CONSTRUCTION OF SHIFT-ORTHOGONAL WAVELETS ...bigorder of the synthesis side, is not arbitrary: we want our new wavelets to exhibit properties that are potentially useful for image

CONSTRUCTION OF SHIFT-ORTHOGONAL WAVELETS USING SPLINES

Michael Unser, Philippe Thévenaz, and Akram Aidroubi

Biomedical Engineering and Instrumentation Program, Bldg. 13, Room 3N17,National Center for Research Resources, National Institutes of Health,

Bethesda, Maryland 20892-5766, USA.

ABSTRACT

We present examples of a new type of wavelet basis functions that are orthogonal across shifts, but not across scales. Theanalysis functions are low order splines (piecewise constant or linear) while the synthesis functions are polynomial splines ofhigher degree The approximation power of these representations is essentially as good as that of the corresponding Battle-Lemarié orthogonal wavelet transform, with the difference that the present wavelet synthesis filters have a much faster decay.This last property, together with the fact that these transformations are almost orthogonal, may be useful for image coding anddata compression.

Keywords: splines, wavelet basis, biorthogonal wavelets, perfect reconstruction filterbanks, approximation properties, imagecoding.

1. INTRODUCTION

So far, researchers in wavelet theory have identified and characterized three primary types of multiresolution bases of L2 (thespace of square integrable functions) 8, 9, 11, 22 The first and earlier type are the orthogonal wavelet bases (e.g., Daubechies, andBattle-Lemarié wavelets) ". The second closely related family are the semi-orthogonal wavelets which span the samemultiresolution subspaces as before, but are not necessarily orthogonal with respect to shifts 19 The versatility of semi-orthogonal wavelet basis allows one to introduce many interesting properties ' 19, and almost any desirable shape , whileretaining the orthogonality property across scales that is inherent to Mallat's construction. A noteworthy example in thiscategory are the B-spline wavelets which exhibit near optimal time-frequency localization 17 The third category are thebiorthogonal wavelets which are constructed using two multiresolution analysis of L2 instead of one, as in the two previouscases 6,21 The advantage of this last category is that the wavelet filters can be shorter; in particular, they can be both FIR andlinear phase, which is typically not possible otherwise.

intra-scale orthogonalityyes no

inter-scale yes orthogonal semi-orthogonalorthogonality no shift-orthogonal bi-orthogonal

Table I: Nomenclature of the various types of wavelet bases

One way to differentiate these various wavelet bases is to look at their orthogonality properties (cf . Table I). Thisclassification naturally leads to the identification of one more type. These are the so-called shift-orthogonal wavelets which areorthogonal to their translates within the same scale, but not across scales. We presented the construction of an example usingsplines in a preliminary report 20•

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In this paper, we extend this previous construction by considering more general spline spaces. A notable difference with ourearlier example is that the present B-spline basis functions are not centered about the origin. This modification is necessary tocover the case of even degree splines, which leads to new wavelets that are anti-symmetric. We should emphasize that there is apractical motivation behind the present construction. Our choice of spline basis functions, especially the idea of using higherorder of the synthesis side, is not arbitrary: we want our new wavelets to exhibit properties that are potentially useful for imagecoding and data compression 22• First, they are very nearly orthogonal. Remember that orthogonality is required for thequantization error in the transformed domain to be equivalent to the error in the reconstructed image domain. Here,orthogonality with respect to shifts is consistent with the idea of independent channel processing (scalar quantization). We aregiving up orthogonality with respect to dilations, but in a very controlled fashion by varying the angle between the analysis andsynthesis spaces (cf. Section 2.2). Second, all basis functions are either symmetric or anti-symmetric. Consequently, thesewavelet transforms can be computed using mirror signal extensions, one of the most efficient techniques for reducing boundaryartifacts 15 In contrast, non-linear phase wavelets such as Daubechies' can only be implemented using periodic signalextensions. Third, we will see that some of the wavelet filters decay faster than their orthogonal (Battle-Lemarié) counterparts.Hopefully, this should reduce the spreading of coding errors and ringing artifacts. Finally, our new wavelets have excellentapproximation properties because they are constructed using very smooth functions (splines). This means that they will providea very accurate low resolution approximation of smooth surfaces. Thus, we can expect most of the finer scale waveletcoefficients to be close to zero in slowly varying image regions, which is advantageous for zero-tree wavelet coding 12• lastaspect of the problem is perhaps the strongest reason for using higher order splines on the synthesis side; a rigorous theoreticaljustification can be found elsewhere14.

2. SPLINE SPACES

Our construction uses two spline multiresolution analyses of L2. The analysis and synthesis functions will be splines ofdegree ni and 2, respectively. Typically, we will select n=O or ni=1, and n2>n1. We will start by specifying these splinesubspaces of L2 explicitly.

2.1 Spline spacesThe spline multiresolution space of degree n at resolution i, %' , is defined as

s(x)E7 s(2'x)EV0 = s0(x)=c(k)q(x—k)IcEl , (1)( kEZ J

where V0 is the basic space of splines of degree n, that is, the subspace of functions that are (n- 1) continuously differentiable andare polynomial of degree n in each interval [k,k + 1) k Z. The generating function p'(x) is Schoenberg's causal B-spline ofdegree n, which is obtained from the (n+1)-fold convolution of the indicator function in the unit interval [0,1). The B-spline ofdegree n satisfies the two-scale relation 19

(2)kEZ

where h"(k) is the binomial filter of order n+1 whose transfer function is

h(k) (Z, Hn(z)=.[1J. (3)

A crucial quantity in the specification of such a subspace is the autocorrelation sequencea'(k) = (q"(x—k),q(x)) = b2''(k), (4)

which is a sampled centered B-spline'8 of degree 2n+1 denoted by b21+l(k). It can be shown that the discrete Fourier transformof the sequence d1(k) is bounded as follows:

466/SP!E Vol. 2825


V(i) E [O,2ic), 0 < A &(o) 1, (5)where A = kEZsj(2 + k) =(2 1 )2fl+2 .kEz2k + 1)_(22) is strictly positive. The constants A= 4 and B =1 are thecorresponding Riesz (or frame) bounds. This inequality ensures that the various spline spaces are well-defined (closed)subspaces of L2 and that the construction actually makes sense.

Another important property is that splines of degree n have an order of approximation L=n+1. This is equivalent to say thatfor any L-times differentiable functionf, we have the following error bound 13, 14

Iii — 'fII C . (2)' . IIfI (6)where JL represents any bounded projection operator (orthogonal or not) onto the spline space Vp) at resolution 2' , C is aconstant that depends on p" and the type of projector, and IfII is the L2-norm of the Lth derivative off. In other words, theerror decays like O(a') a the scale a = 2' gets sufficiently small. Thus, we should expect higher order splines to result in asmaller approximation error, but the price to pay is an increase in complexity. The fact that the projector F does not need to beorthogonal also stresses the importance of the synthesis space which determines the rate of approximation. The exact type ofprojector will depend on our choice of analysis space. In the sequel, we will consider the oblique projector into %2 in adirection perpendicular to V'.

2.2 Angles between the spline spaces

Let p and P2 be any given two admissible analysis and synthesis scaling functions, respectively. Then, the largest angle°12 between the analysis and synthesis spaces Vp1) and %'p2) is given by 16

cos912 ess inf a12(o)I , (7)(OE[O,lt) jâ1i(co) . â22((O)

where the functions â.o) are the discrete Fourier transforms of the autocorrelation sequences

a1(k) = (p1(x —k),q1(x)) = (q' * p)(k), (i,j = 1,2) , (8),

where p(x) = ip (—x). In addition, the (oblique) projection into V(p2) perpendicular to %'p1) is well defined if and only ifcosO12 > (that is, if —ic/2 < 012 < icI2) .

In our case, we select p = pfh and p2 P . Simplifying the notation and dropping the order parameter n, we can determinethe analysis and synthesis autocorelations sequences

aii(k)=p21(k+ 2n1+1) (9)

a22(k) = 2n2+1(k+ 2n2+lJ (10)

which is equivalent to (4) with n = n1 and n = n2 ,respectively. Similarly, it is not difficult to show that the cross-correlation isgiven by

ai2(k)=12+1(k_(n2

_ni)) (11)

These are all finite sequences that can be determined numerically for any given n1 and n2 .It is then straightforward to determinethe angle between the various spline spaces using (7). The result of these calculations for all combination of splines up to thedegree 7 are given in Table II. In the present context, these worst case cosine values are useful indicators of the loss oforthogonality across scale for our hybrid spline wavelet transforms. In general, the angle is less then itI2 in absolute value (i.e.,cosO12 >0) only when (n2 — n1) is even, that is, when the degrees n1 and n2 are both odd, or both even.

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Having defined these new basis functions, we can now write the projection 1 of a functionf E L into %2 perpendicular toli7Zi

Ff(x) = i,k (16)kEZ

where we use the standard short form notation 4j =2' ' 2(xI2' —k).

3.2 Construction of shift-orthogonal spline wavelets

We now consider the characterization of the corresponding wavelets at resolution level i=1 . The synthesis wavelet xi(x I2)must satisfy the following conditions:

(i) iji(x/ 2) V2 (i.e., ir(x/ 2) is a spline of degree n2);(ii) ((x I 2), (p"1 (xl2 — k))=0 because W(x/ 2) is perpendicular to 1'7';(iii) (2'2(x/2),2"2(x/2 —k))=ö[kJ (intra-scaje orthogonality).

It turns out that there is a unique function ii that verifies all those conditions; it is given by

ijr(x12)= [12 q(k) pfl2 (x — k), (17)JceZ

where the sequences q and p are defined as follows

q(k +1) = (_1)c (T * a12)(k) (18)—1/2

p(k)=V.([q*qT*a22}2) (k). (19)

The operators [k2 and [k2 denote up-sampling and down-sampling by a factor of two, respectively, and qT(k) =q(—k). Thesequence h1(k) is the binomial filter in (3) with =i; the corresponding correlation sequences are given by (10) and (9). Notethat this wavelet is nothing else as the orthogonalized version of the basic wavelet defined by Abry and Aldroubi 2, which isgiven by (17) with p=identity. The framebound conditions on the basic wavelet imply that the operator p is a well-definedconvolution operator from 12 into 12.

The dual analysis wavelet ic(x I 2) must satisfy a similar set of conditions:(i) iiJ(x/2) V01 (i.e., t(x/2) is a spline ofdegree ni);(ii) (t(x/ 2), pfl2 (xl 2 —k)) = 0 because ii(x/ 2) is perpendicular to 1i2;(iii) (2"2it(x/ 2), 2"2J(x/ 2 —k))=[kJ (bi-orthogonality).

After some algebraic manipulations, we obtain the similar expression

ii(x/ 2) =[1t2 (k) pZ (x—k) , (20)kEZ

where

(k +1) = (_1)k (' * a)(k) (21)

(k) = 2 .(p * [qT * * aJJ1(k) (22)

with p and q defined in (19) and (18); the sequences h and a are the time-reversed version of binomial filter in (3) with =2and the cross-correlation sequence (9). When the maximum angle 12between the analysis and synthesis spaces 1'' and %2 isless than 90 degrees (cf. Table II), the corresponding wavelet subspaces and oblique projection operators are well-defined .Hence, the convolution and square-root inverses in (19) and (22) are well-posed, and the resulting digital filters p and arestable and invertible. The corresponding dual pairs of scaling functions and wavelets for n i=0 and n2=2 are shown in Fig. 1.One can observe that the basis functions are piecewise constant on the analysis side and piecewise quadratic with a first order ofcontinuity on the synthesis side. Interestingly, the analysis function has a very fast decay and is reasonably close to a B-splineof degree 0. Also note that for the particular case =n2, the present construction yields the Battle-Lemarié spline waveletswhich are completely orthogonal

SPIE Vol. 2825 / 469


1

0.750.5

0.25

-0.25-0.5

-0.75-1

SYNTHESIS

1

0.750.5

0.25

—0.25

—0.5

—0.75

—1

ANALYSIS

Fig. 1 : Dual sets of quadratic spline and piecewise constant wavelets and scaling functions at the first resolution level : (a) shift-orthogonalquadratic spline wavelet ;(b) dual piecewise constant wavelet; (c) orthogonal quadratic spline Battle-Lemarié scaling function; and (d) dualpiecewise constant analysis scaling function. The basis functions in (a) and (b) (resp., (c) and (d)) are quadratic splines with knots at theintegers (resp., at the even integers).

4. WAVELET TRANSFORM AND FILTERBANK ALGORITHM

Let %42 be the complementary wavelet space of 1' in %' but perpendicular to V' ; i.e., V =142 + 12 with 142J..VIt is not difficult to show that {i1Jk =2"2W(x/2' — k)}kEz iS an orthogonal basis of l42 . Since it is well known that UIEZ 2is dense in L2, it follows that the set {Wjk}(k)Ez2 is an unconditional basis of L, and that every function f e L can berepresented by its shift-orthogonal wavelet expansion

f(x) = (23)iEZ kEZ

The special feature of this decomposition is that the basis functions are orthogonal with respect to shifts (index k ), but not acrossscales or dilations (index i). We should note, however, that the residual correlations across scales should not be too significantbecause the angles between the various spline spaces are relatively small (cf. Table II). For these reasons, we can expect theshift-orthogonal decomposition (23) to provide essentially the same type of energy compaction as the corresponding orthogonalBattle-Lemarid wavelet transform.

The wavelet transform (23) can be implemented iteratively using a standard tree-structured perfect reconstructionfilterbank22. The corresponding symmetric analysis and synthesis filters (h,) and (h,g), respectively, are defined as follows:

V° — s 10

(a) : 2"2(x/2)

II

0.0.

0.

—5 1___J

[j

10

(b) : 2"2r(x/2)

II—0.

1

0.8

0.6

0.4

0.2

—0.2

(C) : 2"24(x/2)

5 10—5 ______0

(d) : 2"24(x/2)

470 1 SP1E Vol. 2825


h(k) =*((x/2),(x+ k))

(k) = *(fr(xI2),(x+k)). (24)

h(k) =

g(k) = i(xI2),(x-k))The easiest way to determine these filters is to perform the appropriate change of coordinate system and express (x I 2) andiji(x/ 2) (resp., (x I 2) and i.r(x/ 2)) in terms of the integer shifts of 4 (resp., 4). This provides an explicit characterization oftheir impulse response in the signal domain. We have chosen here to present frequency domain formulas because these turn outto be the most useful in practice. Our results are summarized as follows:

H(e) := ) = (w) • I a22 (25)'a22(2o)

fI(e°) := I;(O)) = i(co) • a12(0)) • /a22(2co) (26)a12(2w) \j a22(°))

G(e3) := (w)= • h1(w+ it) . fr(2o)•â12(o + it) •jw) (27)

O(e') := 1(w) = ejW•(w+ ) • 1

(28)a12(2co)a22(o) p(2w)

where the auxiliary filters (2w) and a(O) are defined as follows

2p(2w)= (29)

-'Iq) +q((O + it)

a() 1(w + ic) • kui2(w + ic)12 . (30)

All filters are infinite but decay exponentially fast.In order to compute a truncated version of a filter's impulse response, the simplest approach is to evaluate its transfer

function at the discrete frequencies o =2ici I N, i = 0,. . ., N — 1 , where N is chosen sufficiently large to avoid aliasing in thesignal domain. The impulse response is then determined by using an N-point inverse FFT. The first 20 filter coefficients forn1=1 and fl23 (cubic splines) are given in Table HI. The lowpass filter h is the same as the Battle-Lemarié filter described byMallat Interestingly, the wavelet synthesis filter g decays significantly faster, and turns out to be very similar to a Battle-Lemarié filter of degree 1 (n=1), which is also given for comparison.

4. CONCLUSION

We have presented a new class of hybrid spline wavelet transforms. The motivation behind this proposal was to use lowerorder analysis functions, while essentially preserving the approximation and orthogonality properties of higher order Battle-Lemarié wavelets. An important consideration was also to design wavelets that are either symmetric or anti-symmetric, a featurethat is desirable in many applications. In contrast with previous semi- and bi-orthogonal constructions, we have only relaxed theorthogonality constraint in-between resolution levels. The basis functions are still orthogonal within a given wavelet channel (orscale), a property that is quite desirable for quantization purposes. The advantage over the Battle-Lemarié family is that thewavelet synthesis filter decays substantially faster, a property that could be useful for reducing ringing artifacts in coding

SPIE Vol. 2825/471


applications. All the underlying basis functions (including duals) have been characterized explicitly in the time domain(polynomial spline representation). Such direct formulas are in general not available for other wavelet bases, except for the classof semi-orthogonal splines considered by us earlier 19

k h(k) (k—1) h(k) g(k+1) Battle-Lemarién=1

0 0.82382 0.772627 0.76613 0.8166 0.817646

0.399749 -0.435933 0.433923 -0.398208 -0.397297-0.073438 -0.0543491 -0.0502017 -0.0688421 -0.069101

-0.0559962 0.113795 -0.110037 0.0532619 0.0519453

0.0174812 0.0321753 0.0320809 0.0174535 0.016971

0.0118749 -0.0443877 0.0420684 -0.0104416 -0.00999059

-0.00187877 -0.0152844 -0.0171763 -0.00411726 -0.00388326

-0.00235935 0.0184738 -0.0179823 0.00228748 0.00220195-0.00130068 0.00673909 0.00868529 0.000960346 0.00092337 1

0.000166395 -0.00785314 0.00820148 -0.000556564 -0.000511636

0.0013526 -0.00292239 -0.00435384 -0.000247084 -0.000224296

0.000257732 0.00336119 -0.00388243 0.00013056 0.000122686

-0.000916596 0.00126143 0.00218671 0.0000590167 0.0000553563

-0.000236368 -0.00144343 0.00188213 -0.0000338138 -0.0000300112

0.000534489 -0.000543627 -0.00110374 -0.0000157658 -0.0000138188

0.000155065 0.000620628 -0.000927199 8.13544 106 744435 106

-0.000292241 0.000234175 0.000559937 3.80939 106 3.4798 106

-0.0000892527 -0.000267052 0.000462115 -2.17513 106 -1.86561 106

0.000153515 -0.000100845 -0.000285384 -1.04198 i06 -8.82258 i07

0.0000484485 0.000114942 -0.000232347 5.29128 i07 4.71223 iø

-0.0000788157 0.000043426 0.000146042 2.52873 i07 2.24913 iø

Table Ill: Filter coefficients for the cubic spline shift-orthogonal wavelet transform withnl = 1 and n2 = 3. The filters are all symmetric. The last column displays the Battle-Lemariéwavelet filter of degree one (n1 =n2 =1).

References1. P. Abry and A. Aidroubi, "Designing multiresolution analysis-type wavelets and their fast algorithms", J. FourierAnalysis andApplications, Vol. 2,

No. 2, pp. 135-159, 1995.2. P. Abry and A. Aidroubi, "Semi- and bi-orthogonal MRA-type wavelet design and their fast algorithms", Proc. SPIE Conf WaveletApplications in

Signal and Image Processing III, San Diego, CA, pp. 452-463, July 9-14, 1995.3. A. Aldroubi, "Oblique projections in atomic spaces", Proc. Amer. Math. Soc., Vol. 124, No. 7, pp. 2051-2060, July 1996.4. A. Aldroubi and M. Unser, "Families of multiresolution and wavelet spaces with optimal properties", Numerical Functional Analysis and

Optimization, Vol. 14, No. 5-6, pp. 417-446, 1993.5. G. Battle, "A block spin construction of ondelettes. Part I: Lemarid functions", Commun. Math. Phys., Vol. 110, pp. 601-615, 1987.6. A. Cohen, I. Daubechies and J.C. Feauveau, "Bi-orthogonal bases of compactly supported wavelets", Comm. Pure AppI. Math., Vol. 45, pp. 485-

560, 1992.7. I. Daubechies, "Orthogonal bases of compactly supported wavelets', Comm. Pure AppI. Math., Vol. 41, pp. 909-996, 1988.8. I. Daubechies, Ten lectures on wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.

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CONSTRUCTION OF SHIFT-ORTHOGONAL WAVELETS ...bigorder of the synthesis side, is not arbitrary: we want our new wavelets to exhibit properties that are potentially useful for image