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The Journal of Fourier Analysis and Applications

Volume 4, Issue 3, 1998

Factor ing Wavelet Transforms into Lifting Steps

lngrid Daubechies and Wim Sweldens

C o m m u n i c a t e d by John J. Benede t to

Research Tutorial

ABSTRACT. This article is essentially tutorial in nature. We show how any discrete wavelet transform or

two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula SL(n; R[Z, z-I]) = E(n; R[z, z-l])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.

1. Introduct ion

Over the last decade several constructions of compactly supported wavelets originated both from mathematical analysis and the signal processing community. The roots of critically sampled wavelet transforms are actually older than the word "wavelet" and go back to the context of sub- band filters, or more precisely quadrature mirror filters [35, 36, 42, 50, 51, 52, 53, 55, 57, 59]. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions [13, 18, 21, 22, 34]. In the mid-eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer pro- vided the connection between subband filters and wavelets [30, 31, 34]; this led to new constructions, such as the smooth orthogonal and compactly supported wavelets [ 16]. Later many generalizations

Math Subject Classifications. 42C15, 42C05, 19-02. Keywords and Phrases. Wavelet, lifting, elementary matrix, Euclidean algorithm, Laurent polynomial. Acknowledgements and Notes. Page 264.

1998 Birkh~iuser Boston. All righLs reserved ISSN 1069-5869

248 Ingrid Daubechies and Wim Sweldens

to the biorthogonal or semiorthogonal (pre-wavelet) case were introduced. Biorthogonality allows the construction of symmetric wavelets and thus linear phase filters. Examples are the construc- tion of semiorthogonal spline wavelets [1, 8, I0, 11, 49], fully biorthogonal compactly supported wavelets [12, 56], and recursive filter banks [25].

Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building blocks are known. One of these is lifting. The original motivation for developing lifting was to build second generation wavelets, i.e., wavelets adapted to situations that do not allow translation and dilation like non-Euclidean spaces. First generation wavelets are all translates and dilates of one or a few basic shapes; the Fourier transform is then the crucial tool for wavelet construction. A construction using lifting, on the contrary, is entirely spatial and therefore ideally suited for building second generation wavelets when Fourier techniques are no longer available. When restricted to the translation and dilation invariant case, or the "first generation," lifting comes down to well-known ladder type structures and certain factoring algorithms. In the next few paragraphs, we explain lifting and show how it provides a spatial construction and allows for second generation wavelets; later we focus on the first generation case and the connections with factoring schemes.

The basic idea of wavelet transforms is to exploit the correlation structure present in most real life signals to build a sparse approximation. The correlation structure is typically local in space (time) and frequency; neighboring samples and frequencies are more correlated than ones that are far apart. Traditional wavelet constructions use the Fourier transform to build the space-frequency localization. However, as the following simple example shows, this can also be done in the spatial domain.

Consider a signal x = (Xk)k~Z with Xk ~ R. Let us split it in two disjoint sets which are called the polyphase components: the even indexed samples Xe = (X2k)k~Z, or "evens" for short, and the odd indexed samples xo = (X2k+l)k~Z, or "odds." Typically these two sets are closely correlated. Thus it is only natural that given one set, e.g., the even, one can build a good predictor P for the other set. Of course the predictor need not be exact, so we need to record the difference or detail d:

d = x e - P ( x o ) 9

Given the detail d and the odd, we can immediately recover the even as

Xe = P (x,,) + d .

I f P is a good predictor, then d approximately will be a sparse set; in other words, we expect the first order entropy to be smaller for d than for xo. Let us look at a simple example. An easy predictor for an odd sample x2k+l is simply the average of its two even neighbors; the detail coefficient then is

dk = x2k+l - (x2k + x 2 k + l ) / 2 .

From this we see that if the original signal is locally linear, the detail coefficient is zero. The operation of computing a prediction and recording the detail we will call a lifting step. The idea of retaining d rather than Xo is well known and forms the basis of so-called DPCM methods [26, 27]. This idea connects naturally with wavelets as follows. The prediction steps can take care of some of the spatial correlation, but for wavelets we also want to get some separation in the frequency domain. Right now we have a transform from (Xe, xo) to (xe, d). The frequency separation is poor since xe is obtained by simply subsampling so that serious aliasing occurs. In particular, the running average of the Xe is not the same as that of the original samples x. To correct this, we propose a second lifting step, which replaces the evens with smoothed values s with the use of an update operator U applied to the details:

s = Xe + U(d) .

Again this step is trivially invertible: given (s, d) we can recover Xe as

x , = s - U ( d ) ,

Factoring Wavelet Transforms into Lifting Steps 249

and then Xo can be recovered as explained earlier. This illustrates one of the built-in features of

lifting: no matter how P and U are chosen, the scheme is always invertible and thus leads to critically sampled perfect reconstruction filter banks. The block diagram of the two lifting steps is

given in Figure 1.

X

Xe , (

P

Xo

, d

t, $

FIGURE 1. Block diagram of predict and update lifting steps.

Coming back to our simple example, it is easy to see that an update operator that restores the correct running average, and therefore reduces aliasing, is given by

Sk -~ X2k q- (dk-I + d r ) / 4 .

This can be verified graphically by looking at Figure 2.

d k/4 dk

i i i i i i I I I

2 k - 2 2 k - 1 2k 2 k + 1 2 k + 2 2 k + 3 2 k + 4

FIGURE 2. Geometric interpretation for piecewise linear predict and update lifting steps. The original signal is drawn in bold. The wavelet coefficient dk is computed as the difference of an odd sample and the average of the two neighboring evens. This corresponds to a loss dk/2 in area drawn in grey. To preserve the running average this area has to be redistributed to the even locations resulting in a coarser piecewise linear signal sk drawn in thin line. Because the coarse scale is twice the fine scale and two even locations are affected, dk/4, i.e, one quarter of the wavelet coefficient, has to be added to the even samples to obtain the sk. Then the thin and bold lines cover the same area. (For simplicity we assumed that the wavelet coefficients dk-I and dk+t are zero.)

This simple example, when put in the wavelet framework, turns out to correspond to the biorthogonal (2,2) wavelet transform of [12], which was originally constructed using Fourier argu- ments. By the construction above, which did not use the Fourier transform but instead reasoned using only spatial arguments, one can easily work in a more general setting. Imagine for a moment that

250 Ingrid Daubechies and Wire Sweldens

the samples were irregularly spaced. Using the same spatial arguments as above, we could then see that a good predictor is of the form fl x2k + (1 - /~ ) x2k+l where the/~ varies spatially and depends on the irregularity of the grid. Similarly spatially varying update coefficients can be computed [46]. This thus immediately allows for a (2,2) type transform for irregular samples. These spatial lifting steps can also be used in higher dimensions (see [45]) and lead, e.g., to wavelets on a sphere [40] or more complex manifolds.

Note that the idea of using spatial wavelet constructions for building second generation wavelets has been proposed by several researchers:

9 The lifting scheme is inspired by the work of Donoho [19] and Lounsbery et al. [29]. Donoho [ 19] shows how to build wavelets from interpolating scaling functions, while Louns- bery et al. built a multiresolution analysis of surfaces using a technique that is algebraically the same as lifting.

9 Dahmen and collaborators, independently