 · · ·dun9(u1, . . . , un)

Furthermore, let the functions f1, f2, . . . , fn be extendable to continuously dif-ferentiable mappings defined on [0, 1]n. Then the extension functions f =( f1, f2, . . . , fn) define a diffeomorphism of [0, 1]n

/L where (10) is always valid.

PROOF. 9 is a continuous function. L is a closed subset with Lebesque-measure 0. L can be excluded from the aforementioned integrations withoutchanging the results. The specific structure of these functions simplifies thedeterminant of the Jacobian of f to exactly one product

∂ f1(u1, . . . , un)∂u1

∂ f2(u2, . . . , un)∂u2

. . .∂ fn(un)∂un

.

The rest is similar to proof of Theorem 1.

Algorithm II:

(II.1) Given an abstract surface S defined on [0, 1]n where9(u1, u2, . . . , un) satisfies thepropositions of Theorem 2. Furthermore, let x(u1, u2, . . . , un) be an embedding ofS in Rm.

(II.2) In accordance with Theorem 2, a diffeomorphism f (u1, . . . , un) =( f1(u1, . . . , un), . . . , fn(u1, . . . , un)) of [0, 1]n can be constructed.



Fig. 4. The case n = 3, projections are included.

(II.3) Compute the inverse transform f −1(u1, . . . , un).(II.4) Let D be a well-distributed set in Rn. The image of D under the trans-

form x( f −1(u1, . . . , un)) forms a well-distributed set lying on the aforementionedembedding in Rm.

As an illustrative example, for the unit ball in n-dimensional Euclideanspace, the standard parameterization is given by:

f (u1, . . . , un) = un(sin(2πu1) sin(πu2) · · · sin(πun−1),cos(2πu1) sin(πu2) · · · sin(πun−1) · · · cos(πun−2) sin(πun−1), cos(πun−1))

Then, one can show that the following formula is valid:

9(u1, . . . , un)2 = 4(π2u2

n

)n−1 sin2(πu2) sin4(πu3) · · · sin2(n−2)(πun−1).

Applying Theorem 2 leads to the functions:

fk (u1, . . . , un) =∫ uk

0 duk sink−1 (πuk)∫ 10 duk sink−1 (πuk)

for k = 1, 2, . . . , n− 1

fn (u1, . . . , un) = unn

The functions fk depend only on uk and can easily be inverted. Figure 4demonstrates the case n = 3.

It is worthwhile to note that the alternative approach of deriving a well-distributed set based on the surrounding unit n-cube suffers from serious lim-itations. The content of such an n-cube is 1, the content of an n-dimensionalunit ball is

√π/

20(n

2 + 1)which is on the order of n−n/2. In other words, the

probability to hit the n-ball is extremely small when n is beyond 10.

6. NUMERICAL IMPLEMENTATION

The algorithms for low-discrepancy sampling presented earlier apply to a con-tinuous description of the abstract surfaces. The algorithms require as an inputthe definition of a Riemanian metric on the surface to be sampled and an em-bedding for it. Using these definitions, the area-preserving diffeomorphism can



be constructed following the steps discussed in the earlier sections. Low-discrepancy points can be generated using any of the available methods (Sobol,Halton, etc.) and mapped in accordance with the area-preserving diffeomor-phism.

In practice, numerical approximations have to be used at many stages ofAlgorithms I and II. For example, in one scenario, although each function fihas a closed form solution, the inverse may not be available, or maybe hard tocompute. The numerical computation of the inverse is not hard, though. Theinverse n-tuple (u1, . . . , un) can be computed using a linear algorithm. Noticingthat fi is only a function of the parameters u1, . . . , ui and that the functionsare monotonic:

(1) Start with i = 1. Solve f1(u1) = x1for u1.(2) For i = 2 to m. Solve f1(u1, . . . , ui) = xi, given u1 to ui−1. 1-dimensional

solvers can be used.

In some situations, 9 (u, v) may not have closed forms for the required inte-grals. In those cases, the functions fi can be computed numerically and inter-polators used in conjunction with 1-dimensional solvers to compute the inversefunctions.

In other cases, the surface is given by a discrete set of points that representthe continuous function. An image is a typical example, where the continuousimage function I (u, v) is represented by a discrete map (the pixel map). Thediffeomorphism can still be constructed for such cases, but will be given by adiscrete map as well (sampling of the function).

Let us consider surfaces in R3. Generalized surfaces can be treated in asimilar fashion. To obtain the diffeomorphism in the discrete case, the functionsE(u, v), F (u, v) and G(u, v) should be computed numerically. This can be doneby noticing that E(u, v) is the variation when dv is set to zero, and F (u, v) thevariation when du is set to zero. G(u, v) is the variation when both du and dv areallowed to change, subtracting E(u, v) and G(u, v). If the embedding is sampledin a regular grid, the variations can be calculated using standard numericaldifferentiation (or even just subtraction).

Based on the numerical maps of E, F and G, the map of the area func-tion 9(u, v) can be constructed. The integrals in (9) can be solved numerically,using standard methods, such as the Backwards Euler method. The numer-ical solution will yield numerical maps for the functions f (u, v) and g (u, v).The monotonic nature of the diffeomorphism makes numerical inversion of thefunctions a simple procedure that relies on standard interpolation. Given valuesfor ( f , g ), the corresponding (u, v) pair can be numerically obtained by linearsearching and interpolation of the f (u, v) and g (u, v) maps.

7. APPLICATIONS

7.1 Integration

The most straightforward applications are calculations of integrals of smoothfunctions defined on Riemannian geometries embedded in Rm. The replacement



of integrals by sums of function values of well-chosen sampling points offers thesame advantages as those based on low-discrepancy sets defined in unit cubes.

7.2 Image Processing

Another application field is image processing. Assuming, the intensity of animage is described by I (u, v) where (u, v) fills the unit square. Furthermore, letI (u, v) be sufficiently smooth in u and v. Such an image can be reinterpreted asa surface (u, v, I (u, v)) and Theorem 1 can be applied. The Riemannian metricbelonging to I (u, v) is

ds2 =(

1+(∂ I (u, v)∂u

)2)

du2 + 2∂ I (u, v)∂u

∂ I (u, v)∂v

du dv

+(

1+(∂ I (u, v)∂v

)2)

dv2

Therefore,

9(u, v) =√

1+(∂ I (u, v)∂u

)2

+(∂ I (u, v)∂v

)2

(11)

The resulting diffeomorphism ( f (u, v), g (u, v)) represents information con-tained in I (u, v). In particular, it encodes image-related information such assharp local transitions (edges). See also Figure 5. In applications, the integralsare replaced with sums of function values. More importantly, the inverse diffeo-morphism in conjunction with appropriate sampling strategies in [0, 1]2 allowsan induced sampling of a given image content.

Algorithm III:

(III.1) Compute (11) where I is a given image (e.g., grey-levels).(III.2) Determine the inverse diffeomorphism according to Theorem 1.(III.3) Choose a low-discrepancy sequence in [0, 1]2.(III.4) Compute the image sequence of this low-discrepancy set under the diffeomor-

phism mentioned in (III.2).

The intensity values of I at these points describe the image content in anefficient manner and can be used to compress an image or to characterize itscontent. To a certain extent, the resulting description can be regarded as a gen-eralized feature detector that is highly optimized for a specific image. Potentialapplications can be found in pattern matching where a given template shouldbe characterized by a small amount of pixels, to speed up successive searchoperations. The use of low-discrepancy sets offers the advantages mentionedbefore.

Figure 5 demonstrates the effect of Algorithm III on a given image. Thesmall rectangles represent the sampling points computed by Algorithm III.The algorithms identify edges and other significant points.



Fig. 5. Effect of Algorithm III on an image.

7.3 Testing and Exploration

A typical scenario in nondestructive testing or during search routines is basedon well-chosen locations where measurements are taken. Usually, the searchspace is large and a specific point operation is computationally or otherwiseexpensive. Efficient scanning strategies are necessary.

We assume here that an efficient search strategy for an n-dimensional cube isavailable. Such a strategy could be based on low-discrepancy sets, homogenousgrids, or other approaches. With the aid of Theorem 2, an induced strategy canbe derived.

As an example, a search problem in fiber optics is chosen. Given an array offibers where a second array of theses fibers must be aligned accurately. Typi-cally, lasers are applied and the intensity of these beams can be measured toalign the arrays (see Figure 6 for details). If the first array is regarded as fixedwith unknown position and orientation, the second array can be moved to estab-lish an alignment. A simple model of this scenario consists of 3 position degreesof freedom (x, y , and z) in combination with 3 orientation degrees of freedom(pitch α, roll β, and yaw angle γ ). This results in a 6-dimensional search spacewhere further constraints add complexity to this space. For example,

x2 + y2 + z2 ≤ P and (α − α0)2 + (β − β0)2 + (γ − γ0)2 ≤ O, (12)



Fig. 6. Simplified model of an alignment problem in fiber optics. Real applications are morecomplicated.

where P (position) and O (orientation) are known parameters and α0, β0,and γ0 are given. A more general system uses constraints in form ofFk(x, y , z, α, β, γ ) ∈ Mk for k = 1, . . . , N where Mk are manifolds in R6. Incase of (12), the two separate position and orientation spaces can (after nor-malization) be treated separately and an induced metric can be constructed.Homogeneous sets in R6 are mapped onto sets with similar properties in thesearch space. Topologically, the resulting search space is the product of twoballs. In a separate article, we will investigate low-discrepancy sets in the Liegroup SO(3) of all rotations in R3. This Lie group is of fundamental importancein robotics and 3-dimensional image processing.

7.4 Networks

Tenenbaum et al. [2000] and Roweis and Saul [2000] report methods of con-structing manifolds representing large data sets. The algorithms describe dataglobally where local properties are used. The procedures are based on near-est neighbors. Included is the determination of the dimensionality of the dataunder consideration.

Given a large set of discrete points in a manifold. Assuming, a well-distributed subset of points of this graph has to be determined. Such a setshould treat all parts of the point set equally well. The following algorithm canbe applied.

Algorithm IV:

(IV.1) Apply Theorem 2 to generate a low-discrepancy set in this manifold.(IV.2) Determine nodes and edges of the graph that minimize the distance to this set of

the manifold. To this end, appropriate thresholds must be given.



If the number of points of the subset under construction is unknown in ad-vance the algorithm can stop whenever a certain goal is achieved, for example,any point of the underlying point set is in a neighborhood of a point constructedso far.

7.5 Probability Distributions

Sometimes, one is interested in a well-chosen point set where an underlyingprobability distribution is known. For example, find a point distribution in[0, 1]2 where the probability distribution p(u, v) is known. The connectionbetween this problem and an abstract surface is given by

ds2 = du2 + p(u, v)2 dv2.

Using Theorem 1, appropriately chosen sets of points can be derived fromequally distributed sequences in [0, 1]2. The case of probability distributions inhigher dimensional spaces or even manifolds can be treated analogously.

ACKNOWLEDGMENTS

The authors wish to thank Prof. Harald Niederreiter for reading the manuscriptand for providing feedback. We also wish to thank Prof. John Rice for numeroussuggestions and improvements regarding an earlier version of this article.

REFERENCES

CORPUT, J. G. V. D. 1935. Nederl. Akad. Wetensch. Proc. Ser. B 38 813, 1058.GRAY, A. 1998. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC

Press.HALTON, J. H. 1960. On the efficiency of certain quasi-random sequences of points in evaluating

multi-dimensional integrals. Numer. Math. 2, 84–90.KOCIS, L. AND WHITEN, W. J. 1997. Computational investigations of low-discrepancy sequences.

ACM Trans. Math. Softw. 23, 2, 266–294.NIEDERREITER, H. 1992. Random number generation and quasi-Monte Carlo methods. In CBMS-

NSF Regional Conference Series in Applied Mathematics, No. 63, SIAM.RICHTMYER, R. D. 1951. The evaluation of definite integrals and quasi-Monte Carlo method based

on the properties of algebraic numbers. Report LA-1342, Los Alamos Scientific Laboratory, LosAlamos, NM.

RICHTMYER, R.D. 1958. A non-random sampling method, based on congruences for Monte Carloproblems. Report NYO-867, New York, Institute of Mathematical Sciences, New York University.

ROWEIS, S. T. AND SAUL, L. K. 2000. Nonlinear dimensionality reduction by locally linear embed-ding. Science, 290, 2323–2326.

SOBOL, I. M. 1967. On the distribution of points in a cube and the approximative evaluation ofintegrals. USSR Comput. Math. Mathemat. Phys. 7, 86–112.

TENENBAUM, J. B., DE SILVA, V., AND LANGFORD, J. C. 2000. A global geometric framework for non-linear dimensionality reduction. Science, 290, 2319–2322.

WENZEL, L., RAJAGOPAL, R., AND NAIR, D. 2001. Low-discrepancy curves and efficient samplingstrategies. Manuscript.

Received May 2001; revised April 2002 and November 2002; accepted November 2002


Induced well-distributed sets in Riemannian spaces

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