Spherical Wavelets: Efﬁciently Representing Functions …jdl/bib/appearance/wavelets/shroder95.pdfSpherical Wavelets: Efﬁciently Representing Functions on the Sphere Peter Schrod¨

Spherical Wavelets:Efficiently Representing Functions on the Sphere

Peter Schroder��

Wim Sweldens��

University of South Carolina

Abstract

Wavelets have proven to be powerful bases for use in numericalanalysis and signal processing. Their power lies in the fact thatthey only require a small number of coefficients to represent gen-eral functions and large data sets accurately. This allows compres-sion and efficient computations. Classical constructions have beenlimited to simple domains such as intervals and rectangles. In thispaper we present a wavelet construction for scalar functions definedon the sphere. We show how biorthogonal wavelets with customproperties can be constructed with the lifting scheme. The basesare extremely easy to implement and allow fully adaptive subdivi-sions. We give examples of functions defined on the sphere, suchas topographic data, bidirectional reflection distribution functions,and illumination, and show how they can be efficiently representedwith spherical wavelets.CR Categories and Subject Descriptors: I.3.0 [Computer Graphics]:General; G.1.0 [Numerical Analysis]: General – Numerical Algorithms;G.1.1 Interpolation – Smoothing; G.1.2 Approximation – Nonlinear Ap-proximation.

Additional Key Words and Phrases: wavelets, sphere.

1 Introduction

1.1 Wavelets

Over the last decade wavelets have become an exceedingly pow-erful and flexible tool for computations and data reduction. Theyoffer both theoretical characterization of smoothness, insights intothe structure of functions and operators, and practical numericaltools which lead to faster computational algorithms. Examples oftheir use in computer graphics include surface and volume illumi-nation computations [16, 29], curve and surface modeling [17], andanimation [18] among others. Given the high computational de-mands and the quest for speed in computer graphics, the increasingexploitation of wavelets comes as no surprise.

While computer graphics applications can benefit greatly fromwavelets, these applications also provide new challenges to theunderlying wavelet technology. One such challenge is the con-struction of wavelets on general domains as they appear in graphicsapplications.

Classically, wavelet constructions have been employed on infi-nite domains (such as the real line R and plane R2). Since mostpractical computations are confined to finite domains a number ofboundary constructions have also been developed [5]. However,

�Department of Mathematics.�Department of Computer Science.�Department of Computer Science, Katholieke Universiteit Leuven, Belgium.

Figure 1: The geodesic sphere construction starting with the icosa-hedron on the left (subdivision level 0) and the next 2 subdivisionlevels.

wavelet type constructions for more general manifolds have onlyrecently been attempted and are still in their infancy.

Our work is inspired by the ground breaking work of Lounsberyet al.[20, 19] (hereafter referred to as LDW). While their primarygoal was to efficiently represent surfaces themselves we examinethe case of efficiently representing functions defined on a surface,and in particular the case of the sphere.

Although the sphere appears to be a simple manifold, techniquesfrom R2 do not easily extend to the sphere. Wavelets are no excep-tion. The first construction of wavelets on the sphere was introducedby Dahlke et al.[6] using a tensor product basis where one factoris an exponential spline. To our knowledge a computer imple-mentation of this basis does not exist at this moment. A continuouswavelet transform and its semi-discretization were proposed in [13].Both these approaches make use of a �� parameterization of thesphere. This is the main difference with our method, which isparameterization independent.

Aside from being of theoretical interest, a wavelet constructionfor the sphere leading to efficient algorithms, has practical appli-cations since many computational problems are naturally stated onthe sphere. Examples from computer graphics include: manipula-tion and display of earth and planetary data such as topography andremote sensing imagery, simulation and modeling of bidirectionalreflection distribution functions, illumination algorithms, and themodeling and processing of directional information such as envi-ronment maps and view spheres.

In this paper we describe a simple technique for constructingbiorthogonal wavelets on the sphere with customized properties.The construction is an incidence of a fairly general scheme referredto as the lifting scheme [27, 28].

The outline of the paper is as follows. We first give a briefreview of applications and previous work in computer graphics in-volving functions on the sphere. This is followed by a discussionof wavelets on the sphere. In Section 3 we explain the basic ma-chinery of lifting and the fast wavelet transform. After a sectionon implementation, we report on simulations and conclude with adiscussion and suggestions for further research.

1.2 Representing Functions on the Sphere

Geographical information systems have long had a need to repre-sent sampled data on the sphere. A number of basic data structuresoriginated here. Dutton [10] proposed the use of a geodesic sphere

Figure 2: Recursive subdivision of the octahedral base shape asused by LDW for spherelike surfaces. Level 0 is shown on the leftfollowed by levels 2 and 4.

construction to model planetary relief, see Figure 1 for a picture ofthe underlying subdivision. More recently, Fekete [12] describedthe use of such a structure for rendering and managing spherical geo-graphic data. By using hierarchical subdivision data structures theseworkers naturally built sparse adaptive representations. There alsoexist many non-hierarchical interpolation methods on the sphere(for an overview see [22]).

An important example from computer graphics concerns the rep-resentation of functions defined over a set of directions. Perhapsthe most notable in this category are bidirectional reflectance distri-bution functions (BRDFs) and radiance. The BRDF,

�� ,describes the relationship at a point

�� on a surface between incomingradiance from direction

�� and outgoing radiance in direction�� .

It can be described using spherical harmonics, the natural extensionof Fourier basis functions to the sphere, see e.g. [30]. These basisfunctions are globally supported and suffer from some of the samedifficulties as Fourier representations on the line such as ringing. Toour knowledge, no fast (FFT like) algorithm is available for spheri-cal harmonics. Westin et al. [30] used spherical harmonics to modelBRDFs derived from Monte Carlo simulations of micro geometry.Noting some of the disadvantages of spherical harmonics, Gondeket al.[15] used a geodesic sphere subdivision construction [10, 12]in a similar context.

The result of illumination computations, the radiance � �� ,is a function which is defined over all surfaces and all directions.For example, Sillion et al. [26] used spherical harmonics to modelthe directional distribution of radiance. As in the case of BRDFrepresentations, the disadvantages of using spherical harmonics torepresent radiance are due to the global support and high cost ofevaluation. Similarly no locally controlled level of detail can beused.

In finite element based illuminations computations wavelets haveproven to be powerful bases, see e.g. [24, 3]. By either reparame-terizing directions over the set of visible surfaces [24], or mappingthem to the unit square [3], wavelets defined on standard domains(rectangular patches) were used.

Mapping classical wavelets on some parameter domain onto thesphere by use of a parameterization provides one avenue to con-struct wavelets on the sphere. However, this approach suffers fromdistortions and difficulties due to the fact that no globally smoothparameterization of the sphere exists. The resulting wavelets arein some sense “contaminated” by the parameterization. We willexamine the difficulties due to an underlying parameterization, asopposed to an intrinsic construction, when we discuss our construc-tion.

We first give a simple example relating the compression of sur-faces to the compression of functions defined on surfaces.

1.3 An Example

LDW constructs wavelets for surfaces of arbitrary topological typewhich are parameterized over a polyhedral base complex. For thecase of the sphere they employed an octahedral subdivision domain

Σ Σ0.51 10.5 -0.5-0.5

Figure 3: A simple example ofrefinement on the line. The ba-sis functions at the top can beexpressed as linear combina-tions of the refined functions atthe bottom.

(see Figure 2). In this framework a given goal surface such as theearth is parameterized over an octahedron whose triangular facesare successively subdivided into four smaller triangles. Each vertexcan now be displaced radially to the limit surface. The resultingsequence of surfaces then represents the multiple levels of detailrepresentation of the final surface.

As pointed out by LDW compressing surfaces is closely relatedto compressing functions on surfaces. Consider the case of the unitsphere and the function

�� cos2 � with��

S2.We can think of the graph of this function as a surface over thesphere whose height (displaced along the normal) is the value ofthe function

�. Hence an algorithm which can compress surfaces

can also compress the graph of a scalar function defined over somesurface.

At this point the domain over which the compression is definedbecomes crucial. Suppose we want to use the octahedron . Definethe projection ! : #"%$ S2,

� � ! �'& �(� &%)+*,&-*. We then have

˜��'& �.� �� ! �'& �,� with&/� . Compressing

�� with waveletson the sphere is now equivalent to compressing ˜��'& � with waveletsdefined on the octahedron. While

�is simply a quadratic function

over the sphere, ˜� is considerably more complicated. For examplea basis over the sphere which can represent quadratics exactly (seeSection 3.4) will trivially represent

�. The same basis over the

octahedron will only be able to approximate ˜� .This example shows the importance of incorporating the un-

derlying surface correctly for any construction which attempts toefficiently represent functions defined on that surface. In case ofcompression of surfaces themselves one has to assume some canon-ical domain. In LDW this domain was taken to be a polyhedron.

By limiting our program to functions defined on a fixed surface(sphere) we can custom tailor the wavelets to it and get more effi-ciency. This is one of the main points in which we depart from theconstruction in LDW.

2 Wavelets on the Sphere

2.1 Second Generation Wavelets

Wavelets are basis functions which represent a given function atmultiple levels of detail. Due to their local support in both spaceand frequency, they are suited for sparse approximations of func-tions. Locality in space follows from their compact support, whilelocality in frequency follows from their smoothness (decay towardshigh frequencies) and vanishing moments (decay towards low fre-quencies). Fast ��0 � algorithms exist to calculate wavelet coeffi-cients, making the use of wavelets efficient for many computationalproblems.

In the classic wavelet setting, i.e., on the real line, wavelets aredefined as the dyadic translates and dilates of one particular, fixedfunction. They are typically built with the aid of a scaling function.Scaling functions and wavelets both satisfy refinement relations(or two scale relations). This means that a scaling function orwavelet at a certain level of resolution (1 ) can be written as a linearcombination of scaling basis functions of the same shape but scaledat one level finer (level 132 1), see Figure 3 for an example.

The basic philosophy behind second generation wavelets is tobuild wavelets with all desirable properties (localization, fast trans-form) adapted to much more general settings than the real line, e.g.,

Functions�� primal scaling functions

˜� �� dual scaling functions� �� primal wavelets˜� �� dual wavelets

Biorthogonality relationships� � �� ˜� �� "!$#��%� �� and ˜� �� are biorthogonal� � �� &� ˜� � � � � � �'!(#�� # �� and ˜� � � � � � are biorthogonal� � �� ˜� �� "! 0 ) �&* ˜+ �� &� ˜�,�� "! 0+ � * ˜) �

Vanishing moment relations˜� �� has - vanishing moments �,�� reprod. polyn. degree ./-� �� has 0- vanishing moments ˜�,�� reprod. polyn. degree .10-

Refinement relations�� !32547698": � ;1 <>= �� ?� 4 � �@; 1

� 4 scaling function refinement eq.

˜�� ! 2547698": � ;1 < ˜= �� ?� 4 ˜� �@; 1

� 4 dual scaling function refinement eq.� �� ! 2A4B698": � ;1 <DC �� 4 � �@;

1� 4 wavelet refinement equation

˜� �� ! 2 4B698": � ;1 < ˜C �� 4 ˜� �@; 1

� 4 dual wavelet refinement equation) � ! clos span E �� FG�HI�J�K��L with ) 0 the coarsest space+ � ! clos span E � �� MFG�NO�P��L with+

0 the coarsest space) �RQ + � ! ) � ; 1 wavelets encode difference between

levels of approximation

Wavelet transformsS �� ! �7T � ˜�,�� scaling function coefficientU?�� ! �7T � ˜� �� wavelet coefficient

Forward Wavelet Transform (Analysis)S �� ! 2 4B698": � < ˜= �� %� 4 S � ; 1� 4 scaling function coeff., fine to coarseU?�� !(2A4B69VW: � < ˜C �� 4 S �@;1� 4 wavelet coeff., fine to coarse

Inverse Wavelet Transform (Synthesis)S �@;1� 4 ! 2 � 698": � < = �� %� 4 S �� scaling function coeff., coarse to fineX 2 � 6%VW: � < C �� J� 4 U?��

Table 1: Quick reference to the notation and some basic relation-ships for the case of second generation biorthogonal wavelets.

wavelets on manifolds. In order to consider wavelets on a surface,we need a construction of wavelets which are adapted to a measureon the surface. In the case of the real line (and classical con-structions) the measure is Y,Z , the usual translation invariant (Haar)Lebesgue measure. For a sphere we will denote the usual areameasure by Y�[ . Adaptive constructions rely on the realization thattranslation and dilation are not fundamental to obtain the waveletswith the desired properties. The notion that a basis function can bewritten as a finite linear combination of basis functions at a finer,more subdivided level, is maintained and forms the key behind thefast transform. The main difference with the classical wavelets isthat the filter coefficients of second generation wavelets are not thesame throughout, but can change locally to reflect the changing(non translation invariant) nature of the surface and its measure.

Classical wavelets and the corresponding filters are constructedwith the aid of the Fourier transform. The underlying reason is thattranslation and dilation become algebraic operations after Fouriertransform. In the setting of second generation wavelets, translationand dilation can no longer be used, and the Fourier transform thusbecomes worthless as a construction tool. An alternative construc-tion is provided by the lifting scheme.

2.2 Multiresolution Analysis

We first introduce multiresolution analysis and wavelets and setsome notation. For more mathematical detail the reader is referredto [9]. All relationships are summarized in Table 1.

Consider the function space L2 \ L2 ] S2 ^ Y�[�_ , i.e., all functions

of finite energy defined over S2. We define a multiresolution anal-ysis as a sequence of closed subspaces >acb L2, with d3e 0, sothat

I `>aHbf`>a ; 1, (finer spaces have higher index)II g a%h 0 `>a is dense in L2,

III for each d , scaling functions ija�k l with mcnpo ] d�_ exist so thatq i a�k lOr msn/o ] d�_�t is a Riesz basis1 of ` a .Think of o ] d>_ as a general index set where we assume that o ] d�_bo ] dvu 1 _ . In the case of the real line we can take o ] d>_ \ 2 w a Z,while for an interval we might have o ] d>_ \ q

0 ^ 2 w a ^?xGx?x%^ 1 y 2 w a t .Note that, unlike the case of a classical multiresolution analysis, thescaling functions need not be translates or dilates of one particularfunction. Property (I) implies that for every scaling function i a�k l ,coefficients

q%z a�k l9k { t exist so that

ija�k l \ 2 { z a�k l|k {9iRa ; 1 k { x (1)

Thez a�k l9k { are defined for d1e 0, m}n~o ] d�_ , and �Hn~o ] d�u 1 _ .

Each scaling function satisfies a different refinement relation. Inthe classical case we have

z a�k l9k { \ z { w 2 l , i.e., the sequencesz a�k l|k {

are independent of scale and position.Each multiresolution analysis is accompanied by a dual multires-

olution analysis consisting of nested spaces ˜>a with bases given bydual scaling functions ˜iRa�k l , which are biorthogonal to the scalingfunctions: �

i a�k l ^ ˜i a�k l �� \N� l9k l � for m ^ m��jnso ] d�_ ^where

�� \A�� Y�[ is the inner product on the sphere. The

dual scaling functions satisfy refinement relations with coefficientsq ˜z a�k l9k {�t .In case scaling functions and dual scaling functions coincide,

( iRa�k l \ ˜iRa�k l for all d and m ) the scaling functions form an orthog-onal basis. In case the multiresolution analysis and the dual mul-tiresolution analysis coincide ( `>a \ ˜`>a for all d but not necessarilyi a�k l \ ˜i a�k l ) the scaling functions are semi-orthogonal. Orthog-onality or semi-orthogonality sometimes imply globally supportedbasis functions, which has obvious practical disadvantages. Wewill assume neither and always work in the most general biorthog-onal setting (neither the multiresolution analyses nor the scalingfunctions coincide), introduced for classical wavelets in [4].

One of the crucial steps when building a multiresolution analysisis the construction of the wavelets. They encode the differencebetween two successive levels of representation, i.e., they form abasis for the spaces ��a where >a,��a \ >a ; 1. Consider the set offunctions

q9� a�k � r dce 0 ^"� n�� ] d�_�t , where � ] d�_�b�o ] d�u 1 _is again an index set. If

1. the set is a Riesz basis for L2 ] S2 _ ,2. the set

q9� a�k � r � n�� ] d�_�t is the Riesz basis of ��a ,we say that the

� a�k � define a spherical wavelet basis. Since � a b` a ; 1, we have� a�k � \ 2 { � a�k �Hk {9ija ; 1 k { for � ns� ] d�_ x (2)

An important property of wavelets is that they have vanishingmoments. The wavelets

� a�k � have 0� vanishing moments if 0�independent polynomials �R� , 0 �� 0� exist so that� � a�k � ^ �� \ 0 ^for all dce 0 ^�� n�� ] d�_ . Here the polynomials � � are defined asthe restriction to the sphere of polynomials on R3. Note that inde-pendent polynomials on R3 can become dependent after restrictionto the sphere, e.g.,

q1 ^ Z 2 ^�� 2 ^�� 2 t .

1A Riesz basis of some Hilbert space is a countable subset E T �|L so that everyelement

Tof the space can be written uniquely as

T !P2 � � � T � , and positive

constants ¡ and ¢ exist with ¡/£ T £ 2 � 2 � F � � F 2 �¤¢�£ T £ 2.

For a given set of wavelets we have dual basis functions ˜�� which are biorthogonal to the wavelets, or � � �� ˜� ��

for � � �� 0��! �#" �� $�! �%�&" . This implies

� ˜�� ' �� ( �� ˜' �� ( � �� )�

0 for�*�+�, �-" and . �0/1 �-" ,

and for 2 � L2 we can write the expansion

2 �43 �� ˜� �� 5� 2 � �� 43 �� 76 �� (3)

Given all of the above relationships we can also write the scalingfunctions

'��81� 9

as a linear combination of coarser scaling functionsand wavelets using the dual sequences (cf. Eqs. (1,2))'��8

1� 9 �:3 ( ˜; �� (<� 9='�� (�> 3 � ˜? �� 9 � �� A@

If not stated otherwise summation indices are understood to runover . �B/1 �#" , C �7/1 � > 1 " , and

�D�E�! �#" .Given the set of scaling function coefficients of a function 2 ,F=GIH � ( � ��2 � ˜

'�� ( �J . ��/1 LK "M whereK

is some finest resolutionlevel, the fast wavelet transform recursively calculates the

F 6 �� J0 NO�QP KR�-�D�B�, �-"M , and

F=G0� ( J . �S/1 0 "M , i.e., the coarser

approximations to the underlying function. One step in the fastwavelet transform computes the coefficients at a coarser level (� )from the coefficients at a finer level (� > 1)G �� ( �:3 9 ˜;-�� (<� 9 G ��8 1

� 9and 6 �� 3 9 ˜? �� )� 9 G ��8 1

� 9 @A single step in the inverse transform takes the coefficients at thecoarser levels and reconstructs coefficients at a finer levelG ��8

1� 9 �43 ( ; �� (=� 9 G �� (T> 3 � ? �� 9 6 �� 5@

3 Wavelet Construction and Transform

We first discuss the lifting scheme [27, 28]. After the introduc-ing of the algebra we consider two important families of waveletbases, interpolating and generalized Haar. At the end of this sectionwe give a concrete example which shows how the properties of agiven wavelet basis can be improved by lifting it and lead to bettercompression.

Lifting allows us to build our bases in a fully biorthogonal frame-work. This ensures that all bases are of finite (and small) supportand the resulting filters are small and easy to derive. As we will seeit is also straightforward to incorporate custom constraints into theresulting wavelets.

3.1 The Lifting Scheme

The whole idea of the lifting scheme is to start from one basicmultiresolution analysis, which can be simple or even trivial, andconstruct a new, more performant one, i.e., the basis functions aresmoother or the wavelets have more vanishing moments. In casethe basic filters are finite we will have lifted filters which are alsofinite.

We will denote coefficients of the original multiresolution anal-ysis with an extra superscript U (from old or original), starting withthe filters

; U�� (=� 9 , ˜; U�� (<� 9 , ? U�� (<� 9 , and ˜? U�� (=� 9 . The lifting scheme nowstates that a new set of filters can be found as; �� (<� 9 � ; U�� (=� 9 � ? �� 9 � ? U�� 9IV 3 (�W �� (=� � ; �� (<� 9X�

˜? U�� 9 � ˜? �� 9 � ˜;-�� (=� 9 � ˜; U�� (=� 9 > 3 �YW �� (<� � ˜? �� 9 �and that, for any choice of

F W �� (=� � M , the new filters will automat-ically be biorthogonal, and thus lead to an invertible transform.The scaling functions

' �� 9are the same in the original and lifted

multiresolution analysis, while the dual scaling function and primalwavelet change. They now satisfy refinement relations� �� Z3 9 ? U�� 9 '��8 1

� 9 V 3 ( W �� (=� ��'�� ( (4)

˜'�� ( �Z3 9 ˜; U�� (=� 9 ˜

'��81� 9-> 3 �YW �� (<� � ˜� �� @

Note that the dual wavelet has also changed since it is a linearcombination (with the old coefficients ˜? U ) of a now changed dual

scaling function. Equation (4) is the key to finding theF W �� (=� � J .IM

coefficients. Since the scaling functions are the same as in theoriginal multiresolution analysis, the only unknowns on the righthand side are the W �� (=� � . We can choose them freely to enforcesome desired property on the wavelets

� �� . For example, in case

we want the wavelet to have vanishing moments, the condition thatthe integral of a wavelet multiplied with a certain polynomial [R\ iszero can now be written as

0�:3 9 ? U�� 9 � '��8 1

� 9�� [�\ V 3 ( W �� (<� � � '�� (]� [�\ @For a fixed � and

�, this is a linear equation in the unknownsF W �� (=� � J .IM . If we choose the number of unknown coefficientsW �� (=� � equal to the number of equations ^ , we need to solve a linear

system for each � and�

of size ^ _� . A priori we do not knowif this linear system can always be solved. We will come back tothis later.

The fast wavelet transform after lifting can be written as6 �� a3 9 ˜? U�� 9 G ��8 1� 9

G �� ( � 3 9 ˜; U�� (<� 9 G ��8 1� 9 > 3 �YW �� (=� � 6 ��

i.e., as a sequence of two steps. First the old dual high and lowpass filters. Next the update of the old scaling function coefficientswith the wavelet coefficients using the

F W �� (=� � J .IM . The inversetransform becomesG ��8

1� 9 �43 ( ; U�� (=� 9cb G �� ( V 3 � W �� (<� � 6 �� edI> 3 � ? U�� )� 9 6 �� A@

Instead of writing everything as a sequence of two steps involvingF W �� (=� � J .IM we could have formed the new filters;

and ˜? firstand then applied those in a single step. Structuring the new filtersas two stages, however, simplifies the implementation considerablyand is also more efficient.

Remarks:1. The multiple index notation might look confusing at first sight,

but its power lies in the fact that it immediately corresponds tothe data structure of the implementation. The whole transformcan also be written as one giant sparse matrix multiplication,but this would obscure the implementation ease of the liftingscheme.

2. Note how the inverse transform has a simple structure directlyrelated to the forward transform. Essentially the inverse trans-form subtracts exactly the same linear combination of waveletcoefficients from

G �� (as was added in the forward transform.

3. It is also possible to keep the dual scaling function fixed and putthe conditions on the dual wavelet. The machinery is exactlythe same provided one switches primals and duals and thustoggles the tildes in the equations. We refer to this as the duallifting scheme, which employs coefficients ˜W �� (<� � . It allows usto improve the performance of the dual wavelet. Typically, thenumber of vanishing moments of the dual wavelet is importantto achieve compression. Also, the lifting scheme and the duallifting scheme can be alternated to bootstrap one’s way up to adesired multiresolution analysis (cakewalk construction).

4. The construction in LDW can be seen as a special case of thelifting scheme. They use the degrees of freedom to achievepseudo-orthogonality (i.e., orthogonality between scaling func-tion and wavelets of one level within a small neighborhood)starting from an interpolating wavelet. The lifting scheme ismore general in the sense that it uses a fully biorthogonal settingand that it can start from any multiresolution analysis with finitefilters. The pseudo-orthogonalization requires the solution oflinear systems which are of the size of the neighborhood (typi-cally 24 by 24). Since many wavelets may in fact be the samecaching of matrix computations is possible.

5. After finishing this work, the authors learned that a similar con-struction was obtained independently by Dahmen and collabo-rators. We refer to the original papers [2, 8] for details.

wavelets

scaling functions

scaling functions

j

j+1

j

wavelets

scaling functions

scaling functions

wavelets

scaling functions

scaling functions

Lazy wavelet on the real line

wavelets

scaling functions

scaling functions

j

j+1

j

Interpolating wavelet on the real line

Primals Duals Primals Duals

Figure 4: For the Lazy wavelet all primals are Kronecker functions(1 at the origin, 0 otherwise), while all duals are unit pulses (Diracdistributions). Going from a finer to a coarser scale is achievedby subsampling with the missing samples giving the wavelet spaces(��

2 � Z, � �� 2 � � Z � 1 � 2

�, and � ��

��). The well

known linear B-splines as primal scaling and wavelet functionswith Diracs as duals can be reached with dual lifting ( ˜� ��

�1 � 2 � �� 2 �� 1 � � � 1 � 2 � �"! 2 ��"� 1 � � ), resulting in ˜# ��

� � �%$'&(��and ˜) ��

�*&1 � 2 � �%$ &,+*&

2 � 1 � �-� �%$ &,+.��&1 � 2 � �%$ &,+ �

2 �" 1 � .

6. Evidently, the lifting scheme is only useful in case one has aninitial set of biorthogonal filters. In the following sections wewill discuss two such sets.

3.2 Fast Lifted Wavelet Transform

Before describing the particulars of our bases we give the gen-eral structure of all transforms. Forward (analysis) and inverse(synthesis) transforms are always performed level wise. The for-mer begins at the finest level and goes to the root while the latterstarts at the root and descends to the leaf level. /�021�3�4�5�6'5�7 com-putes the unlifted wavelet coefficients at the parent level while/�0�1'3�4�5�6�5'7�7 performs the lifting if the basis is lifted otherwise it isempty. Similarly, 8�4�0'9�:�;25�6'5�7 performs the inverse lifting, if any,while 8�4'0�9�:�;25'6�5�7�7 computes the scaling function coefficients atthe child level.

/'0�1�3�4�5'6�5<2=�> 3';�?�;�3A@B3�;'1�C�3�;�?2;�3D9 =D>�='= 9�3�;�?2;�3/'0�1�3�4�5'6�5�7FEG3';�?�;�32H/'0�1�3�4�5'6�5�7�7FEI3�;�?�;�3JH

8�4�0�9�:�;J5�6�5<2=�> 3';�?�;�3A@ >�='= 9�3�;�?2;�3D9 = 3�;'1�C�3�;�?2;�38�4�0�9�:�;J5�6�5�7FEI3�;�?�;�3JH8�4�0�9�:�;J5�6�5�7�7�EG3�;�?�;'32H

The transforms come in two major groups: (A) Lifted from theLazy wavelet: this involves interpolating scaling functions and avertex based transform; (B) Lifted from the Haar wavelet: thisinvolves a face based transform. We next discuss these in detail.

3.3 Interpolating Scaling Functions

We first give a trivial example of a wavelet transform: the Lazywavelet [27, 28]. The Lazy wavelet transform is an orthogonaltransform that essentially does not compute anything. However, it isfundamental as it is connected with interpolating scaling functions.The filters of the Lazy fast wavelet transform are given asKFL

M� �� N� ˜KFL�� N

� � �� N and O L�� P� N�

Õ LM� �P� N� � �P� NRQ

Consequently, the transform does not compute anything, it onlysubsamples the coefficients. Figure 4 (left) illustrates this idea forthe case of the real line.

Scaling functions S # M� �T �*U

0 V �XWY��GZare called in-

terpolating if a set of points S�� T �XU

0 V �[W\��GZwith

34e

e

v1

1

2v

2

f1

e

f

2e

m

Figure 5: Neighbors used in ourbases. Members of the index setsused in the transforms are shownin the diagram (

+ W � �� ,

S�] 1 VG] 2 VI^ 1 VI^ 2 V _ 1 V"_ 2 V"_ 3 V"_ 4Z`�a�

� ).

� �� M! 1 � � exists, so thatb`� V ��cdW,��

: # �� M� �fe

�� ge QAn example for such functions on the real line is shown on theright side of Figure 4. In case of interpolating scaling functions, wecan always take the dual scaling functions to be Dirac distributions,˜# ��

� � �P� � � � & � M� ��, which are immediately biorthogonal (see

the dual scaling functions on the right of Figure 4). This leadsto trivial inner products with the duals, namely evaluation of thefunction at the points � �� .

The set of filters resulting from interpolating scaling functionsand Diracs as their formal dual, can be seen as a dual lifting of theLazy wavelet. This implies that

K�� fe

� � �� fe ,KM� ��

�˜� �� ,

Õ �� P� ��h&

˜� �� , Õ � � �P� �ie� � �P� �ie . The wavelets are given by)

�� # M! 1 � � and the dual wavelets by

˜) �� %$j& � M! 1 � �

�k&(l� ˜� �� %$j& � M� �

�Q

The linear B-spline (right side of Figure 4) can be seen to be thedual lifting of the Lazy wavelet. Since we applied dual lifting theprimal wavelet does not yet have a vanishing moment.

Below we present other choices for the filter coefficientsK�� .

Typically one can choose the ˜� �� to insure that ˜) �� has vanish-ing moments (this will lead to the Quadratic scheme), or that # �� is smooth (this will lead to the Butterfly scheme).

At this point we have an interpolating multiresolution analysis,which was dually lifted from the Lazy wavelet. A disadvantageof this multiresolution analysis is that the functions cannot provideRiesz bases for L2. The dual functions do not even belong to L2.This is related to the fact that the wavelet does not have a vanishingintegral since it coincides with a scaling function. Consequently,unconditional convergence of the expansion (3) is not guaranteed.One can now apply the primal lifting scheme to try to overcomethis drawback by ensuring that the primal wavelet has at least 1vanishing moment. Note that this is only a necessary and not asufficient condition. This yields

˜K M� �� N� � �� N �

l�

� �� Õ M� �P� NO �� P� N� � �P� N

&ml��

KM� �� N Q

The resulting wavelet can be written as)M� �

� # M! 1 � �&(l

�� # �� Q (5)

In the situation in Figure 4 setting � ��

1 � 4 � �P� �"! 2 �� 1 �1 � 4 � �P� �� 2 �� 1 results in

)M� � having a vanishing integral. This

choice leads us to the well known�2 V 2 � biorthogonal wavelet of [4].

3.4 Vertex Bases

Up to this point we have treated all index sets involved in thevarious filters as abstract sets. We now make these index sets moreconcrete. In order to facilitate our description we consider all indexsets as defined locally around a given site � M! 1 � � . A diagram isgiven in Figure 5. The index of a given site is denoted

+nW � ��and all the neighboring vertices ( � �� with

�oW,��) needed in the

transform have indices ] , ^ , and _ respectively. To give some moreintuition to these index sets recall wavelets on the real line as inFigure 4. In that case the set

��0�qp,r

would consist of all integers,while � �%&

1�spa+

would contain the odd and��%&

1�tpu�

theeven integers. For vertex based schemes we may think of thesites

+vW � �� as always living on the midpoint of some parent

edge (these being the “odd” indices), while the endpoints of agiven edge form the “even” indices (

��), and their union��

1

gives the set of all indices. Foreach � the filters only range over some small neighborhood. Wewill refer to the elements in these neighborhoods by a local namingscheme (see Figure 5),

��. For example, the site �

lies in between the elements of�� "!

1 # ! 2 $ .For all vertex bases the unlifted scaling coefficients are simply

subsampled during analysis and upsampled during synthesis, whilethe wavelet coefficients involve some computation.%'&(')+*�,'-','.0/21�35467�8�9��

: :<;>= ? :� :<;2@ 1 = ?6 � �8�A�� : BC;>= � :� :<;2@ 1 = ��DFE ?"G'HJI ˜K ;>= ?"= � :<;2= ?L+*'&'M'NPOP,'-','.Q.0/>1�35467�8�9��

: :<;2@ 1 = ? :� :<;>= ?6 � �8�A�� : :<;2@ 1 = � :� BP;>= �� E ?"G'HJI ˜K ;>= ?"= � :<;2= ?

We now give the details of the wavelet coefficient computations.

Lazy: As mentioned above the Lazy wavelet does nothing butsubsampling. The resulting analysis and synthesis steps then be-come BP;>= � :

� :<;2@ 1 = � and :<;2@ 1 = � :� BP;2= ��R

respectively. The corresponding stencil encompasses no neighbors,i.e., the sums over ˜K ;>= ?+= � are empty.

Linear: This basic interpolatory form uses the stencil��S��"!

1 # ! 2 $ (see Figure 5) for analysis and synthesisBP;>= � :� :<;2@ 1 = ��D 1 T 2 � :<;>@ 1 = U 1 � :<;2@ 1 = U 2 :<;2@ 1 = � :� BC;>= �� 1 T 2 � :<;>= U 1 � :<;2= U 2 #

respectively. Note that this stencil does properly account for thegeometry provided that the � sites at level

�V�1 have equal geodetic

distance from the�"!

1 # ! 2 $ sites on their parent edge. Here ˜K ;2= U 1 = ��˜K ;>= U 2 = � � 1 T 2.

Quadratic: The stencil for this basis is given by��W��"!

1 # ! 2 #YX 1 #ZX 2 $ (see Figure 5) and exploits the degrees of freedomimplied to kill the functions [ 2, \ 2, and ] 2 (and by implication theconstant function [1]). Using the coordinates of the neighbors ofthe involved sites a small linear system results^__` 1 1 1 1a 2; > =U 1 a 2; > =U 2 a 2; > =b 1 a 2; > =b 2c 2; 2 =U 1 c 2; > =U 2 c 2; > =b 1 c 2; > =b 2d 2; 2 =U 1 d 2; 2 =U 2 d 2; > =b 1 d 2; > =b 2

e + ffg^__` ˜h ; > =U 1 = �

˜h ; > =U 2 = �˜h ; > =b 1 = �˜h ; > =b 2 = �

e + ffg �^__` 1a 2; 2 @ 1 = �c 2; 2 @ 1 = �d 2; 2 @ 1 = �

e + ffgSince [ 2 � \ 2 � ] 2 � 1 this system is singular (but solvable) andthe answer is chosen so as to minimize the

�2 norm of the resulting

filter coefficients. Note that this is an instance of dual lifting witheffective filters ˜K ;2= ?"= �i�ij ;2= ?"= �k� D ˜l ;>= � = ? .Butterfly: This is the only basis which uses other than immediateneighbors (all the sites

��denoted in Figure 5). Here ˜K U 1 � ˜K U 2 �

1 T 2, ˜K b 1 � ˜K b 2 � 1 T 8, and ˜Knm 1 � ˜Knm 2 � ˜Knm 3 � ˜Knm 4 �oD 1 T 16.It is inspired by a subdivision scheme of Dyn et al. [11] for theconstruction of smooth surfaces.

3.5 Lifting Vertex Bases

All of the above bases, Lazy, Linear, Quadratic, and Butterfly canbe lifted. In this section we use lifting to assure that the wavelet hasat least one vanishing moment. It does not improve the ability of thedual wavelet to annihilate more functions. Consequently the ability

of the bases to compress is not increased, but smaller error resultswhen using them for compression (see the example in Section 3.8and the results in Section 5). We propose wavelets of the formp ;>= �q�ir ;2@ 1 = ��D K ;>= U 1 = ��r ;>= U 1 D K ;2= U 2 = �sr ;2= U 2 R (6)

In words, we define the wavelet at the midpoint of an edge as a linearcombination of the scaling function at the midpoint (

�t�1 # � ) and

two scaling functions on the coarser level at the two endpoints ofthe parent edge (

� # ! 1 = 2). The weights K ;>= ?+= � are chosen so that theresulting wavelet has a vanishing integralK ;2= ?"= �k�vu ;2@ 1 = � T 2 u ;2= ? with

u ;2= ? ��wS2

r ;2= ?�xQy RDuring analysis lifting is a second phase (at each level

�) after theBP;2= � computation, while during synthesis it is a first step followed

by the regular synthesis step (Linear, Quadratic, or Butterfly as givenabove). The simplicity of the expressions demonstrates the powerof the lifting scheme. Any of the previous vertex basis waveletscan be lifted with the same expression. The integrals

u ;2= ? can beapproximated on the finest level and then recursively computed onthe coarser levels (using the refinement relations).%'&(')+*�,'-','.'.</>1�3546 � �8�A�� : z : ;>= U 1 �{� K ;2= U 1 = � B ;>= �:<;>= U 2 �{� K ;2= U 2 = � BC;>= �L+*'&'M'NPOP,'-','.</>1�3546 � �8�A�� : z : ;>= U 1 D|� K ;2= U 1 = � B ;>= �:<;>= U 2 D|� K ;2= U 2 = � BC;>= �

For the interpolating case in the previous section, the scalingfunction coefficients at each level are simply samples of the functionto be expanded (inner products with the ˜

r�} = ? ). In the lifted case thecoefficients are defined as the inner product of the function to beexpanded with the (new) dual scaling function. This dual scaling

Figure 6: Images of the graphs of all vertex based wavelets. On theleft is the scaling function (or unlifted wavelet) while the right showsthe lifted wavelet with 1 vanishing moment. From top to bottom:Linear, Quadratic, and Butterfly. Positive values are mapped to alinear red scale while negative values are shown in blue. The grayarea shows the support.

dual scalingprimal scaling

Figure 7: Example Haar scaling functions on a triangular subdivi-sion. On the left are primal functions each of height 1. On the rightare the biorthogonal duals each of height �� 1. Disjoint baseshave inner product of 0 while overlapping (coincident supports)lead to an inner product of 1. (For the sphere all triangles arespherical triangles.)

3

T1T0

T3

T2

children Bio-Haar 1 Bio-HaarBio-Haar 2

Figure 8: The Bio-Haar wavelets. Note that the heights of thefunctions are not drawn to scale.

function is only defined as the limit function of a non-stationarysubdivision scheme. The inner products at the finest level thereforeneed to be approximated with a quadrature formula, i.e., a linearcombination of function samples. In our implementation we use asimple one point quadrature formula at the finest level.

Figure 6 shows images of the graphs of all the vertex based �functions for the interpolating and lifted case.

3.6 The Generalized Haar Wavelets and Face Bases

Consider spherical triangles resulting from a geodesic sphere con-struction � �� S2 with �� (note that the face based � ��are not identical to the vertex based � �� defined earlier). Theysatisfy the following properties:

1. S2 �� "!#�%$ � �� and this union is disjoint, i.e., the � �� provide a simple cover of S2 for every � ,

2. for every � and � , � �� can be written as the union of 4 “child”triangles � �%& 1

� ' .Let �� be the spherical area of a triangle and define the scalingfunctions and dual scaling functions as

( �� *),+.-�/ 0 and ˜( �� 1�2�� %� � � 1 ),+.-�/ 043Here ),+ is the function whose value is 1 for 56� � and 0 otherwise.The fact that the scaling function and dual scaling function arebiorthogonal follows immediately from their disjoint support (seeFigure 7). Define the 7 � � L2 as

7 � � clos span 8 ( �� :9 �;�6� ��<=3The spaces 7 � then generate a multiresolution analysis of L2

� S2 � .Now fix a triangle � �%�?> . For the construction of the general-

ized Haar wavelets, we only need to consider the set of children� �%&1� ' @

0�1�2�3 of � �� > . We call these bases the Bio-Haar functions

(see Figure 8). The wavelets ( A � 1 B 2 B 3) are chosen as

� �%� CD� 2 ��("��& 1� CFEHGI�%&

1� CKJLGI�%&

1�0(,�%&

1�0� B

so that their integral vanishes. A set of semi-orthogonal dualwavelets is then given by

˜� �%� C � 1 J 2 � ˜( ��& 1� C E ˜( �%�?> �3

These bases are inspired by the construction of orthogonal Haarwavelets for general measures, see [14, 21] where it is shown thatthe Haar wavelets form an unconditional basis.

The Bio-Haar wavelets have only 1 vanishing moment, but us-ing the dual lifting scheme, we can build a new multiresolution

1

2

Bio-Haar

Bio-Haar

3Bio-Haar

Aunts and parent

Figure 9: Illustration of the dual lifting of the dual Bio-Haarwavelets. New dual wavelets can be constructed by taking linearcombinations of the original dual Bio-Haar wavelets and parentlevel dual scaling functions. Each such linear combination is signi-fied by a row. Solving for the necessary weights ˜M �� L� C requires thesolution to a small matrix problem whose right hand side encodesthe desired constraints.

analysis, in which the dual wavelet has more vanishing moments.Let � �%� �N@ 4

�5�6 be the neighboring triangles of � �%�?> (at level � ), and� C � 8POQB 4 B 5 B 6 < . The new dual wavelets are

˜� �%� C � 1 J 2 � ˜( �%& 1� C E ˜( �� > �"ESR �L�=�UT ˜M �� C ˜( �� 3

Note that this is a special case of Equation (4). The coefficients˜M �� L� C can now be chosen so that ˜� �� C has vanishing moments.Figure 9 illustrates this idea. In the left column are the three dualBio-Haar wavelets created before. The following four columnsshow the dual scaling functions over the parent and aunt triangles� �%� �N@U>�

4�5�6. Each row signifies one of the linear combinations.

Similarly to the Quadratic vertex basis we construct dually liftedBio-Haar wavelets which kill the functions 5 2, V 2, W 2, and thus 1.This leads to the equations

R �L� �UT ˜M �%� �L� CKX ˜( �%� � BZY\[ � 1 J 2 X ˜( �%& 1� C E ˜( �%�?> B�Y\[

with Y � 5 2 BV 2 BZW 2 B 1. The result is a 4 ] 4 singular (but solvable)matrix problem for each A � 1 B 2 B 3. The unknowns are the ˜M �%� �L� Cwith � � O4B 4 B 5 B 6 and the entries of the linear system are momentsof dual scaling functions. These can be computed recursively fromthe leaf level during analysis.

The Bio-Haar and lifted Bio-Haar transforms compute the scal-ing function coefficient during analysis at the parent triangle asa function of the scaling function coefficients at the children andpossibly the scaling function coefficients at the neighbors of theparent triangle (in the lifted case). The three wavelet coefficientsof the parent level are stored with the children �

1, � 2, and �3 for

convenience in the implementation. During synthesis the scalingfunction coefficient at the parent and the wavelet coefficients storedat children �

1, � 2, and �3 are used to compute scaling function

coefficients at the 4 children.As before, lifting is a second step during analysis and modifies

the wavelet coefficients. During synthesis lifting is a first stepbefore the inverse Bio-Haar transform is calculated.

^=_�`=aLbdc=e=c=f f�g�hji"kl A��6m �� : n �� C Eo�pR �� UT ˜M �%� �L� Crq �� s�b=_=t u�vQc=e=c f�g�hji"kl A��6m �� : n �� Cxw �pR �� UT ˜M �%� �L� Crq ��

3.7 Basis Properties

The lifting scheme provides us with the filter coefficients needed inthe implementation of the fast wavelet transform. To find the basisfunctions and dual basis functions that are associated with them, weuse the cascade algorithm. To synthesize a scaling function ��

0 � � 0

one simply initializes the coefficient � � 0 � �� 0 . The inverse

wavelet transform starting from level 0 with all wavelet coefficients�� with �� 0 set to zero then results in � � � � coefficients whichconverge to function values of ��

0 � � 0 as �� . In case thecascade algorithm converges in L2 for both primal and dual scalingfunctions, biorthogonal filters (as given by the lifting scheme) implybiorthogonal basis functions.

One of the fundamental questions is how properties, such asconvergence of the cascade algorithm, Riesz bounds, and smooth-ness, can be related back to properties of the filter sequences. Thisis a very hard question and at this moment no general answer isavailable to our knowledge. We thus have no mathematical proofthat the wavelets constructed form an unconditional basis exceptin the case of the Haar wavelets. A recent result addressing thesequestions was obtained by Dahmen [7]. In particular, it is shownthere which properties in addition to biorthogonality are needed toassure stable bases. Whether this result can be applied to the basesconstructed here needs to be studied in the future.

Regarding smoothness, we have some partial results. It is easyto see that the Haar wavelets are not continuous and that the Linearwavelets are. The original Butterfly subdivision scheme is guaran-teed to yield a � 1 limit function provided the connectivity of thevertices is at least 4. The modified Butterfly scheme that we useon the sphere, will also give � 1 limit functions, provided a locallysmooth ( � 1) map from the spherical triangulation to a planar tri-angulation exists. Unfortunately, the geodesic subdivision we usehere does not have this property. However, the resulting functionsappear visually smooth (see Figure 6). We are currently workingon new spherical triangulations which have the property that theButterfly scheme yields a globally � 1 function.

In principle, one can choose either the tetrahedron, octahedron, oricosahedron to start the geodesic sphere construction. Each of themhas a particular number of triangles on each level, and thereforeone of them might be more suited for a particular application orplatform. The octahedron is the best choice in case of functionsdefined on the hemisphere (cfr. BRDF). The icosahedron will leadto the least area imbalance of triangles on each level and thus to(visually) smoother basis functions.

3.8 An Example

We argued at the beginning of this section that a given waveletbasis can be made more performant by lifting. In the section oninterpolating bases we pointed out that a wavelet basis with Diracsfor duals and a primal wavelet, which does not have 1 vanishingmoment, unconditional convergence of the resulting series expan-sions cannot be insured anymore. We now give an example on thesphere which illustrates the numerical consequences of lifting.

Consider the function �� for � � ��"!#�$�!%� &'�)( S2.

This function is everywhere smooth except on the great circle �� 0, where its derivative has a discontinuity. Since it is largely smoothbut for a singularity at 0, it is ideally suited to exhibit problems inbases whose primal wavelet does not have a vanishing moment.Figure 10 shows the relative * 1 error as a function of the number ofcoefficients used in the synthesis stage. In order to satisfy the sameerror threshold the lifted basis requires only approximately 1 + 3 thenumber of coefficients compared to the unlifted basis.

1E+01 1E+02 1E+03 1E+04

number of coefficients

1E-04

1E-03

1E-02

1E-01

rela

tive

l1 e

rror

LinearLinear lifted

Figure 10: Relative * 1 error as a function of the number ofcoefficients for the example function ��

� � �� and (lifted)Linear wavelets. With the same number of coefficients the error issmaller by a factor of 3 or conversely a given error can be achievedwith about 1 + 3 the number of coefficients if the lifted basis is used.

4 Implementation

We have implemented all the described bases in an interactive appli-cation running on an SGI Irix workstation. The basic data structureis a forest of triangle quadtrees [10]. The root level starts with 4(tetrahedron), 8 (octahedron), or 20 (icosahedron) spherical trian-gles. These are recursively subdivided into 4 child triangles each.Naming edges after their opposite vertex, and children after the ver-tex they retain (the central child becomes , 0) leads to a consistentnaming scheme throughout the entire hierarchy. Neighbor findingis a simple -.� 1 � (expected cost) function using bit operations onedge and triangle names to guide pointer traversal [10]. A vertexis allocated once and any level which contains it carries pointers toit. Each vertex carries a single � and � slot for vertex bases, whileface bases carry a single � and � slot per spherical triangle. Ouractual implementation carries other data such as surface normalsand colors used for display, function values for error computations,and copies of all � and � values to facilitate experimentation. Theseare not necessary in a production system however.

Using a recursive data structure is more memory intensive (dueto the pointer overhead) than a flat, array based representation ofall coefficients as was used by LDW. However, using a recursivedata structure enables the use of adaptive subdivision and resultsin simple recursive procedures for analysis and synthesis and asubdivision oracle. For interactive applications it is straightforwardto select a level for display appropriate to the available graphicsperformance (polygons per second).

In the following subsections we address particular issues in theimplementation.

4.1 Restricted Quadtrees

In order to support lifted bases and those which require stencilsthat encompass some neighborhood the quadtrees produced needto satisfy a restriction criterion. For the Linear vertex bases (liftedand unlifted) and the Bio-Haar basis no restriction is required. ForQuadratic and lifted Bio-Haar bases no neighbor of a given face maybe off by more than 1 subdivision level (every child needs a properset of “aunts”). For the Butterfly basis a two-neighborhood mustnot be off by more than 1 subdivision level. These requirements areeasily enforced during the recursive subdivision. The fact that weonly need “aunts” (as opposed to “sisters”) for the lifting schemeallows us to have wavelets on adaptively subdivided hierarchies.This is a crucial departure from previous constructions, e.g., tree

Basis Analysis Synthesis Lifted Basis Analysis Synthesis

Linear 3.59 3.55 Linear 5.85 5.83

Quadratic 21.79 21.00 Quadratic 24.62 24.68

Butterfly 8.43 8.42 Butterfly 10.64 10.62

Bio-Haar 4.31 6.09 Bio-Haar 42.43 36.08

Table 2: Representative timings for wavelet transforms beginningwith 4 spherical triangles and expanding to level 9 (220 faces and219 � 2 vertices). All timings are given in seconds and measured onan SGI R4400 running at 150MHz. The initial setup time (allocatingand initializing all data structures) took 100 seconds.

wavelets employed by Gortler et al.[16] who also needed to supportadaptive subdivision.

4.2 Boundaries

In the case of a hemisphere (top 4 spherical triangles of an octahedralsubdivision), which is important for BRDF functions, the issuesassociated with the boundary need to be addressed. Lifting ofvertex bases is unchanged, but the Quadratic and Butterfly schemes(as well as the lifted Bio-Haar bases) need neighbors, which maynot exist at the boundary. This can be addressed by simply usinganother, further neighbor instead of the missing neighbor (across theboundary edge) to solve the associated matrix problem. It implicitlycorresponds to adapting filter coefficients close to the boundaryas done in interval constructions, see e.g. [5]. This constructionautomatically preserves the vanishing moment property even at theboundary. In the implementation of the Butterfly basis, we tooka different approach and chose in our implementation to simplyreflect any missing faces along the boundary.

4.3 Oracle

One of the main components in any wavelet based approximationis the oracle. The function of the oracle is to determine which co-efficients are important and need to be retained for a reconstructionwhich is to meet some error criterion. Our system can be driven intwo modes. The first selects a deepest level to which to expand allquadtrees. The storage requirements for this approach grow expo-nentially in the depth of the tree. For example our implementationcannot go deeper than 7 levels (starting from the tetrahedron) ona 32MB Indy class machine without paging. Creating full trees,however, allows for the examination of all coefficients throughoutthe hierarchies to in effect implement a perfect oracle. The secondmode builds sparse trees based on a deep refinement oracle. Inthis oracle quadtrees are built depth first exploring the expansionto some (possibly very deep) finest level. On the way out of therecursion a local �� is performed and any subtrees whosewavelet coefficients are all below a user supplied threshold aredeallocated. Once the sparse tree is built the restriction criterion isenforced and the (possibly lifted) analysis is run level wise.

The time complexity of this oracle is still exponential in thedepth of the tree, but the storage requirements are proportionalto the output size. With extra knowledge about the underlyingfunction more powerful oracles can be built whose time complexityis proportional to the output size as well.

4.4 Transform Cost

The cost of a wavelet transform is proportional to the total numberof coefficients, which grows by a factor of 4 for every level. Forexample, 9 levels of subdivision starting from 4 spherical trianglesresult in 220 coefficients (each of � and � ) for face bases and 219 �2 (each of � and � ) for vertex bases. The cost of analysis andsynthesis is proportional to the number of basis functions, whilethe constant of proportionality is a function of the stencil size.

Table 2 summarizes timings of wavelet transforms for all the newbases. The initial setup took 100 seconds and includes allocationand initialization of all data structures and evaluation of the � 9 � � .Since the latter is highly dependent on the evaluation cost of thefunction to be expanded we used the constant function 1 for thesetimings. None of the matrices which arise in the Quadratic, andBio-Haar bases (lifted and unlifted) was cached, thus the cost ofsolving the associated 4 � 4 matrices with a column pivoted QR (forQuadratic and lifted Bio-Haar) was incurred both during analysisand synthesis. If one is willing to cache the results of the matrixsolutions this cost could be amortized over multiple transforms.

We make three main observations about the timings: (A) Liftingof vertex bases adds only a small extra cost, which is almost entirelydue to the extra recursions; (B) the cost of the Butterfly basis is onlyapproximately twice the cost of the Linear basis even though thestencil is much larger; (C) solving the 4 � 4 systems implied byQuadratic and lifted Bio-Haar bases increases the cost by a factorof approximately 5 over the linear case (note that there are twice asmany coefficients for face bases as for vertex bases).

While the total cost of an entire transform is proportional to thenumber of basis functions, evaluating the resulting expansion ata point is proportional to the depth (log of the number of basisfunctions) of the tree times a constant dependent on the stencil size.The latter provides a great advantage over such bases as sphericalharmonics whose evaluation cost at a single point is proportional tothe total number of bases used.

5 Results

In this section we report on experiments with the compression of aplanetary topographic data set, a BRDF function, and illuminationof an anisotropic glossy sphere.

Most of these experiments involved some form of coefficientthresholding (in the oracle). In all cases this was performed asfollows. Since all our bases are normalized with respect to theL � norm, L2 thresholding against some user supplied threshold �becomes

if � �� supp �� "! �$#��%� � � : & 0 'Furthermore � is scaled by � max ��( �$) min ��( �*� for the given function( to make thresholding independent of the scale of ( .

5.1 Compression of Topographic Data

In this series of experiments we computed wavelet expansions oftopographic data over the entire earth. This function can be thoughtof as both a surface, and as a scalar valued function giving height(depth) for each point on a sphere. The original data, ETOPO5from the National Oceanographic and Atmospheric Administrationgives the elevation (depth) of the earth from sea level in meters at aresolution of 5 arc minutes at the equator. Due to the large size ofthis data set we first resampled it to 10 arc minutes resolution. Allexpansions were performed starting from the tetrahedron followedby subdivision to level 9.

Figure 11 shows the results of these experiments (left and mid-dle). After computing the coefficients of the respective expansionsat the finest level of the subdivision an analysis was performed.After this step all wavelet coefficients below a given threshold werezeroed and the function was reconstructed. The thresholds weresuccessively set to 2 +-, for ./& 0 #0'1'0'2# 17 resulting in the number ofcoefficients and relative 3 1 error plotted (left graph). The error wascomputed with a numerical quadrature one level below the finestsubdivision to insure an accurate error estimation. The results areplotted for all vertex and face bases (Linear, Quadratic, Butterfly,Bio-Haar, lifted and unlifted). We also computed 3 2 and 3 � errornorms and the resulting graphs (not shown) are essentially identical

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Figure 11: Relative�1 error as a function of the number of coefficients used during the reconstruction of the earth topographic data set (left

and middle) and BRDF function (right). The six vertex bases and two face bases perform essentially the same for the earth. On the left withfull expansion of the quadtrees to level 9 and thresholding. In the middle the results of the deep refinement oracle to level 10 with only asparse tree construction. The curves are identical validating the refinement strategy. On the right the results of deep refinement to level 9 forthe BRDF. Here the individual bases are clearly distinguished.

(although the��

error stays initially high before falling off due todeep canyon features). The plot reaches to about one quarter ofall coefficients. The observed regime is linear as one would expectfrom the bases used.

The most striking observation about these error graphs is the factthat all bases perform similar. This is due to the fact that the un-derlying function is non-smooth. Consequently smoother bases donot perform any better than less performant ones. However, whendrawing pictures of highly compressed versions of the data set thesmoother bases produce visually better pictures (see Figure 12).Depending on the allowed error the compression can be quite dra-matic. For example, 7 200 coefficients are sufficient to reach 7%error, while 119 000 are required to reach 2% error.

In a second set of experiments we used the deep refinementoracle (see Section 4.3) to explore the wavelet expansion to 10levels (potentially quadrupling the number of coefficients) withsuccessively smaller thresholds, once again plotting the resultingerror in the middle graph of Figure 11. The error as a functionof coefficients used is the same as the relationship found by theperfect oracle. This validates our deep refinement oracle strategy.Memory requirements of this approach are drastically reduced. Forexample, using a threshold of 2 � 9 during oracle driven refinementto level 10 resulted in 4 616 coefficients and consumed a total of27MB (including 10MB for the original data set). Lowering thethreshold to 2 � 10 yielded 10 287 coefficients and required 43MB(using the lifted Butterfly basis in both cases).

Finally Figure 12 shows some of the resulting adaptive data setsrendered with RenderMan using the Butterfly basis and a pseudocoloring, which maps elevation onto a piecewise linear color scale.Total runtime for oracle driven analysis and synthesis was 10 min-utes on an SGI R4400 at 150MHz.

5.1.1 Comparison with LDW

The earth data set allows for a limited comparison of our re-sults with those of LDW. They also compressed the ETOPO5 dataset using pseudo orthogonalized (over a 2 neighborhood) Linearwavelets defined over the octahedron. They subdivide to 9 levels(on a 128MB machine) which corresponds to twice as many coef-ficients as we used (on a 180MB machine), suggesting a storageoverhead of about 3 in our implementation. It is hard to comparethe quality of the bases without knowing the exact basis used orthe errors in the compressed reconstruction. However, LDW reportthe number of coefficients selected for a given threshold (741 for

Figure 12: Two views of the earth data set with 15 000 and 190 000coefficients respectively using the Butterfly basis with

�1 errors of

0 � 05 and 0 � 01 respectively. In the image on the left coastal re-gions are rather smoothed since they contain little height variation(England and the Netherlands are merged and the Baltic sea isdesertificated). However, such spiky features as the Cape VerdeIslands off the coast of Africa are clearly preserved.

0 � 02, 15 101 for 0 � 002, and 138 321 for 0 � 0005). Depending on thebasis used we generally select fewer coefficients (6 000 � 15 000 for0 � 002 and 28 000 � 65 000 for 0 � 0005). As timings they give 588seconds (on a 100 MHz R4000) for analysis which is significantlylonger than our smoothest basis (lifted Butterfly). Their recon-struction time ranges from 75 (741 coefficients) to 1 230 (138 058coefficients) seconds which is also significantly longer than ourtimes (see Table 2). We hypothesize that the timing and storagedifferences are largely due to their use of flat array based data struc-tures. These do not require as much memory, but they are morecompute intensive in the sparse polygonal reconstruction phase.

5.2 BRDF Compression

In this series of experiments we explore the potential for efficientlyrepresenting BRDF functions with spherical wavelets. BRDF func-tions can arise from measurements, simulation, or theoretical mod-els. Depending on the intended application different models maybe preferable. Expanding BRDF functions in terms of locally sup-ported hierarchical functions is particularly useful for wavelet basedfinite element illumination algorithms. It also has obvious appli-cations for simulation derived BRDF functions such as those ofWestin et al. [30] and Gondek et al. [15].

Figure 13: Graphs of adaptive oracle driven approximations of theSchlick BRDF with 19, 73, and 203 coefficients respectively (leftto right), using the lifted Butterfly basis. The associated thresholdswere 2

� 3, 2� 6, and 2

� 9 respectively, resulting in relative�1 errors

of 0 � 35, 0 � 065, and 0 � 015 respectively.

The domain of a complete BRDF is a hemisphere times a hemi-sphere. In our experiments we consider only a fixed incomingdirection and expand the resulting function over all outgoing direc-tions (single hemisphere). To facilitate the computation of errorswe used the BRDF model proposed by Schlick [23]. It is a simplePade approximant to a micro facet model with geometric shad-owing, a microfacet distribution function, but no Fresnel term. Ithas roughness ( �� 0 � 1 , where 0 is Dirac mirror reflection and1 perfectly diffuse) and anisotropy (�� 0 � 1 , where 0 is Diracstyle anisotropy, and 1 perfect isotropy) parameters. To improvethe numerical properties of the BRDF we followed the suggestionof Westin et al. [30] and expanded cos �� 0 �� .

In the experiments we used all 8 bases but specialized to thehemisphere. The parameters were ��! �" 3, � � 0 � 05, and � 1.The results are summarized in Figure 11 (rightmost graph). It showsthe relative

�1 error as a function of the number of coefficients used.

This time we can clearly see how the various bases differentiatethemselves in terms of their ability to represent the function withinsome error bound with a given budget of coefficients. We makeseveral observations

- all lifted bases perform better than their unlifted versions,confirming our assertion that lifted bases are more performant;

- increasing smoothness in the bases (Butterfly) is more im-portant than increasing the number of vanishing moments(Quadratic);

- dual lifting to increase dual vanishing moments increases com-pression ability dramatically (Bio-Haar and lifted Bio-Haar);

- overall the face based schemes do not perform as well as thevertex based schemes.

Figure 13 shows images of the graphs of some of the expansions.These used the lifted Butterfly basis with an adaptive refinementoracle which explored the expansion to level 9 (i.e., it examined219 coefficients). The final number of coefficients and associatedrelative

�1 errors were (left to right) 19 coefficients (

�1� 0 � 35), 73

coefficients (�1� 0 � 065), and 203 coefficients (

�1� 0 � 015). Total

runtime was 170 seconds on an SGI R4400 at 150MHz.

5.3 Illumination

To explore the potential of these bases for global illumination algo-rithms we performed a simple simulation computing the radianceover a glossy, anisotropic sphere due to two area light sources. Weemphasize that this is not a solution involving any multiple reflec-tions, but it serves as a simple example to show the potential ofthese bases for hierarchical illumination algorithms. It also servesas an example of applying a finite element approach to a curvedobject (sphere) without polygonalizing it.

Figure 14 shows the results of this simulation. We used the liftedButterfly basis and the BRDF model of Schlick with � � 0 � 05, � 0 � 05, and an additive diffuse component of 0 � 005. Two arealight sources illuminate the red sphere. Note the fine detail in thepinched off region in the center of the “hot” spot and also at the

Figure 14: Results of an illumination simulation using the liftedButterfly basis. A red, glossy, anisotropic sphere is illuminatedby 2 area light sources. Left: solution with 2 000 coefficients(�1� 0 � 017). Right: solution with 5 000 coefficients (

�1� 0 � 0035)

north pole where all “grooves” converge.

6 Conclusions and Future Directions

In this paper we have introduced two new families of wavelets on thesphere. One family is based on interpolating scaling functions andone on the generalized Haar wavelets. They employ a generalizationof multiresolution analysis to arbitrary surfaces and can be derivedin a straightforward manner from the trivial multiresolution analysiswith the lifting scheme. The resulting algorithms are simple andefficient. We reported on the application of these bases to thecompression of earth data sets, BRDF functions and illuminationcomputations and showed their potential for these applications. Wefound that

- for smooth functions the lifted bases perform significantlybetter than the unlifted bases;

- increasing the dual vanishing moments leads to better com-pression;

- smoother bases, even with only one vanishing moment, tendto perform better for smooth functions;

- our constructions allow non-equally subdivided triangulationsof the sphere.

We believe that many applications can benefit from these waveletbases. For example, using their localization properties a numberof spherical image processing algorithms, such as local smoothingand enhancement, can be realized in a straightforward and efficientway [25].

While we limited our examination to the sphere, the constructionpresented here can be applied to other surfaces. In the case ofthe sphere enforcing vanishing polynomial moments was naturalbecause of their connection with spherical harmonics. In the case ofa general, potentially non-smooth (Lipschitz) surface, polynomialmoments do not necessarily make much sense. Therefore, onemight want to work with local maps from the surface to the tangentplane and enforce vanishing moment conditions in this plane.

Future research includes- the generalization to arbitrary surfaces,

- the incorporation of smoother ( # 2) subdivision schemes asrecently introduced by Dyn et al. (personal communication,1995),

- the use of these bases in applications such as the solution ofdifferential and integral equations on the sphere as needed in,e.g., illumination or climate modeling.

Acknowledgments

The first author was supported by DEPSCoR Grant (DoD-ONR)N00014-94-1-1163. The second author was supported by NSFEPSCoR Grant EHR 9108772 and DARPA Grant AFOSR F49620-93-1-0083. He is also Senior Research Assistant of the NationalFund of Scientific Research Belgium (NFWO). Other support camefrom Pixar Inc. We would also like to thank Princeton Univer-sity and the GMD, Germany, for generous access to computingresources. Help with geometric data structures was provided byDavid Dobkin. Finally, the comments of the referees were veryhelpful in revising the paper.

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