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MULTIDIMENSIONAL INTEGRAL INVERSION, WITH APPLICATIONS IN SHAPE RECONSTRUCTION

SIAM J.S CI.C OMPUT.c 2005Society for Industrial and Applied Mathematics Vol.27,No.3,pp.1058–1070

MULTIDIMENSIONAL INTEGRAL INVERSION,WITH APPLICATIONS IN SHAPE RECONSTRUCTION?ANNIE CUYT?,GENE GOLUB?,PEYMAN MILANFAR§,AND BRIGITTE VERDONK?Abstract.In shape reconstruction,the celebrated Fourier slice theorem plays an essential role. It allows one to reconstruct the shape of a quite general object from the knowledge of its Radon transform[S.Helgason,The Radon Transform,Birkh¨a user Boston,Boston,1980]—in other words from the knowledge of projections of the object.In case the object is a polygon[G.H.Golub, https://www.docsj.com/doc/04542279.html,anfar,and J.Varah,SIAM https://www.docsj.com/doc/04542279.html,put.,21(1999),pp.1222–1243],or when it de?nes a quadrature domain in the complex plane[B.Gustafsson,C.He,https://www.docsj.com/doc/04542279.html,anfar,and M.Putinar,Inverse Problems,16(2000),pp.1053–1070],its shape can also be reconstructed from the knowledge of its moments.Essential tools in the solution of the latter inverse problem are quadrature rules and formal orthogonal polynomials.

In this paper we show how shape reconstruction from the knowledge of moments can also be realized in the case of general compact objects,not only in two but also in higher dimensions.To this end we use a less-known homogeneous Pad′e slice property.Again integral transforms—in our case the multivariate Stieltjes transform and univariate Markov transform—formal orthogonal polynomials in the form of Pad′e denominators,and multidimensional integration formulas or cubature rules play an essential role.

We emphasize that the new technique is applicable in all higher dimensions and illustrate it through the reconstruction of several two-and three-dimensional objects.

Key words.shape,multidimensional,inverse problem,moment problem

AMS subject classi?cations.65D32,65F20,41A21,44A60

DOI.10.1137/030601703

1.Problem statement.The problem of reconstructing a function and/or its domain given its moments is encountered in many areas.Several applications from diverse areas such as probability and statistics[10],signal processing[18],computed tomography[16,17],and inverse potential theory[4,19](magnetic and gravitational anomaly detection)can be cited,to name just a few.We can expound on some of these applications in a bit more detail.Consider the following diverse set of examples:?A region of the plane can be regarded as the domain of a probability density function.In this case,the problem is that of reconstructing this density func-tion and/or approximating its domain from measurements of its moments[10].

?Tomographic(line integral)measurements of a body can be converted into moments from which an approximation to its density and boundary can be extracted[17].

?Measurements of exterior gravitational?eld induced by a body of uniform mass can be turned into moment measurement,from which the shape of the region may be reconstructed[19].

?Measurements of an exterior magnetic?eld induced by a body of uniform

?Received by the editors November5,2003;accepted for publication(in revised form)March29, 2005;published electronically December30,2005.

https://www.docsj.com/doc/04542279.html,/journals/sisc/27-3/60170.html

?Department of Mathematics and Computer Science,University of Antwerp,Middelheimlaan1, Antwerp,B2020,Belgium(annie.cuytua.ac.be,brigitte.verdonk@ua.ac.be).

?Department of Computer Science,Stanford University,Stanford,CA94305(golub@sccm. https://www.docsj.com/doc/04542279.html,).

§Electrical Engineering Department,University of California at Santa Cruz,Santa Cruz,CA 95064(milanfar@https://www.docsj.com/doc/04542279.html,).

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INTEGRAL INVERSION FOR SHAPE RECONSTRUCTION1059 magnetization can yield measurement of the moments of the region from which the shape of the region may be determined[19].

?Measurements of thermal radiation made outside a uniformly hot region can yield moment information,which can subsequently be inverted to give the shape of the region[19].

In fact,aside from the general case where the density inside the body may not be uniform,the set of inverse problems for uniform density regions related to general elliptical equations can all be cast as moment problems which fall within the scope of application of the results of this paper.

Although the reconstruction of a shape from its Radon transform is well under-stood,the reconstruction of a shape from its moments is a problem that has only partially been solved.For instance,when the object is a polygon[11],or when it de?nes a quadrature domain in the complex plane[12],it has been proved that its shape can be exactly reconstructed from the knowledge of its moments.Both results deal with particular two-dimensional shapes.For general n-dimensional shapes no inversion algorithm departing from the moments is known.In order to explain the type of result we are looking for,we brie?y repeat the inversion formula based on a shape’s projections provided by the Radon transform.

The Radon transform R

ξ(f)of a square-integrable n-variate function f( x)with

x=(x1,...,x n)is de?ned as(for ease of notation we drop the dependence of f in the notation)

ξ(u)=

R n

f( x)δ( ξ x?u)d x

with|| ξ||=1and ξ· x=u an(n?1)-dimensional manifold orthogonal to ξ.When n=2, ξis fully determined by an angleθand is given by

Rθ(u)=

+∞

?∞ +∞

?∞

f(t,s)δ(t cosθ+s sinθ?u)dt ds.

For n=3, ξis determined by anglesθandφand

Rθ,φ(u)=

f(t,s,v)δ(t cosφcosθ+s cosφsinθ+v sinφ?u)dt ds dv.

Making use of the celebrated Fourier slice theorem,one obtains,for instance,that the one-dimensional Fourier transform of Rθ(u),

F1(Rθ)(z)= +∞

?∞

Rθ(u)exp(?2πi zu)du,

equals the two-dimensional Fourier transform of the function f restricted to the straight line(z cosθ,z sinθ):

F2(f)(z cosθ,z sinθ)= +∞

?∞

+∞

?∞

f(t,s)exp(?2πi z(t cosθ+s sinθ))dt ds

(1)

=F1(Rθ)(z).

When f(t,s)is the characteristic function of a compact set A in the complex plane, then(1)allows one to reconstruct A,departing from the Radon transform Rθ(u),by

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taking the inverse two-dimensional Fourier transform of F1(Rθ).In higher dimensions the procedure is completely analogous[14].

Our aim is to establish a similar type of relationship,making use of moment information instead of projections.To this end we need to introduce a few tools.

2.Univariate Markov transform and Pad′e approximant.A Markov func-tion g(z)is de?ned to be a function with an integral representation of the form

g(z)=

f(u)

1+zu

du,?∞

(2)

where f(u)is nontrivial and positive and the moments

c i=

a u i f(u)du,i=0,1,...,

(3)

are?nite.The function g is called the Markov transform of f and is also denoted by g=M1(f).A Markov series is de?ned to be a series of the form

∞

i=0(?1)i c i z i

(4)

which is derived by a formal expansion of(2).It is well known that the one-dimensional Markov moment problem is determinate.

Given a series of the form(4),one can construct Pad′e approximants of this series as follows.With the moments c i introduced in(3),one computes coe?cients a0,...,a m+k and b0,...,b m such that for

p m+k,m(z)=m+k

i=0

a i z i,

q m+k,m(z)=

i=0

b i z i,

the series expansion of(gq m+k,m?p m+k,m)(z)satis?es ∞

i=0d i z i=

∞

i=0

(?1)i c i z i

q m+k,m(z)?p m+k,m(z)=O(z2m+k+1).

(5)

In other words,the2m+k+2coe?cients a0,...,a m+k and b0,...,b m are determined from the2m+k+1conditions d0=0,...,d2m+k=0and an additional normalization condition.The irreducible form of p m+k,m(z)/q m+k,m(z)is denoted by r m+k,m(z)and is called the(m+k,m)Pad′e approximant.It is usually normalized by putting the constant term in the denominator equal to1.

The following property plays a crucial role in our novel shape reconstruction technique.

Theorem1(see[1,p.228]).For the Markov function(2),each sequence of Pad′e approximants{r m+k,m(z)}m∈N with k≥?1converges to(2)for z∈]?∞,?1/b]∪[?1/a,+∞[.The rate of convergence is governed by

lim sup m→∞|g(z)?r m+k,m(z)|1/m≤

1/z+b?

1/z+a

1/z+b+

1/z+a

INTEGRAL INVERSION FOR SHAPE RECONSTRUCTION1061

3.Multivariate Stieltjes transform and homogeneous Pad′e approxi-mant.Pad′e approximants have been generalized to higher dimensions by several authors in di?erent ways.For an overview and comparison of these de?nitions the reader is referred to[8].For our purpose the de?nition given in[7,6]is most useful. Without loss of generality,we repeat it only for bivariate functions,but it can be de?ned in any number of variables.

A bivariate Stieltjes function g(v,w)is de?ned by the integral representation

g(v,w)=

∞

0 ∞

f(t,s)

1+(vt+ws)

dt ds

(6)

with?nite real-valued moments

c ij=

∞

0 ∞

t i s j f(t,s)dt ds.

A formal expansion of(6)provides a bivariate Stieltjes series

∞ i,j=0

i+j

(?1)i+j c ij v i w j.

(7)

The function g is also called the bivariate Stieltjes transform of f and is denoted by g=S2(f).

Given the moments c ij,one can compute the(m+k,m)homogeneous bivariate Pad′e approximant of(7)as follows.First we introduce the homogeneous expressions

A (v,w)=

i+j=

a ij v i w j,

B (v,w)=

i+j=

b ij v i w j

to de?ne the polynomials

p m+k,m(v,w)=(m+k)(m+1)

=(m+k)m

A (v,w),

q m+k,m(v,w)=(m+k+1)m

=(m+k)m

B (v,w).

Second we write down the homogeneous accuracy-through-order conditions

C (v,w)=

i+j=

c ij v i w j,

(8)

∞

i,j=0d ij v i w j=

∞

(?1) C (v,w)

q m+k,m(v,w)?p m+k,m(v,w) =O

v i w j,i+j≥(m+k+2)m+k+1

It has been shown[7,pp.60–61]that a nontrivial solution of these conditions can always be computed.Moreover,all solutions p m+k,m(v,w)/q m+k,m(v,w)deliver the

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same unique irreducible form r m+k,m(v,w),which is called the homogeneous Pad′e approximant of(7).A proper normalization of r m+k,m(v,w)can still be chosen but di?ers most of the time from the univariate normalization q m+k,m(0)=1since the denominator of r m+k,m(v,w)need not start with a constant term.It starts with a homogeneous expression in v and w of as low degree as possible.This homogeneous generalization of the Pad′e approximant is the only one to satisfy the following powerful slice theorem.Although it was pointed out soon after the introduction of homogeneous Pad′e approximants,its full impact was only understood recently[9,3].For the sake of the reader we also repeat the short proof.

Let us de?ne the slice function:

gθ(z)=g(z cosθ,z sinθ),?π/2<θ≤π/2,

(9)

∞

0 ∞

f(t,s)

1+(t cosθ+s sinθ)z

dt ds.

We denote the univariate(m+k,m)Pad′e approximant of gθ(z)as de?ned in(5)by

r(gθ)

m+k,m (z).

Theorem2(see[15,5]).The homogeneous Pad′e approximant r m+k,m(v,w)of g(v,w)satis?es

r m+k,m(z cosθ,z sinθ)=r(gθ)

m+k,m

(z),?π/2<θ≤π/2.

Proof.The univariate rational function of the variable z,parameterized byθ,

r m+k,m(z cosθ,z sinθ)= (m+k)(m+1)

=(m+k)m

i+j=

a ij cos iθsin jθ

z (m+k+1)m

=(m+k)m

i+j=

b ij cos iθsin jθ

= m+k

i+j=(m+k)m+

a ij cos iθsin jθ

z m

i+j=(m+k)m+

b ij cos iθsin jθ

=:p(θ)

m+k,m

(z)

q(θ)

m+k,m

(z)

satis?es

∞

=0(?1) C (cosθ,sinθ)z

q(θ)

m+k,m

(z)?p(θ)

m+k,m

(z)

=gθ(z)q(θ)

m+k,m (z)?p(θ)

m+k,m

(z)=O(z2m+k+1)

with d2m+k+1in(5)being a homogeneous expression in cosθand sinθ.Because of the unicity of the Pad′e approximant,r(gθ)

m+k,m

(z)must equal the irreducible form of

p(θ)

m+k,m

(z)

m+k,m (z)

(10)

which completes the proof.

In other words,restricting the homogeneous Pad′e approximant to the slice

Sθ={(z cosθ,z sinθ)|z∈R}

(11)

INTEGRAL INVERSION FOR SHAPE RECONSTRUCTION1063 is equivalent to computing the univariate Pad′e approximant of the slice function gθ(z). After analyzing the behavior of the homogeneous Pad′e approximant on the slices Sθ, let us have another look at the slice function gθ(z)itself.

Let the square-integrable function f(t,s)be de?ned in a compact region A of the?rst quadrant t≥0,s≥0of the plane.According to a fundamental property of the Radon transform Rθ(u)of f(t,s)[14],the following relation holds for any square-integrable function F(u):

+∞

?∞Rθ(u)F(u)du=

∞

f(t,s)F(t cosθ+s sinθ)dt ds.

(12)

If we take F(u)=1/(1+zu),then

+∞?∞Rθ(u)

1+zu

du=

∞

f(t,s)

1+(t cosθ+s sinθ)z

dt ds=gθ(z).

(13)

Consequently,if f(t,s)is zero outside a compact subset A of the?rst quadrant,then gθ(z)is a Markov function,because Rθ(u)is zero outside a compact support.In addition Theorem1applies.

4.Connection and new results.Making use of the homogeneous Pad′e slice property and the fact that the slice function gθ(z)is a Markov function with f in(2) equal to the Radon transform of f(t,s),it is now easy to obtain the following result for the bivariate Stieltjes transform g(v,w)de?ned by(6).

Theorem 3.Let the function f(t,s)in the Stieltjes transform(6)be square-integrable and zero outside a compact support A?{(t,s):0≤t2+s2≤1}∩

{(t,s):t≥0,s≥0}.Let the slice Sθbe de?ned by(11).Then for all k≥?1 and each?π/2<θ≤π/2,the sequence{r m+k,m(z cosθ,z sinθ)}m∈N converges to g(z cosθ,z sinθ)for|z|<1given by(6).The rate of convergence is governed by

lim sup m→∞|g(z cosθ,z sinθ)?r m+k,m(z cosθ,z sinθ)|1/m≤

1/z+1?

1/z?1

1/z+1+

1/z?1

(14)

Proof.Since f(t,s)is zero outside A and by means of the celebrated Fourier slice formula(12),we know that

S2(f)(z cosθ,z sinθ)=g(z cosθ,z sinθ)=

f(t,s)

1+(t cosθ+s sinθ)z

dt ds

b(θ)

a(θ)Rθ(u)

1+zu

du=M1(Rθ)(z)

with?1≤a(θ)≤b(θ)≤https://www.docsj.com/doc/04542279.html,bining Theorems1and2delivers

b(θ) a(θ)Rθ(u)

1+zu

du=lim

m→∞

r(gθ)

m+k,m

(z)=lim

m→∞

r m+k,m(z cosθ,z sinθ)

for?1/b(θ)

When the compact support A intersects all four quadrants,the theorem still holds.The current formulation is just one way of scaling the problem,without loss of generality.

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To the authors’knowledge,at the time of this writing,no inverse Stieltjes trans-form exists for the transform de?ned by (6).Hence the function f (t,s )must be solved for from the relationship S 2(f )(z cos θ,z sin θ)=M 1(R θ)(z )=lim m →∞

r m +k,m (z cos θ,z sin θ),

(15)

which is an identity of the same type as (1).Note that the switch from Cartesian to polar coordinates is only required to make the transition from the Stieltjes transform to the Markov and Radon transform.More generally,

S 2(f )(v,w )=lim m →∞

r m +k,m (v,w ),

where (v,w )=(z cos θ,z sin θ)with ?π/2<θ≤π/2.In the new reconstruction algorithm,this Markov and Radon transform shall not be computed explicitly.We only want to make use of moment information.We summarize the computations described so far and complete the reconstruction of f (t,s )by solving (15).

5.Shape reconstruction algorithm.From the preceding theory we now sum-marize the bivariate reconstruction algorithm,with the higher-dimensional case being completely similar.Given the moments

c ij = ∞0

∞

f (t,s )t i s j dt ds

(16)we compute,for some k ≥?1and consecutive m ,the homogeneous bivariate Pad′e

approximant r m +k,m (v,w )as a function of the Cartesian coordinates v and w .A fast algorithm for the computation of the multivariate Pad′e approximant is given in [2].The computation of r m +k,m (v,w )requires knowledge of the moments c ij appearing in the expressions C (v,w )given in (8)for =0,...,2m +k ,or in other words the ?rst (2m +k +1)(2m +k +2)/2moments c ij .

In the reconstructions,shape often means compact set A in R 2(or R 3)and then f (t,s )equals the characteristic function δA .In that case

c ij =

t i s j dt ds.Theorem 3guarantees that the approximants r m +k,m (v,w )converge rapidly on

each slice S θto g (v,w )restricted to that slice.A typical value for m is between 3and 7,and hence one needs with k =?1on average between 21and 105moments.Moreover,the moments are not required very accurately.For a rough estimate of a shape as in Figure 2,moments with 2to 3signi?cant digits (a relative error bounded by 5×10?2or 5×10?3)are su?cient.

The Pad′e approximant is then evaluated in a discrete number of points (v j ,w j )inside the unit disc to approximate

g (v j ,w j )≈r m +k,m (v j ,w j ).

The latter constitutes the right-hand side of (17).To speed up the convergence of the Pad′e approximants r m +k,m the points (v j ,w j )can be taken in a disc of radius r <1:Pad′e approximants converge more rapidly in the neighborhood of the origin.At the same time,for each point (v j ,w j )the value g (v j ,w j )can be approximated to high accuracy by a cubature formula,

L i =1

ωi

1+t i v j +s i w j

f (t i ,s i ),

j =0,1,...,

INTEGRAL INVERSION FOR SHAPE RECONSTRUCTION1065 with weightsωi and nodes(t i,s i).The inverse problem of computing f(t i,s i)from

L i=1

ωi

1+t i v j+s i w j

f(t i,s i)≈g(v j,w j)=lim

m→∞

r m+k,m(v j,w j)

(17)

is a(structured)system of linear equations.When applying a quadrature method directly to(16),it is clear that a few dozen moments are not su?cient to retrieve the value of f(t,s)with the accuracy shown in the illustrations below(h=k=2?5gives a resolution of about3200pixels in the unit disk and h=k=2?6a resolution of almost 13,000pixels)!The current technique allows one to write down as many equations as required,by adding evaluations of the Pad′e approximant in points(v j,w j),without increasing the number of required moments c ij.

The linear problem(17)is in general ill-conditioned,and therefore a regulariza-tion technique must be applied.In all of the following examples we have found the technique known as truncated SVD[13]to do an excellent job.After regularization, we solve(17)for f(t i,s i)and identify an approximation for the shape A with

A≈{(t i,s i)|f(t i,s i)≥0.5}.

The threshold0.5is chosen because for the original shape f(t,s)=1inside A and f(t,s)=0outside A.

Since the homogeneous Pad′e approximant can be de?ned analogously in higher dimensions,the procedure for three-dimensional shape reconstruction is entirely sim-ilar.In this case,we are given the moments

c ijk=

∞

0 ∞

∞

f(t,s,v)t i s j v k dt ds dv.

(18)

Here Theorems2and3hold on the slices

Sθ,φ={(z cosφcosθ,z cosφsinθ,z sinφ)|z∈R}, and the three-dimensional version of(13)is

+∞?∞Rθ,φ(u)

1+zu

du=

∞

f(t,s,v)

1+(t cosφcosθ+s cosφsinθ+v sinφ)z

dt ds dv

=lim

m→∞

r m+k,m(z cosφcosθ,z cosφsinθ,z sinφ).

(19)

The homogeneous Pad′e approximant r m+k,m(w,x,y)of the Stieltjes transform

∞0 ∞

∞

f(t,s,v)

1+(tw+sx+vy)

dt ds dv

(20)

is now constructed from the trivariate Stieltjes series representation

∞

i,j,k=0(?1)i+j+k

i+j

i+j+k

i+j

c ijk w i x j y k

for(20).Note that the vector(cosφcosθ,cosφsinθ,sinφ)generating the one-dimen-sional slice Sθ,φbelongs to the three-dimensional unit sphere.

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6.Numerical examples.For each example we give the cubature parameters h and k used in formula (21)or h,k ,and used in (22)and the distance Δbetween the coordinates of the points (v j ,w j )used in the formulation of (17).For these evaluation points we construct a uniform grid (the distance Δin the x -and y -directions is kept equal)that we intersect with the disc of radius r .All points in the intersection are selected.We also list the denominator degree m of the Pad′e approximant (the numerator degree is m ?1),the radius r of the disc in which the points (v j ,w j )are chosen,and the relative accuracy of the moments (here 2?4,2?11,2?14,and 2?24stand for,respectively,2,4,5,and 8signi?cant digits).Even when r <1,the shape A is allowed to cover the entire unit disk.Note that the resolution of the reconstruction is expressed by h and k or h,k ,and ,not by Δ.

6.1.Reconstruction of two-dimensional shapes.We take for f (t,s )in The-orem 3the characteristic function of a compact set A contained in the unit disc and use the simple compound 4-point Gauss-Legendre product rule [20,pp.230–231]for the approximation of g (v j ,w j )in (17):

a +h

b +k

f (t,s )1+tv j +sw j dt ds ≈hk 44

i =1f (t i ,s i )1+t i v j +s i w j

(t 1,s 1)=

a +

3?√3

6h,b +3?√36k ,(t 2,s 2)= a +3?√3

6h,b +3+√

36k

(21)

(t 3,s 3)=

a +

3+√3

6h,b +3?√

36k ,(t 4,s 4)=

a +3+√3

6h,b +3+√

k .

We reconstruct several shapes,such as a simple convex shape like the ellipse in Figures

1and 2,the more di?cult nonconvex lemniscates in Figures 3and 4,and the bone-like Figure 5containing a hole.In each of the illustrations we delimit the original shape in black and show the reconstructed area in gray.The black contour is given only for comparison.Note that the shape’s boundary is unknown in real-life situations where only the shape’s moments are known up to some order and accuracy.For the reconstruction of the two-dimensional shapes we choose h and k in (21)equal to h =k =2?5.

?0.8?0.6?0.4?0.2

00.20.40.60.81

?1?0.8?0.6?0.4?0.200.20.40.60.810

0.10.20.30.40.50.60.70.80.91?1

?0.8?0.6?0.4?0.2

00.20.40.60.81

?1?0.8?0.6?0.4?0.200.20.40.60.810

0.10.20.30.40.50.60.70.80.91.5)Fig. 1.m =7,r =1.00, =2?24,Δ=2?4.Fig. 2.

m =5,r =0.15, =2?4,Δ=2?6.

INTEGRAL INVERSION FOR SHAPE RECONSTRUCTION

1067

?0.8?0.6

?0.4?0.2

00.20.40.60.81

?1?0.8?0.6?0.4?0.200.20.40.60.810

0.10.20.30.40.50.60.70.80.91?1

?0.8?0.6?0.4?0.2

00.20.40.60.81

?1?0.8?0.6?0.4?0.200.20.40.60.810

0.10.20.30.40.50.60.70.80.91A ={(t,s )| (t ?t 0)+(s ?s 0)2+α (t ?0.1)=βFig. 3.

t 0=s 0=0.02,α=β=0.5,m =9,r =0.12, =2?11,Δ=2?6.

Fig. 4.

t 0=s 0=?0.05,α=0.62,β=0.65,m =9,r =0.30, =2?14,Δ=2?5.

?0.8?0.6?0.4?0.2

00.20.40.60.81

?1?0.8?0.6?0.4?0.200.20.40.60.810

0.10.20.30.40.50.60.70.80.91A ={(t,s )| t 2+s 2+0.4225 =0.2401}\{(t,s )||t |<0.8,|s |<0.1}

Fig. 5.m =9,r =0.50, =2?24,Δ=2?4.

Finally,a di?cult two-dimensional shape is presented in Figure 6.Here,for a change,h =k =2?4and the evaluation points (v j ,w j )are placed on a radial grid,while the moments are ?1

?0.5

00.5

00.1

0.2

0.3

0.40.50.60.7

0.8

0.9

A ={(t,s )sin(τ)/3}

Fig. 6.m =10,r =0.5,Δ≈1/12.

6.2.Reconstruction of particular three-dimensional shapes.Since The-orem 3also applies to more general functions f (t,s )than characteristic functions,particular three-dimensional shapes as in Figure 7,namely,with one ?at side,can also be reconstructed by means of the two-dimensional integral inversion technique.Here the positive function f (t,s )de?nes the top surface of the three-dimensional shape while the bottom surface is the domain of f in the (t,s )-plane.Further,the object is cylindrical.The reconstruction of the crater-like object is obtained by plotting the reconstruction of f (t,s )as a function of t and s .The plot is overlaid with a mesh depicting the exact function f (t,s ).

In this example we have also increased the sizes of h and k to 2?4,while keeping the number of points (v j ,w j )rather moderate.

1068 A.CUYT,G.GOLUB,https://www.docsj.com/doc/04542279.html,ANFAR,AND B.

VERDONK

Three-dimensional object with surface f(t,s)=t2+s2for t2+s2≤0.25.

Fig.7.m=8,r=0.5, =2?11,Δ=2?4.

6.3.Reconstruction of three-dimensional shapes.A more general three-dimensional shape,such as the ball in Figure8,can be reconstructed through the solu-tion of the three-dimensional analogue of(17)for its characteristic function f(t,s,v). Here we use the compound8-point Gauss–Legendre product cubature formula given in[20,pp.230–231]:

a+h a b+k

f(t,s,v)

1+tw+sx+vy

dt ds dv≈

i=1

f(t i,s i,v i)

1+t i w j+s i x j+v i y j

, (t i,s i,v i)=

3±

√

h,b+

3±

√

k,c+

3±

√

(22)

Reconstruction of the ball t2+s2+v2≤0.49.

Fig.8.m=6,h=k= =2?2,Δ=2?4.

7.Conclusion.The new technique is able to deal with very general shapes: nonconvex such as in Figures3and4,shapes with nonconnected boundary such as in Figure5,and last but not least higher-dimensional objects such as in Figures7 and8.In Figure7we illustrate that the technique can also be used for more general functions f(t,s)than characteristic functions.

By carrying out a lot of numerical experiments,we have come to the conclusion that the numerical quality of the output delivered by an implementation of the math-ematical property formulated in Theorem3seems to depend on a number of elements. Let us formulate a conclusion and some numerical advice.

When the accuracy in the moment information decreases,then it is recommended to reduce the radius r of the ball from which the points(v j,w j)(in two dimensions) or(w j,x j,y j)(in three dimensions)are selected.This shrinking of the region for the evaluation points improves the quality of the Pad′e approximant.We refer the reader to Figures1and2for a clear illustration.With moments known up to7

INTEGRAL INVERSION FOR SHAPE RECONSTRUCTION

1069

signi?cant digits,a perfect reconstruction is possible with r =1,with a moderate number of moments.With moments known only up to 2signi?cant digits,a good quality reconstruction is possible when selecting the (v j ,w j )inside a disc with far smaller radius.Fortunately,the lack of quality of the moments does not have to be compensated for by their quantity.Note that in Figure 2even fewer moments are used than in Figure 1:m =5versus m =7.

The quality of the reconstruction improves when the shape’s center of gravity is positioned near the origin.To illustrate this we increase the values of t 0and s 0in Figure 3to t 0=s 0=0.1,without altering the other parameters.The reconstruction technique then delivers Figure 9.The same observation is made in [11],the reason for this being that the conditioning of the problem improves.

?0.8?0.6?0.4?0.2

00.20.40.60.81

?1?0.8?0.6?0.4?0.200.20.40.60.810

0.10.20.30.40.50.60.70.80.91Fig.9.m =9,r =0.12, =2,Δ=2?6.

Increasing the degree of the Pad′e denominator has little or no in?uence on the reconstruction.This is due to the fact that the Pad′e approximants converge quite rapidly.Their relative error drops quickly below the relative error on the moments.As a rule of thumb one can bound the relative error of the Pad′e approximant r m ?1,m in the proposed reconstruction technique by 10?m ?1.

REFERENCES

[1]G.A.Baker,Jr.,and P.Graves-Morris ,Pad′e approximants ,2nd ed.,Cambridge University

Press,Cambridge,UK,1996.

[2]S.Becuwe and A.Cuyt ,On the Fast Solution of Toeplitz-Block Linear Systems Arising in

Multivariate Approximation Theory ,Technical report,University of Antwerp,2002.

[3] B.Benouahmane and A.Cuyt ,Properties of multivariate homogeneous orthogonal polyno-mials ,J.Approx.Theory,113(2001),pp.1–20.

[4]M.Brodsky and E.Panakhov ,Concerning a priori estimates of the solution of the inverse

logarithmic potential problem ,Inverse Problems,6(1990),pp.321–330.

[5] C.Chaffy ,Interpolation polynomiale et rationnelle d’une fonction de plusieurs variables com-plexes ,Th`e se,Institut Polytechnique Grenoble,Grenoble,France,1984.

[6] A.A.M.Cuyt ,A comparison of some multivariate Pad′e -approximants ,SIAM J.Math.Anal.,

14(1983),pp.195–202.

[7] A.Cuyt ,Pad′e Approximants for Operators:Theory and Applications ,Lecture Notes in

Math.1065,Springer-Verlag,Berlin,1984.

[8] A.Cuyt ,How well can the concept of Pad′e approximant be generalized to the multivariate

case?,https://www.docsj.com/doc/04542279.html,put.Appl.Math.,105(1999),pp.25–50.

[9] A.Cuyt and D.Lubinsky ,A de Montessus theorem for multivariate homogeneous Pad′e

approximants ,Ann.Numer.Math.,4(1997),pp.217–228.

[10]P.Diaconis ,Application of the method of moments in probability and statistics ,in Moments in

Mathematics (San Antonio,TX,1987),Proc.Sympos.Appl.Math.37,AMS,Providence RI,1987,pp.125–142.

[11]G.H.Golub,https://www.docsj.com/doc/04542279.html,anfar,and J.Varah ,A stable numerical method for inverting shape

from moments ,SIAM https://www.docsj.com/doc/04542279.html,put.,21(1999),pp.1222–1243.

1070 A.CUYT,G.GOLUB,https://www.docsj.com/doc/04542279.html,ANFAR,AND B.VERDONK

[12] B.Gustafsson,C.He,https://www.docsj.com/doc/04542279.html,anfar,and M.Putinar,Reconstructing planar domains from

their moments,Inverse Problems,16(2000),pp.1053–1070.

[13]P.C.Hansen,The truncated SVD as a method for regularization,BIT,27(1987),pp.543–553.

[14]S.Helgason,The Radon Transform,Birkh¨a user Boston,Boston,1980.

[15]J.Karlsson,private communication,1981.

[16]https://www.docsj.com/doc/04542279.html,anfar,W.C.Karl,and A.S.Willsky,A moment-based variational approach to

tomographic reconstruction,IEEE Trans.Image Proc.,5(1996),pp.459–470.

[17]https://www.docsj.com/doc/04542279.html,anfar,G.C.Verghese,W.Karl,and A.S.Willsky,Reconstructing polygons from

moments with connections to array processing,IEEE Trans.Signal Process.,43(1995), pp.432–443.

[18]M.I.Sezan and H.Stark,Incorporation of a priori moment information into signal recovery

and synthesis problems,J.Math.Anal.Appl.,122(1987),pp.172–186.

[19]V.N.Strakhov and M.A.Brodsky,On the uniqueness of the inverse logarithmic potential

problem,SIAM J.Appl.Math.,46(1986),pp.324–344.

[20] A.H.Stroud,Approximate Calculation of Multiple Integrals,Prentice–Hall,Englewood Cli?s,

NJ,1971.

Inversion(倒置法)

Inversion（倒置法） Inversion of W ord Order in Translating W ord-Groups or Phrases（词组或短语的词序倒置） Examples & Drills8.1 Give Chinese equivalents for the following, paying attention to the word order: 1. Four cardinal points: north, south, east, and west. 2. northeast, northwest, southeast, southwest; North and South Poles 3. card-playing, letter-writing, white-washing, absent-minded, open-door policy, time-sharing system, error correcting code 4. contradictions between ourselves and the enemy 5. men and women, young and old; sisters and brothers (cf. boys and girls); dear mother and father 6. the iron and steel industry 7. sooner or later; quick of eye and deft of hand 8. heal the wounded and rescue the dying; life or death; now sink, now emerge 9. the soil and water mixture 10. you, he and I; Lisa, Linda and I; my wife (husband) and I 11. rich and poor 12. food, clothing and housing 13. negative and positive 14. hot and cold (water); hard and soft; old and new 15. back and forth; to and fro 16. by twos and threes 17. right and left 18. loss and gain 19. land and water; fire and water; as well blended as milk and water 20. track and field (events) 21. model worker The coordinating words of an English word-group or phrase are usually arranged according to their shades o f meaning: “each is more impressive than the preceding” (Fowler 1965:92). Chinese word order is often the opposite. Thus a common rule for translation is: English Chinese narrower range →wider range wider range →narrower range less important →more important more important →less important weaker →stronger stronger →weaker Examples: 1. 无地和少地的农民 peasants who had little or no land 2. 丰收年多积累一点，灾荒年或半灾荒年就不积累或少积累一点。

分词作定语解析与练习

分词作定语一．分词的位置 1. 分词前置 We can see the rising sun. 我们可以看到东升的旭日 He is a retired worker.他是位退休的工人 2. 分词后置(i.分词词组；ii. 个别分词如given, left；iii. 修饰不定代词something等) There was a girl sitting there.有个女孩坐在那里 This is the question given.这是所给的问题 There is nothing interesting.没有有趣的东西二．分词的类别 1.过去分词，即动词的-ed形式 2.现在分词，即动词的-ing形式两者的区别： 1. 现在分词表示正在进行的动作，而过去分词表示已完成的动作。 eg：falling leaves fallen leaves developing country developed country 2. 现在分词有主动的含义，过去分词有被动的含义。 eg：I heard someone opening the door. I heard the door opened. 3.现在分词表示它所修饰的名词的性质和特征，过去分词表示它所修饰的名词的状态。 eg：an exciting news an excited boy bored students boring lecture 练习： 1) The first textbook ___ for teaching English as a foreign language came out in the 16th century. A. have written B. to be written C. being written D. written 2）What's the language ____ in Germany? A. speaking B. spoken C. be spoken D. to speak 3）I could say nothing, and___ tears come out to my eyes. A. surprising B. surprised C. exciting D. excited 4)We were ___ to have seen the ____ leader. A. inspired; inspiring B. inspiring; inspiring C. inspired; inspired D. inspiring; inspired 5)Don’t worry, it’s safe to skating on the ___ lake. A. freezing B. frozen C. freeze D. having frozen 答案：DBDAB

Inversion

Inversion 主语和谓语是句子中最主要的成分，它们的语序有两种： Natural Order ：S+V Inverted Order ：V+S： Full Inversion Partial Inversion 完全倒装（Full Inversion）:又称"全部倒装",是指将句子中的谓语动词全部置于主语之前。此结构通常只用与一般现在时和一般过去时。部分倒装（Partial Inversion）:指将谓语的一部分如助动词或情态动词倒装至主语之前,而谓语动词无变化。如果句中的谓语没有助动词或情态动词，则需添加助动词do, does 或did，并将其置于主语之前。英语句子的倒装一是由于语法结构的需要而进行的倒装,二是由于修辞的需要而进行的倒装。前一种情况,倒装是必须的,否则就会出现语法错误;后一种情况,倒装是选择性的,倒装与否只会产生表达效果上的差异。 Ⅰ.Full Inversion 1. here, there, now, then, thus等副词置于句首, 谓语动词常用be, come, go, lie, run。 There goes the bell. Then came the chairman. Here is your letter. 2. 表示运动方向的副词或地点状语置于句首，谓语表示运动的动词。 Out rushed a missile from under the bomber. Ahead sat an old woman. 注意：上述全部倒装的句型结构的主语必须是名词，如果主语是人称代词则不能完全倒装。 Here he comes. Away they went. 3. 状语或表语位于句首时的倒装为了保持句子平衡或使上下文衔接紧密，有时可将状语或表语置于句首，句中主语和谓语完全倒装： Among these people was his friend Jim. 他的朋友吉姆就在这些人当中。 By the window sat a young man with a magazine in his hand. 窗户边坐着一个年轻人，手里拿着一本杂志。注意：在表语置于句首的这类倒装结构中，要注意其中的谓语应与其后的主语保持一致，而不是

优质课孔雀东南飞教案

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孔雀东南飞公开课一等奖教案设计说课讲解

孔雀东南飞公开课一等奖教案设计

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