Moved here

**Abstract:**
We examine the central issues of invertibility, stability,
artifacts, and frequency-domain characteristics in the construction
of non-linear analogs of the wavelet transform. The lifting
framework for wavelet construction motivates our analysis and
provides new insight into the problem. We describe a new type of
non-linearity for use in constructing non-linear transforms: a set
of linear predictors that are chosen adaptively using a non-linear
selection function. We also describe how earlier families of
non-linear filter banks can be extended through the use of
prediction functions operating on a causal neighborhood. We present
preliminary results for a synthetic test image.

**Status:**
Proceedings of the 31st Asilomar Conference on Signals, Systems,
and Computers, Volume 1, pp 662-667, 1997.

**BiBTeX entry:**

@inproceedings{cdsb:asil97, author = {R. Claypoole and G. Davis and W. Sweldens and R. Baraniuk}, title = {Nonlinear wavelet transform for image coding}, booktitle = {Proceedings of the 31st Asilomar Conference on Signals, Systems, and Computers, Volume 1}, pages = {662-667}, year = {1997} }

**Abstract:**
Invertible wavelet transforms that map integers to integers are
important for lossless representations. In this paper, we present an
approach to build integer to integer wavelet transforms based up on
the idea of factoring wavelet transforms into lifting steps. This
allows the construction of an integer version of every wavelet
transform. We demonstrate the use of these transforms in lossless
image compression.

**Status:**
International Conference on Image Processing (ICIP), Vol. I, pp. 596-599.

**Note:**
This is a short conference version of the paper
``Wavelet Transforms that Map Integers to
Integers.''

**BiBTeX entry:**

@inproceedings{cdsy:icip97, author = {R. Calderbank and I. Daubechies and W. Sweldens and B.-L. Yeo}, title = {Losless Image Compression using Integer to Integer Wavelet Transforms}, booktitle = {International Conference on Image Processing (ICIP), Vol. I}, publisher = {IEEE Press}, pages = {596-599}, year = {1997} }

**Abstract:**
We study the smoothness of the limit function for one dimensional
unequally spaced interpolating subdivision schemes. The new grid
points introduced at every level can lie in irregularly spaced
locations between old, adjacent grid points and not only
midway as is usually the case. For the natural generalization of
the four point scheme introduced by Dubuc and Dyn, Levin, and
Gregory, we show that, under some geometric restrictions, the limit
function is always C^{1}; under slightly stronger restrictions
we show that the limit function is almost C^{2}, the same regularity
as in the regularly spaced case.

**Status:**
Preprint, Department of Mathematics, Princeton University, 1997.
(to appear in Constructive Approximation)

**BiBTeX entry:**

@article{dgs:regir, author = {I. Daubechies and I. Guskov and W. Sweldens}, title = {Regularity of Irregular Subdivision}, journal = {Constructive Approximation}, volume = 15, pages = {381-426}, year = 1999 }

**Applet:** Igor wrote an
applet
to illustrate irregular interpolating subdivision.

**Abstract:**
We describe a multiresolution representation for meshes based on
subdivision. Subdivision is a natural extension of the existing
patch-based surface representations. At the same time subdivision
algorithms can be viewed as operating directly on polygonal meshes,
which makes them a useful tool for mesh manipulation. Combination of
subdivision and smoothing algorithms of Taubin allows us to construct
a set of algorithms for interactive multiresolution editing of complex
meshes of arbitrary topology. Simplicity of the essential algorithms
for refinement and coarsification allows to make them local and
adaptive, considerably improving their efficiency. We have built a
scalable interactive multiresolution editing system based on such
algorithms.

**Status:**
Computer Graphics Proceedings (SIGGRAPH 97), pp 259-268, 1997.

**BiBTeX entry:**

@article{zss:sig97, author = {D. Zorin and P. Schr{\"o}der and W. Sweldens}, title = {Interactive Multiresolution Mesh Editing}, journal = {Computer Graphics Proceedings (SIGGRAPH 96)}, publisher = {ACM Siggraph}, pages = {259-269}, year = {1997}, }

Compressed PostScript with images (3.4Mb),

PDF with images (5.0Mb).

Moved here

**Abstract:** Invertible wavelet transforms that map integers to
integers have important applications in lossless coding. In this
paper we present two approaches to build integer to integer wavelet
transforms. The first approach is to adapt the precoder of Laroia
et al., which is used in information transmission; we combine it
with expansion factors for the high and low pass band in subband
filtering. The second approach builds upon the idea of factoring
wavelet transforms into so-called lifting steps. This allows the
construction of an integer version of every wavelet transform.
Finally, we use these approaches in a lossless image coder and
compare the results to the literature.

**Status:**
Applied and Computational Harmonic Analysis (ACHA), Vol. 5, Nr. 3,
pp. 332-369, 1998.

**BiBTeX entry:**

@article{cdsy:integer, author = {R. Calderbank and I. Daubechies and W. Sweldens and B.-L. Yeo}, title = {Wavelet transforms that map integers to integers}, journal = {Appl. Comput. Harmon. Anal.}, volume = 5, number = 3, pages = {332-369}, year = 1998 }

**Abstract:**
We present LIFTPACK: A software package written in C for fast
calculation of 2D biorthogonal wavelet transforms using the lifting
scheme. The lifting scheme is a new approach for the construction
of *biorthogonal wavelets* entirely in the spatial domain,
i.e., independent of the Fourier Transform. Constructing wavelets
using lifting consists of three simple phases: the first step or
Lazy wavelet *splits* the data into two subsets, even and
odd, the second step calculates the wavelet coefficients (high
pass) as the failure to *predict* the odd set based on the
even, and finally the third step *updates* the even set using
the wavelet coefficients to compute the scaling function
coefficients (low pass). The predict phase ensures polynomial
cancelation in the high pass (vanishing moments of the dual
wavelet) and the update phase ensures preservation of moments in
the low pass (vanishing moments of the primal wavelet). By varying
the order, an entire family of transforms can be built. The
lifting scheme ensures fast calculation of the forward and inverse
wavelet transforms that only involve FIR filters. The transform
works for images of arbitrary size with correct treatment of the
boundaries. Also, all computations can be done in-place.

**Status:**
In M. Unser, A. Aldroubi, and A. F. Laine, editors,
Wavelet Applications in Signal and Image
Processing IV, pp. 396-408, Proc. SPIE 2825, 1996.

**BiBTeX entry:**

@inproceedings{fps:spie96, author = {G. Fern\'{a}ndez and S. Periaswamy and Wim Sweldens}, title = {{LIFTPACK}: {A} software package for wavelet transforms using lifting}, booktitle = {Wavelet Applications in Signal and Image Processing IV}, editor = {M. Unser and A. Aldroubi and A. F. Laine}, publisher = {Proc.\ SPIE~2825}, pages = {396-408}, year = {1996} }

**Abstract:**
Subdivision is a powerful paradigm for the generation of surfaces of
arbitrary topology. Given an initial triangular mesh the goal is to
produce a smooth and visually pleasing surface, whose shape is
controlled by the initial mesh. Of particular interest are
interpolating schemes since they match the original data exactly,
and are crucial for fast multiresolution and wavelet techniques.
Dyn, Gregory, and Levin introduced the Butterfly
scheme, which is known to yield C1 surfaces in
the topologically regular setting. Unfortunately it exhibits various
degeneracies in case of an irregular topology, leading to
undesirable artifacts. We examine these failures and derive an
improved scheme, which retains the simplicity of the Butterfly
scheme, is interpolating, and results in smoother surfaces.

**Status:**
Computer Graphics Proceedings (SIGGRAPH 96), pp 189-192, 1996.

**BiBTeX entry:**

@article{zss:sig96, author = {D. Zorin and P. Schr{\"o}der and W. Sweldens}, title = {Interpolating Subdivision for Meshes with Arbitrary Topology}, journal = {Computer Graphics Proceedings (SIGGRAPH 96)}, publisher = {ACM Siggraph}, pages = {189-192}, year = {1996}, }

Compressed PostScript with images (377Kb),

Compressed PostScript without images (25Kb),

PostScript without images (74Kb),

PDF with images (82Kb).

**Abstract:**
In this concluding article, we want to look ahead and see what the
future can bring to wavelet research. We try to find a common
denominator for ``wavelets'' and identify promising research
directions and challenging problems.

**Status:**
Proc. of the IEEE, vol. 84, nr. 4, pp. 680-685, 1996.

**BiBTeX entry:**

@article {swe:future, author = {W. Sweldens}, title = {Wavelets: {W}hat Next?}, journal = {Proc. IEEE}, volume = 84, number = 4, pages = {680-685}, year = {1996} }

**Abstract:**
In this paper we present the basic idea behind the lifting scheme, a
new construction of biorthogonal wavelets which does not use the
Fourier transform. In contrast with earlier papers we introduce
lifting purely from a wavelet transform point of view and only
consider the wavelet basis functions in a later stage. We show how
lifting leads to a faster, fully in-place implementation of the
wavelet transform. Moreover, it can be used in the construction of
second generation wavelets, wavelets that are not necessarily
translates and dilates of one function. A typical example of the
latter are wavelets on the sphere.

**Status:**
In A. F. Laine and M. Unser, editors,
Wavelet Applications in Signal and Image
Processing III, pp. 68-79, Proc. SPIE 2569, 1995.

**BiBTeX entry:**

@inproceedings {swe:spie95, author = {W. Sweldens}, title = {The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions}, booktitle = {Wavelet Applications in Signal and Image Processing III}, editor = {A. F. Laine and M. Unser}, publisher = {Proc.\ SPIE~2569}, year = {1995}, pages = {68-79} }

**Abstract:**
In this paper, we give a brief introductory tour to the lifting scheme,
an new method to construct wavelets. We show its advantages over
classical constructions and give pointers to the literature.

**Status:**
Zeitschrift für Angewandte Mathematik und Mechanik, vol. 76
(Suppl. 2), pp. 41-44, 1996.

**BiBTeX entry:**

@article {swe:iciam95, title = {Wavelets and the lifting scheme: {A} 5 minute tour}, author = {W. Sweldens}, journal = {Z. Angew. Math. Mech.}, volume = {76 (Suppl. 2)}, pages = {41-44}, year = 1996 }

**Abstract:**
We present the lifting scheme, a simple construction of second
generation wavelets, wavelets that are not necessarily translates and
dilates of one fixed function. Such wavelets can be adapted to
intervals, domains, surfaces, weights, and irregular samples. We show
how the lifting scheme leads to a faster, in-place calculation of the
wavelet transform. Several examples are included.

**Status:**
Siam J. Math. Anal, Vol. 29, Nr. 2, pp 511-546, 1997.

**BiBTeX entry:**

@article {swe:lift2, author = {W. Sweldens}, title = {The lifting scheme: {A} construction of second generation wavelets}, journal = {SIAM J. Math. Anal.}, number = 2, volume = 29, pages = {511-546}, year = {1997} }

**Abstract:**
Wavelets are a powerful tool for planar image processing. The
resulting algorithms are straightforward, fast, and efficient. With
the recently developed spherical wavelets this framework can be
transposed to spherical textures. We describe a class of processing
operators which are diagonal in the wavelet basis and which can be
used for smoothing and enhancement. Since the wavelets (filters) are
local in space and frequency, complex localized constraints and
spatially varying characteristics can be incorporated easily. Examples
from environment mapping and the manipulation of topography/bathymetry
data are given.

**Status:**
In P. Hanrahan and W. Purgathofer, editors,
Rendering Techniques '95, pp. 252-263, Springer Verlag,
Wien, New York, 1995,

**BiBTeX entry:**

@incollection {sch-swe:env, author = {P. Schr\"oder and W. Sweldens}, title = {Spherical wavelets: {T}exture processing}, booktitle = {Rendering Techniques '95}, editor = {P. Hanrahan and W. Purgathofer}, publisher = {Springer Verlag}, address = {Wien, New York}, month = {August}, year = 1995, pp = {252-263} }

PostScript without images (431Kb),

Compressed PostScript without images (79Kb)

Compressed PostScript with images (558Kb)

PDF with images (287Kb),

Row 1 Column 1 Original Bar (JPEG 44Kb)

Row 1 Column 2 Bar enhanced 1 (JPEG 88Kb)

Row 1 Column 3 Bar enhanced 2 (JPEG 119Kb)

Row 2 Column 1 Bar blurred 1 (JPEG 21Kb)

Row 2 Column 2 Bar blurred 2 (JPEG 13Kb)

Row 2 Column 3 Bar blurred 3 (JPEG 11Kb)

Row 3 Column 1 Butterfly spherical wavelet (JPEG 16Kb)

Row 3 Column 2 Coastlines locations (JPEG 44Kb)

Row 3 Column 3 Coastline wavelet coefficient (JPEG 65Kb)

Row 4 Column 1 Earth (JPEG 36Kb)

Row 4 Column 2 Earth approximated (JPEG 30Kb)

Row 4 Column 3 Soccer trophy (JPEG 34Kb)

**Abstract:**
Given a complete separable sigma-finite measure space (X,Sigma,mu) and
nested partitions of X, we construct unbalanced Haar-like wavelets on
X that form an unconditional basis for Lp(X,Sigma,mu) where 1 < p <
infinity. Our construction and proofs build upon ideas of Burkholder
and Mitrea. We show that if (X,Sigma,mu) is not purely atomic, then
the unconditional basis constant of our basis is (max(p,q)-1). We
derive a fast algorithm to compute the coefficients.

**Status:**
J. Fourier Anal. Appl., Vol. 3, Nr. 4, pp. 457-474, 1997.

**BiBTeX entry:**

@article {gir-swe:haar, author = {M. Girardi and W. Sweldens}, title = {A new class of unbalanced {H}aar wavelets that form an unconditional basis for ${L_p}$ on general measure spaces}, journal = {J. Fourier Anal. Appl.}, volume = 3, number = 4, page = {457-474}, year = 1997 }

**Abstract:**
Wavelets have proven to be powerful bases for use in numerical
analysis and signal processing. Their power lies in the fact that
they only require a small number of coefficients to represent general
functions and large data sets accurately. This allows compression and
efficient computations. Classical constructions have been limited to
simple domains such as intervals and rectangles. In this paper we
present a wavelet construction for scalar functions defined on the
sphere. We show how biorthogonal wavelets with custom properties can
be constructed with the lifting scheme. The bases are extremely easy
to implement and allofw fully adaptive subdivisions. We give examples
of functions defined on the sphere, such as topographic data,
bi-directional reflection distribution functions, and illumination,
and show how they can be efficiently represented with spherical
wavelets.

**Status:**
Computer Graphics Proceedings (SIGGRAPH 95), pp. 161-172, 1995

**BiBTeX entry:**

@article{sch-swe:sphere, author = {Peter Schr{\"o}der and Wim Sweldens}, title = {Spherical Wavelets: {E}fficiently Representing Functions on the Sphere}, journal = {Computer Graphics Proceedings (SIGGRAPH 95)}, year = 1995, publisher = {ACM Siggraph}, pages = {161-172} }

Linear scaling function (JPEG 20Kb)

Linear wavelet (JPEG 23Kb)

Quadratic scaling function (JPEG 26Kb)

Quadratic wavelet (JPEG 29Kb)

Butterfly scaling function (JPEG 26Kb)

Butterfly wavelet (JPEG 22Kb)

BRDF with 19 wavelets (JPEG 17Kb)

BRDF with 73 wavelets (JPEG 16Kb)

BRDF with 203 wavelets (JPEG 16Kb)

Earth coarse (JPEG 81Kb)

Earth fine (JPEG 101Kb)

Glossy sphere with 2000 wavelets (JPEG 40Kb)

Glossy sphere with 5000 wavelets (JPEG 40Kb)

**Abstract:**
We present the lifting scheme, a new idea of constructing compactly
supported wavelets with compactly supported duals. The lifting scheme
provides a simple relationship between all multiresolution analyses
with the same scaling function. It isolates the degrees of freedom
remaining after fixing the biorthogonality relations. Then one has
full control over these degrees of freedom to custom-design the
wavelet for a particular application. It also leads to a faster
implementation of the fast wavelet transform. We illustrate the use
of the lifting scheme in the construction of wavelets with
interpolating scaling functions.

**Status:**
Appl. Comput. Harmon. Anal, vol. 3, nr. 2, pp. 186-200, 1996.

**BiBTeX entry:**

@article{swe:lift1, author = {W. Sweldens}, title = {The lifting scheme: {A} custom-design construction of biorthogonal wavelets}, journal = {Appl. Comput. Harmon. Anal.}, volume = 3, number = 2, pages = {186-200}, year = 1996 }

**Abstract:**
In this paper we discuss smooth local trigonometric bases. We present
two generalizations of the orthogonal basis of Malvar and
Coifman-Meyer: biorthogonal and equal parity bases. These allow
natural representations of constant and, sometimes, linear components.
We study and compare their approximation properties and applicability
in data compression. This is illustrated with numerical examples.

**Status:**
J. Fourier Anal. Appl., vol. 2, nr. 2, pp. 109-103, 1995.

**BiBTeX entry:**

@article {jaw-swe:bio, author = {B. Jawerth and W. Sweldens}, title = {Biorthogonal smooth local trigonometric bases}, journal = {J. Fourier Anal. Appl.}, volume = 2, number = 2, pages = {109-133}, year = 1995 }

**Abstract:**
In this paper we show how to construct wavelets adapted to a weighted
inner product.

**Status:**
Proceedings of the 14th Imacs World Congress.

**BiBTeX entry:**
**Files:**
PostScript (147Kb) or
Compressed PostScript (40Kb) or
PDF (142Kb).

**Abstract:**
We discuss smooth local trigonometric bases and their applications to
signal compression. In image compression, these bases can reduce the
blocking effect that occurs in JPEG. We present and compare two
generalizations of the original construction of Malvar, Coifman and
Meyer: biorthogonal and equal parity bases. These have the advantage
that constant and linear components, respectively, can be represented
efficiently. We show how they reduce blocking effects and improve the
signal to noise ratio.

**Status:**
Optical Engineering, vol. 33, nr. 7, pp. 2125-2135, 1994.

**BiBTeX entry:**

@article {jaw-liu-swe:signal, author = {B. Jawerth and Y. Liu and W. Sweldens}, title = {Signal compression with smooth local trigonometric bases}, journal = {Optical Engineering}, volume = 33, number = 7, year = 1994, pages = {2125-2135} }

**Abstract:**
In this paper we present several techniques to calculate
the wavelet coefficients of a function from its samples.
Interpolation, quadrature formulae and filtering methods are
discussed and compared.

**Status:**
1993 Proceedings of the Statistical Computing Section, pp. 20-29,
American Statistical Association, 1993.

**BiBTeX entry:**

@inproceedings {swe-pie:sanf, author = {W. Sweldens and R. Piessens}, title = {Wavelet Sampling Techniques}, booktitle = {1993 Proceedings of the Statistical Computing Section}, publisher = {American Statistical Association}, pages = {20-29}, year = 1993 }

**Abstract:**
In this paper we show how wavelets can be used for data
segmentation. The basic idea is to split the data into smooth segments
that can be compressed separately. A fast algorithm that uses
wavelets on closed sets and wavelet probing is presented.

**Status:**
In A. F. Laine, editor,
Wavelet Applications in Signal and Image Processing,
pp. 266-276, Proc. SPIE 2034, 1993.

**BiBTeX entry:**

@inproceedings {den-jaw-pet-swe:segment, author = {B. Deng and B. Jawerth and G. Peters and W. Sweldens}, title = {Wavelet probing for compression based segmentation}, booktitle = {Wavelet Applications in Signal and Image Processing}, editor = {A. F. Laine}, publisher = {Proc.\ SPIE~2034}, year = {1993}, pages = {266-276} }

**Abstract:**
We present ideas on how to use wavelets in the solution of boundary
value ordinary differential equations. Rather than using classical
wavelets, we adapt their construction so that they become
(bi)orthogonal with respect to the inner product defined by the
operator. The stiffness matrix in a Galerkin method then becomes
diagonal and can thus be trivially inverted. We show how one can
construct an O(N) algorithm for various constant and variable
coefficient operators.

**Status:**
In N. D. Melson et al., editors,
Sixth Copper Mountain Conference on Multigrid Methods,
NASA Conference Publication 3224, pp. 259-273, 1993.

**BiBTeX entry:**

@inproceedings{jaw-swe:copper, author = {B. Jawerth and W. Sweldens}, title = {Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations}, editor = {N. D. Melson and T. A. Manteuffel and S. F. McCormick}, booktitle = {Sixth Copper Mountain Conference on Multigrid Methods}, publisher = {NASA Conference Publication 3224}, pages = {259-273}, year = {1993} }

**Authors:** Wim Sweldens and Robert Piessens

**Abstract:**
This paper deals with asymptotic error expansions of orthogonal
wavelet approximations of smooth functions. Two formulae are derived
and compared. As already known, the error decays as O(2^(-jN))$ where
j is the multiresolution level and $N$ is the number of vanishing
wavelet moments. It is shown that the most significant term of the
error expansion is proportional to the N-th derivative of the function
multiplied with an oscillating function. This result is used to
derive asymptotic interpolating properties of the wavelet
approximation. Also a numerical extrapolation scheme based on the
multiresolution analysis is presented.

**Status:**
Numer. Math., vol. 68, nr. 3, pp. 377-401, 1994.

**BiBTeX entry:**

@article {swe-pie:error2, author = {W. Sweldens and R. Piessens}, title = {Asymptotic error expansions of wavelet approximations of smooth functions {\rm {I}{I}}}, journal = {Numer. Math.}, volume = {68}, number = 3, pages = {377-401}, year = 1994, }

**Abstract:**
In this paper we present an overview of wavelet based multiresolution
analyses. First, we briefly discuss the continuous wavelet transform
in its simplest form. Then, we give the definition of a
multiresolution analysis and show how wavelets fit into it. We take a
closer look at orthogonal, biorthogonal and semiorthogonal wavelets.
The fast wavelet transform, wavelets on an interval, multidimensional
wavelets and wavelet packets are discussed. Several examples of
wavelet families are introduced and compared. Finally, the essentials
of two major applications are outlined: data compression and
compression of linear operators.

**Status:**
SIAM Rev., vol.36, nr.3, pp.377-412, 1994.

**BiBTeX entry:**

@article {jaw-swe:overview, author = {B. Jawerth and W. Sweldens}, title = {An overview of wavelet based multiresolution analyses}, journal = SIAM Rev., volume = 36, number = 3, pages = {377-412}, year = 1994 }

**Abstract:**
In many applications concerning wavelets, inner products of functions
f(x) with wavelets and scaling functions have to be calculated. This
paper involves the calculation of these inner products from function
evaluations of f(x). Firstly, one point quadrature formulae are
presented. Their accuracy is compared for different classes of
wavelets. Therefore the relationship between the scaling function
phi(x), its values at the integers and the scaling parameters h_k is
investigated. Secondly, multiple point quadrature formulae are
constructed. A method to solve the nonlinear system coming from this
construction is presented. Since the construction of multiple point
formulae using monomials is ill-conditioned, a modified,
well-conditioned construction using Chebyshev polynomials is
presented.

**Status:**
SIAM J. Num. Anal., vol. 31, nr. 4, pp. 2140-2164, 1994.

**BiBTeX entry:**

@article {swe-pie:quaderr, author = {W. Sweldens and R. Piessens}, title = {Quadrature Formulae and Asymptotic Error Expansions for wavelet approximations of smooth functions}, journal = SIAM J. Numer. Anal., volume = 31, number = 4, pages = {1240-1264}, year = {1994} }

Last modified: Mon Jan 5 15:08:01 EST 1998