Nov 15, 1996 - Ten years ago, Deslauriers and Dubuc introduced a process for interpolating data ob- served at .... j is near 0 { and many coe cients `lower down' { where j large. ..... (a): D=2 Refinement of Kronecker Sequence and shifts. â4.
Deslauriers-Dubuc: Ten Years After D.L. Donoho and Thomas P.Y. Yu Stanford University November 15, 1996 Abstract
Ten years ago, Deslauriers and Dubuc introduced a process for interpolating data observed at the integers, producing a smooth function de ned on the real line. In this note we point out that their idea admits many fruitful generalizations including: Interpolation of other linear functionals of f (not just point values), yielding other re nement schemes and biorthogonal wavelet transforms; Interpolation of vector-valued data, yielding vector re nement schemes and multiwavelets; Interpolation of nonlinear functionals, yielding nonlinear re nement schemes and nonlinear wavelet transforms. Interpolation by other families than polynomials, yielding e.g. segmented interpolation and segmented transforms. We refer to a variety of recent work on these schemes.
1 Deslauriers-Dubuc Interpolation
Suppose we are given the point-values (k = f (k) : k 2 Z) of a function f evaluated at the integers. We wish to interpolate, getting a function f~(x) de ned for x 2 R, obeying f~(k) = k . Dubuc (1986) and Deslauriers and Dubuc (1987,1989) introduced a multiscale re nement technique for this problem. For integer L 0, let D = 2L + 1 be an odd integer > 0. They obtained such a function f~ by interpolating the data at the integers to a function de ned on the binary rationals by repeated application of a two-scale re nement transformation. If f~ has already been de ned at all binary rationals with denominator 2j , j 0, their process extends it to all binary rationals with denominator 2j+1 , i.e. all points halfway between previously de ned points. Speci cally, to de ne the function at (k + 1=2)=2j when it is already de ned at all k=2j , we t a polynomial j;k to the data (k0 =2j ; f~(k0 =2j )) for k0 2 f(k ? L)=2j ; : : : ; (k + L + 1)=2j g { this polynomial is unique { and set f~((k + 1=2)=2j ) j;k ((k + 1=2)=2j ): For D = 1 this is just linear interpolation, pure and simple. For D = 3; 5; 7; : : :, Deslauriers and Dubuc showed that this scheme de nes a function which is uniformly continuous at the rationals and hence has a unique continuous extension to the reals. They showed that this extension is actually C R for R = R(D) an increasing function of D = 2L + 1. We illustrate the basic calculation of the local polynomial j;k in Figure 1 using L = 1 and so D = 3. We illustrate the sequence of re nements in Figure 2 by showing the sequence of re nements of a Kronecker sequence (k(0) ) de ned by k(0) = 1fk=0g . The `x' markers display the given data and the `o' markers display the successive generations of interpolated data 1
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Let denote the result of re ning the Kronecker sequence; this depends on the order D de ning the interpolation scheme and so can be denoted = (D) . Figure 3 shows results for D = 1; 3; 7; 11. The function (D) is often called the fundamental function of the DD interpolation scheme. The terminology is justi ed by the fact that if DD re nement is applied to data (k )k2Z , the result of re nement is a superposition of and its shifts:
f~(x) =
X (x ? k):
k2Z
k
This of course follows from the linearity and shift invariance of the re nement process. As discussed above, the Deslauriers-Dubuc fundamental function is compactly supported and C R for R = R(D).
2 Surprises The DD re nement scheme is simple and beautiful; in fact the simplicity is deceptive. On the one hand, to understand the smoothness properties of (D) requires entry into a considerable literature on properties of solutions to re nement schemes; we do not pursue this question in this paper, and instead merely note the existence of work such as [6, 12, 1]. On the other hand, the DD re nement scheme admits of certain very interesting generalizations which we would like to describe in this paper. Moreover, the scheme is connected to the rapidly growing literature on wavelet transforms, and by this connection makes it possible to consider various generalizations of wavelet transforms; we also will describe some of these generalizations in this paper. In short, the point of this paper is to explain our belief that the DD re nement scheme is a natural point of departure for de ning a number of new re nement schemes and mathematical transforms. It is pleasure to have the opportunity to describe this belief and to thank Gilles Deslauriers, Serge Dubuc and Brenda MacGibbon for a chance to visit Montreal and describe a part of this work. 3
2.1
Connection with Wavelets
We begin by sketching the connection between DD-Interpolation and Wavelet transforms. As described above, the result of interpolating point samples (f (k))k2Z furnishes a function f~ which depends linearly on f . Hence we can de ne a linear operator P0Pby the equation f~ = P0 f . As described above, we have the explicit representation (P0 f )(x) = k f (k) (x ? k). De ne now j;k (x) = (2j x ? k), a version of the DD fundamental function that is dilated to 2?j and translated to location k=2j . Then de ne the operator Pj by (Pj f )(x) = Pkwidth f (k=2j ) j;k (x). Equivalently, Pj represents the result of DD interpolation with data given at sites in the dyadic grid 2?j Z rather than at integer sites. The key facts about the operator Pj can be expressed by two relations. First, with the DD fundamental function, Pj = (1) for all j = 0; 1; 2; : : :. Second, if is a polynomial of degreePD, Pj = for all j = 0; 1; 2; : : :. Let Vj be the range of Pj , i.e. the L1 closure of all sums k k j;k with nitely many nonzero terms. Then from (1) we have Pj+h j;k = j;k for all h > 0, and so Vj Vj+h for all h > 0. De ne now the di erence operator Qj = Pj+1 ? Pj and the di erence space Wj = span(Qj ). This operator describes the di erence between interpolating a function known at all sites in 2?j?1 Z and only at sites in 2?j Z. Now the range of Qj is in Vj+1 , so it has a representation
Qj f =
Xq
j;k (f ) j +1;k
k
where qj;k (f ) = (Pj+1 (f )(k=2j+1 ) ? Pj (f )(k=2j+1 )). j +1) . Because for nice functions, f = Now obviously Qj f is composed of terms of scale 2?(P limj!1 Pj f and we have the telescoping sum Pj = P0 + jh?=01 (Ph+1 ? Ph ), we write, formally
f = P0 f +
X Q f; j 0
(2)
j
obtaining a decomposition of f into a superposition of contributions from scales 2?0; 2?1 ; ::::. These contributions are parameterized by coecients (0;k )k (the samples of point values at the unit scale) and ner scale coecients (qj;k (f ))j;k . The number of coecients associated with a given interval grows with j like 2j , and the collection of those coecients associated with a given interval can be thought of as a kind of \Pyramid", with a few coecients `near the top' { where j is near 0 { and many coecients `lower down' { where j large. Since (Pj f )(k=2j ) = (Pj+1 f )(2k=2j+1) = f (k=2j ), we have Qj f (k=2j ) = 0, so qj;2k (f ) = 0, and so actually at most half the coecients q in the pyramid representation are nonzero. In short the pyramid representation is redundant. It is possible to obtain a nonredundant wavelet representation which is at least twice as sparse as the pyramid representation. Let j;k = (2j+1 x ? (2k + 1)) denote a version of the DD function dilated and translated to live at the sites (2k + 1)=2j+1 where the corresponding coecients qj;2k+1 are potentially nonzero; these are at (k + 1=2)=2j , halfway between the sites at which Pj is interpolating. Then we can write
Qj f (x) =
X k
j;k j;k (x)
(3)
with j;k = (Qj f )((k +1=2)=2j ). For example, if D = 1 then j;k = f ((k +1=2)=2j ) ? (f (k=2j )+ f ((k + 1)=2j ))=2, while more generally j;k (f ) = (f ? j;k )((k + 1=2)=2j ): (4) Using (3) we can rewrite (2) as
f=
X k
;k j;k +
0
4
XX j 0 k
j;k j;k :
(5)
This is a nonredundant sum of the same form as commonly used in the wavelet transform: a coarse-scale sum of `father wavelets' j;k and a series of ner and ner scale dilates and translates j;k of a `mother wavelet'. The association of a function f with its coecients (( j;k )k ; ( 0;k )k ; ( 1;k )k ; : : :) is a linear operator, well-de ned for every continuous f . As in [7] we call this the Interpolating Wavelet Transform (IWT), because it is built on wavelets which arise from translation and dilation of , and so have various interpolating properties. The wavelet coecient functionals j;k (f ) have an interesting interpretation. By (4) they measure the extent to which the point values of f at a given scale can be predicted by the point values at the next coarser scale. If f is very smooth, of course, then j;k (f ) will be small at large j corresponding to ne scales 2j . Unlike traditional wavelet transforms, this transform is de ned by point values rather than integrals. It is only nicely de ned on spaces of continuous functions; this places limitations on the extent to which it has properties analogous to the traditional wavelet transform. Thus, it is not de ned for arbitrary square integrable functions (such functions do not have precisely de ned point values) and so there is no hope of a \Parseval relation" relating `coecient energy in the wavelet domain' to `energy in the original domain'. Despite this limitation, the IWT has many of the same properties as traditional wavelet transforms. For example, it can characterize smoothness spaces [7]. Suppose 0 < < R(D). Then a bounded function f is Holder- if and only if j j;k j C 2?j ; j 0; k 2 Z: There are in fact a range of norm-equivalence results [7]. Let (; p; q) denote a triple of indices for or a Triebel Space F : then we can de ne norms on the wavelet coecients a Besov Space Bp;q p;q so that ; k kbp;q kf kBp;q
kf kF : k kfp;q p;q
whenever R(D) > > 1=p. Informally, if we have a functional class in the Besov/Triebel scales (this includes Holder and Sobolev classes), and if all members of the class are continuous (so > 1=p), and if the fundamental function is a member of the class (so R(D) > ), then the decay of the interpolating wavelet coecients tells whether f is in the class. In short, starting from the DD re nement scheme, we can de ne a wavelet-like transform, the coecients of the transform have a nice interpretation in terms of agreement with a local polynomial t, and the coecients of this transform characterize smoothness properties. 2.2
Generalization
We can generalize the original DD re nement scheme in two ways. First, we can change the functionals being interpolated; second, we can change the interpolation family.
2.2.1 Choice of Functional Consider the following Functional Interpolation Problem. Let I0;k denote the interval [k; k + 1). Suppose we are given data k = (f jI0;k ) where (f jI ) is a functional of f associated with the behavior of f in the vicinity of the interval I . We wish to nd a function f~ which agrees with the prescribed values of the functionals at scale 0. Consider the following two-scale re nement procedure, for obtaining functional values associated to the ner collection of intervals (Ij+1;k )k starting from values on the coarser collection of intervals (Ij;k )k . [1] (-Interpolation) For each interval Ij;k , nd a polynomial j;k of degree D satisfying the -interpolation condition:
(j;k jIj;k+h ) = j;k+h for ? L h L: 5
(6)
[2] (-Imputation) Obtain imputed values at a ner scale by setting
j+1;2k+h = (j;k jIj+1;2k+h ) for h = 0; 1:
(7)
This is a generalization of the DD re nement procedure, where (f jIj;k ) plays the role which the point value f (k=2j ) played in the original procedure. We intend here that (f jIj;k ) may be either a linear or nonlinear functional, that it may be scalar valued or vector valued. We will discuss all these possibilities in the section to follow. We don't wish to make light of some serious questions that need to be resolved in attempting this generalization. Q1: for a given family of functionals (jIj;k ), is the interpolation problem solvable by polynomials: can one always nd a polynomial that satis es the constraints (6)? Q2: if so, does the process converge when iterated repeatedly, producing a limit function f~? Q3: if so, does the limit function f~ have any useful properties? Q4: does the procedure yield useful fast algorithms?
2.2.2 Choice of Interpolation Family
Another generalization is to go beyond use of polynomials for interpolation. Let G be a nitedimensional parametric family of functions. We can generalize the polynomial interpolation problem to the (; G )-interpolation problem, de ning a new two-scale re nement procedure by tting and imputing locally from G . This procedure aims to obtain functional values associated to the ner collection of intervals (Ij+1;k )k starting from values on the coarser collection of intervals (Ij;k )k . [G1] ((; G )-Interpolation) For each interval Ij;k , nd a element gj;k 2 G satisfying
(gj;k jIj;k+h ) = j;k+h for ? L h L:
(8)
[G2] ((; G )-Imputation) Obtain imputed values at a ner scale by setting
j+1;2k+h = (gj;k jIj+1;2k+h ) for h = 0; 1:
(9)
Here the family G can be composed of functions for which the interpolation problem is solvable. Below we will give an example with so-called `split polynomials'. There are several questions that need to be resolved here to make sense of this so-called generalization. These are precisely analgous to Q1-Q4 mentioned in connection with -interpolation.
3 Average Interpolation
R
Consider the case where (f jIj;k ) = Ave(f jIj;k ) = jIj;k j?1 Ij;k f (x)dx and we use polynomial interpolation methods. Suppose we have an array (a0;k )1 k=?1 which represents averages of a function f on dyadic intervals I0;k . We may synthesize mock-averages at scale j + 1 from scale j by the following Average-Interpolating Re nement procedure. For integer L > 0, let D = 2L be an even integer greater than 0. [A1] (Average-Interpolation). At each site k, nd the polynomial j;k of degree D which generates the same averages in the neighborhood (aj;k ; k0 = k ? L; : : : ; k + L), i.e. 0
k0 = k ? L; : : : ; k + L:
Avej;k j;k = aj;k ; 0
0
[A2] (Average-Imputation). De ne the mock-averages at the next ner scale as averages of the AI polynomial. On the two halves of the interval we get
aj+1;2k+h = Avej+1;2k+h j;k h = 0; 1: 6
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Figure 4 below illustrates the tting of an average-interpolating polynomial. Actually, the average-interpolation procedure is well-de ned. The desired polynomial has D + 1 coecients and is required to satisfy D + 1 constraints; the average interpolation problem can be written in terms of a nonsingular system of linear equations. Figure 5 below illustrates the results of several iterations of re nement. Figure 6 illustrates the fundamental solution ' = '(D) { re nement of a Kronecker sequence (0) a0;k = 1fk=0g { for a range of values of D = 2L. It turns out that this fundamental solution is continuous and has a smoothness R(D) that increases with roughly linearly D. Indeed, there is a special relationship between the fundamental solution ' of average interpolation (D = 2L) and the fundamental solution of Deslauriers-Dubuc interpolation (D = 2L + 1). In fact [8]
0 (x) = '(x + 1) ? '(x): Evidently, with R(2L+1) denoting the smoothness possessed by DD interpolation for D = 2L+1, AI with D = 2L yields one fewer derivative: R(2L) = R(2L + 1) ? 1. Average Interpolation was introduced in Donoho(1993) and Harten(1994). It can be used to construct a wavelet transform, that is a decomposition
f=
X k
;k '0;k +
0
XX j 0 k
j;k j;k ;
(10)
where '0;k is a translated version of the AI fundamental function, and j;k (x) = 2j=2 p(2j x ? k) is a scaled and translated version of the mother wavelet (x) = ('(x + 1=2) ? '(x))= 2. The wavelet coecient functionals j;k (f ) for this transform obey the relation
j;k (f ) = 2?j=2 (Aveff jIj+1;2k g ? Avefj;k jIj+1;2k g) and so they measure the extent to which the averages of f at a given scale cannot be predicted from the averages at the next coarser scale. This transform is well-de ned for all locally integrable functions and the coecients obey various norm-equivalence results. Let (; p; q) denote a triple of indices for a Besov Space Bp;q 7
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: then we can de ne norms on the wavelet coecients so that [8] or a Triebel Space Fp;q k kbp;q kf kBp;q
kf kFp;q : k kfp;q
whenever R(D) > > 1=p ? 1. Informally, the transform can characterise every space in the Besov/Triebel scales which consists of locally integrable functions and which contains '. The transform itself is not orthogonal; as it turns out [8], it is identical to a family of biorthogonal wavelet transforms developed from another point of view by Cohen, Daubechies, and Feauveau (1991).
4 Hermite Interpolating Re nement Now we consider an example of the general -interpolation picture where is vector-valued. Suppose we are given the values of a function and its rst m ? 1 derivatives at the integers:
f (`)(k) = 0`;k ;
k 2 Z; 0 ` < m;
` where f (`) = dxd ` f . Hermite interpolation is the problem of nding a smooth function f~(x) obeying the given conditions. A two-scale re nement scheme can be developed, based on imputing the point-values and derivatives at the half integers f (`)(k=2) = 1`;2k+1 , using local polynomial interpolation from the values at the integers, and then iterating at ner scales. Fix L > 0 and set D~ = (2L +2)m ? 1. The process works as follows, for each k 2 Z.
[H1] (Hermite Interpolation) Find a polynomial ~0;k of degree D~ with the prescribed point values and derivatives 0`;k for k0 in a neighborhood of k: 0
~0(`;k) (k + h) = 0`;k+h ? L h L + 1:
(11)
[H2] (Hermite Imputation) Calculate the point values and derivatives of 0;k at the next ner scale: (12) 1`;2k+1 = ~0(`;k) (k + 1=2); k 2 Z; 0 ` < m: In the case m = 1 this is just the scheme of Deslauriers-Dubuc Interpolation. This scheme was proposed by Merrien [16] and studied by Dyn and Levin [13]. It is known that the HermiteInterpolation problem (11) is well-posed. Precise information about the smoothness of solutions has been obtained by Dyn, Levin, and Yu (1996). There are m fundamental solutions ` to this problem, corresponding to the re nement of the di erent Kronecker sequences (k(l;`) )k;l de ned by kl;` = 1fk=0g 1fl=`g (i.e. re nement of sequences with all point values and derivatives zero except for one speci c point value or derivative in one speci c location). Figure 7 below shows di erent fundamental solutions for m = 2 and two di erent values of L Herve (1994) has shown how this general type of interpolation can be used to construct a multi-interpolating multi-wavelet-type transform. As the reader should by now expect, the coecients of such a transform measure the extent to which values and rst m ? 1 derivatives at points in a grid of a certain scale can be predicted from values and derivatives at points in the next coarser scale grid. The transform is only well de ned on functions which are C m?1 or better; but within those limitations, the obvious generalization of [7] will show that the multi-wavelet coecients have good norm characterization properties. 9
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5 Moment-Interpolating Re nement
Here is another example where is vector-valued. Let m 1 be xed and let `0;0 denote the `-th Legendre polynomial associated to the interval [0; 1]. For example, 00;0 (x) = 1[0;1](x), p 1 0;0 (x) = (x ? 1=2) 12 1[0;1] (x). These are monomials which are localized to the interval [0; 1] and orthogonalized to corresponding localized monomials of lower degree. Let `j;k (x) = 2j=2 `0;0 (2j x ? k) be the same functions transported to `live' on Ij;k . Moment interpolating re nement is an answer to the following problem: Given the local moments of a function f ,
`0;k = hf; `0;k i
k 2 Z; 0 ` < m;
(13)
construct a \smooth" function f~ matching those moments. We may de ne a two-scale re nement operator which, for a given sequence (j;k )k of moment vectors `j;k = hf; `j;k i delivers a \predicted" sequence (j+1;k )k of moment vectors at the next ner scale. Fix L > 0 and set D = (2L + 1)m ? 1. The process works as follows, for each k 2 Z. [1] (Moment Interpolation) Find a polynomial j;k of degree D with the prescribed moments `j;k for k0 in a neighborhood of k: 0
hj;k ; `j;k h i = `j;k +
h
+
? L h L:
(14)
[2] (Moment Imputation) Calculate the moments of j;k at the next ner scale:
`j+1;2k+h = hj;k ; `j+1;2k+h i; h = 0; 1:
(15)
The Moment Interpolation Problem (14) has a solution. It turns out that the vector equations (14) impose exactly D+1 constraints on j;k which is a polynomial of degree D. These constraints can be proven to be linearly independent over the space of polynomials of degree D. The case m = 1 is really just Average Interpolation. There are m fundamental solutions '` to this problem, corresponding to Moment-Interpolating Re nement of the di erent Kronecker 10
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sequences (k(l;`) )k;l de ned by kl;` = 1fk=0g 1fl=`g (i.e. interpolation of sequences of prescribed moments which all vanish except for one prescribed moment at one prescribed site). Figure 8 below shows di erent fundamental solutions for m = 2 and two di erent values of L Donoho, Dyn, Levin, and Yu (1996) have shown how this general type of interpolation is closely connected with Hermite Interpolation, and that the smoothness of solutions to this problem can be derived easily from the smoothness of solutions to the corresponding Hermite Interpolation problem with the same value of m and L. They also showed that the re nement scheme can be used to construct a multi-wavelet-type transform. It is de ned for all locally integrable functions, and the multi-coecients allow characterization of various smoothness spaces. They indicate applications to the study of recursive partitioning schemes.
6 Nonlinear Re nement: Median-Interpolation
Given a function f on an interval I , let med(f jI ) denote a median of f de ned by
med(f jI ) = inf f : m(t 2 I : f (t) ) m(t 2 I : f (t) )g It turns out to be slightly nicer to work with triadic rather than dyadic re nement in this j setting. Suppose we are given a triadic array fmj;k g3k=0?1 of numbers representing the medians of f on the triadic intervals Ij;k = [k3?j ; (k + 1)3?j ):
mj;k = med(f jIj;k ) 0 k 3j : Median-interpolating re nement can be de ned starting from a xed even integer D. The basic idea is, starting from data on medians over coarser-scale triadic blocks to impute medians over ner-scale blocks. There are two steps. [M1] (Median-Interpolation) For each interval Ij;k , nd a polynomial j;k of degree D = 2L satisfy the median-interpolation condition:
med(j;k jIj;k+h ) = mj;k+h for ? L h L: 11
(16)
Median Interpolation Refinement of Kronecker Sequence 1.2
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(a): D=2 Refinement of Kronecker Sequence and shifts(b): D=2 Refinement of Heaviside Sequence 1.5 1.5 1
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(c): Superposition of shifts of (a)
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[M2] (Median-Imputation) Obtain (psuedo-) medians at the ner scale by setting mj+1;3k+h = med(j;k jIj;3k+h ) for h = 0; 1; 2:
(17)
If this can be done, it provides a systematic way to impute behavior at ner scales from behavior at coarse scales. The questions Q1-Q4 of section 2.1 come up naturally. Such questions are studied in [11], where it is shown that for D = 2, they have armative answers. In that paper, the authors develop an e ective median interpolation algorithm and show that it can be used to generate a nonlinear re nement scheme with continuous (in fact almost Lipschitz) solutions. The solution from re ning a Kronecker sequence is depicted in Figure 9. As the re nement scheme is not linear, the result from re ning an arbitrary sequence is not a linear superposition of shifts of this function. Indeed, Figure 10 below depicts the result of re ning a Heaviside P sequence m0;k = 1fk0g , which can be viewed as a superposition of Kronecker sequences `0 (`) , and then compares this with the superposition of separate components. Donoho and Yu have constructed nonlinear pyramid transforms, but not wavelet transforms, starting from this re nement scheme, and used them to analyse heavily non-Gaussian data. In
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such settings the robustness of the median gives an important bene t over various non-robust linear wavelet procedures.
7 Non-Polynomial Re nement: Segmented Interpolation We now give an example where the family G is not polynomial; more properly, it consists of split-polynomials. That is, there is a segmentation point and G consists of polynomials as well as pairs of polynomials of degree D, one of which is thought of as a right-polynomial and one of which is thought of as a left-polynomial. The functionals (f jIj;k ) are just averages Ave(f jIj;k ). Consider, as in [9], the following segmented re nement procedure, with segmentation point 2 [0; 1]. Given a sequence of averages aj;k , 0 k < 2j , we synthesise mock averages at ner scales by the following procedure: [1] At each site k which is more than L sites away from the segmentation point , use the average interpolating procedure to nd the polynomial j;k of degree D = 2L which generates the same averages in the neighborhood (aj;k ; k0 = k ? L; : : : ; k + L). 0
[2] At each site k which is at most L sites away from the segmentation point , we distinguish two cases. [2a] If 62 Ij;k , then nd the polynomial j;k of degree D which generates the same averages in the neighborhood (aj;k )k 2N (k) , where N (k) consists of the D +1 nearest neighbors of k which are all on the same side of the segmentation point as k. [2b] If 2 Ij;k , L R of degree D to then t, by constrained least squares, left and right polynomials j;k j;k the block averages in the neighborhoods on the right and left of the segmentation point, which is L on the left of respectively; the constraint is that the piecewise polynomial j;k and R on the right of should have an average equal to the average aj;k . 0
0
[3] In cases [1] and [2a], de ne the mock-averages at the next ner scale as averages of the polynomial. On the two halves of the interval we get
a j+1;2k+h = Avej+1;2k+h j;k ; h = 0; 1: . In case [2b], the steps are the same, only using the piecewise polynomial j;k
This segmented average-interpolating resolution has a variety of properties; a key one being that if is a piecewise polynomial of degree D, with one knot, at , then the re nement process recovers exactly. Applications of this procedure to determining optimal segmentations are described in [9]. The basic idea is to measure the quality of a segmentation by an `entropy' of the coecients j;k ; the `best' segmentation point then has coecients with the smallest entropy E (( j;k )j;k ).
8 Conclusion When Deslauriers and Dubuc developed their interpolation scheme, they also made available a general pattern of thought that can be applied in other settings. In this brief note we have shown that by generalizing the family of functionals beyond point values and generalizing the family of interpolating functions beyond polynomials, we can obtain a variety of interesting re nement schemes and transforms. We are con dent there will be many other examples illustrating this theme. This note was written following the principle of Reproducible Research. Code to generate the gures can be obtained by pointing a web browser at http://www-stat.stanford.edu/wavelab. 13
9 Acknowledgements This research was partially supported by NSF DMS-92-09130 and DMS-95-05151, by an AFOSR Multi-University-Research Initiative, and by other sponsors. DLD would like to thank Nira Dyn and David Levin of the Mathematics Department at Tel Aviv University for plenty of conversations and interaction during his visit there in 1996.
References [1] Cavaretta, A.S., Dahmen, W., and Micchelli, C.A. (1991) Stationary Subdivision. Mem. Amer. Math. Soc. 453 [2] Cohen, A., Daubechies, I., and Feauveau, J.C. (1990) Biorthogonal bases of compactly supported wavelets. Commun. Pure and Applied Math., 45, pp. 485-560. [3] Deslauriers, G. and Dubuc, S. (1987) Interpolation dyadique. in Fractals, Dimensions nonentieres et applications. Paris: Masson. [4] Deslauriers, G. and Dubuc, S. (1989) Symmetric iterative interpolation processes. Constructive Approximation, 5, 49-68. [5] Dubuc, S. (1986) Interpolation through an iterative scheme. J. Math. Anal. and Appl. 114 185-204. [6] Daubechies, I. and Lagarias, J. (1991) Two-scale di erence equations, I. Global regularity of solutions. SIAM J. Math. Anal 22 1388-1410. [7] Donoho, D.L. (1992). Interpolating Wavelet Transforms. Technical Report, Department of Statisics, Stanford University. http://www-stat.stanford.edu/pub/donoho/iwt.ps [8] Donoho, D.L. (1993). Smooth Wavelet Decompositions with Blocky Coecient Kernels. in Recent Advances in Wavelet Analysis, L. Schumaker and F. Ward, eds. Academic Press. [9] Donoho, D.L. (1994) On Minimum Entropy Segmentation, in Wavelets: Theory, Algorithms and Applications. C.K. Chui, L. Montefusco and L. Puccio, Eds. San Diego: Academic Press. [10] Donoho, D.L., Dyn, N., Levin, D. and Yu, T.P.Y. (1996) \Smooth Multiwavelet Duals of Alpert Bases by Moment-Interpolation, with applications to recursive partitioning", Technical Report, Department of Statisics, Stanford University. http://www-stat.stanford.edu/pub/donoho/moment.ps
[11] Donoho, D.L. and Yu, T.P.Y. (1996) \Robust Nonlinear Wavelet Transforms by Median Interpolation", Technical Report, Department of Statisics, Stanford University. http://www-stat.stanford.edu/pub/donoho/median.ps
[12] Dyn, N., Gregory, J.A., and Levin, D. (1991) Analysis of uniform binary subdivision schemes for curve design. Constructive Approximation 7 127-147. [13] Dyn, N. and D. Levin, Analysis of Hermite-type subdivision schemes, in Approximation Theory VIII, Charles K. Chui and Larry L. Schumaker (eds.), World Scienti c Publishing, 1995. [14] Dyn, N., Levin, D. and Yu, T.P.Y. (1996) \Smoothness of Solutions to Hermite Re nement Schemes", in preparation. [15] Harten, Ami (1994) Multiresolution Representation of Cell-Averaged Data. UCLA Computational and Applied Mathematics Report CAM 94-21. 14
[16] Merrien, J.L. (1992) A family of Hermite Interpolants by Bisection Algorithms, Numerical Algorithms 2, 187-200. [17] Herve, L. (1994) Multiresolution Analysis of Multiplicity d: applications to dyadic interpolation, ACHA 1, 299-315.
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