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Fast wavelet transform

Module by: Veronique Delouille. E-mail the author

One-dimensional wavelet transform

Suppose we are given as signal the projection of a function onto the space Vj+1Vj+1:

P j + 1 f = k s j + 1 , k ϕ j + 1 , k ( x ) , s j + 1 , k = f , ϕ ˜ j + 1 , k . P j + 1 f = k s j + 1 , k ϕ j + 1 , k ( x ) , s j + 1 , k = f , ϕ ˜ j + 1 , k .
(1)

Using the dual refinement equations, we have:

s j , k = f , ϕ ˜ j , k = f , l h ˜ l ϕ j + 1 , 2 k + l = k h ˜ l - 2 k s j + 1 , l , s j , k = f , ϕ ˜ j , k = f , l h ˜ l ϕ j + 1 , 2 k + l = k h ˜ l - 2 k s j + 1 , l ,
(2)

where the coefficients sjksjk are called scaling coefficients, since they are related to scaling functions. Similarly, the wavelet or detail coefficientsdjkdjk are obtained as

d j k = f , ψ ˜ j k = k g ˜ l - 2 k s j + 1 , l . d j k = f , ψ ˜ j k = k g ˜ l - 2 k s j + 1 , l .
(3)

The coefficients sjksjk and djkdjk are obtained from sj+1,lsj+1,l by `moving average' schemes, using the filter coefficients {h˜l}{h˜l} and {g˜l}{g˜l} as `weights', with the exception that these moving averages are sampled only at the even integers, i.e. a downsampling is performed. Such transform allows, once we have computed sJ,ksJ,k=f,ϕ˜J,k=f,ϕ˜J,k for a fine level JNJN, to compute sjksjk and djkdjk for all coarser levels j<Jj<J without evaluating the integrals.

Suppose now we are given the values of ff at n=2Jn=2J equispaced design points. The scaling functions ϕ˜J,k,k=0,...,2J-1ϕ˜J,k,k=0,...,2J-1, are compactly supported and localized around 2-Jk2-Jk. Hence the coefficients f,ϕ˜J,kf,ϕ˜J,k are weighted and scaled average of ff on a neighborhood of 2-Jk2-Jk which becomes smaller as JJ tends to infinity. Consequently, it makes sense to replace the integral f,ϕ˜J,kf,ϕ˜J,k by the (scaled) value of ff at 2-Jk2-Jk. More complicate quadrature formulae have been developed in [8], [6], [7].

With sj:={sjk;k=0,...,2j-1}sj:={sjk;k=0,...,2j-1} and dj:={djk;k=0,...,2j-1}dj:={djk;k=0,...,2j-1}, the forward (or analyzing) wavelet transform given by Equation 2-Equation 3 can be rewritten as

s j = H ˜ j * s j + 1 and d j = G ˜ j * s j + 1 , s j = H ˜ j * s j + 1 and d j = G ˜ j * s j + 1 ,
(4)

where H˜j*H˜j* denotes the Hermitian conjugate of H˜jH˜j.

The inverse (or synthesis) transform is found by using the primal refinement equations and the fact that Vj+1=VjWjVj+1=VjWj.

P j + 1 f = l s j + 1 , l ϕ j + 1 , l = k s j , k ϕ j , k + k d j , k ψ j , k = k s j , k l h l ϕ j + 1 , 2 k + l + k d j , k l g l ϕ j + 1 , 2 k + l = l ϕ j + 1 , l k h l - 2 k s j , k + k g l - 2 k d j k , P j + 1 f = l s j + 1 , l ϕ j + 1 , l = k s j , k ϕ j , k + k d j , k ψ j , k = k s j , k l h l ϕ j + 1 , 2 k + l + k d j , k l g l ϕ j + 1 , 2 k + l = l ϕ j + 1 , l k h l - 2 k s j , k + k g l - 2 k d j k ,
(5)

from which it follows that

s j + 1 , l = k h l - 2 k s j k + k g l - 2 k d j k . s j + 1 , l = k h l - 2 k s j k + k g l - 2 k d j k .
(6)

In matrix form, we have

s j + 1 = H j s j + G j d j . s j + 1 = H j s j + G j d j .
(7)

In the finite and classical setting, the matrices HjHj, GjGj, H˜jH˜j and G˜jG˜j are of size 2j+1×2j2j+1×2j. Moreover, if the basis functions are compactly supported, the four filters (hlhl, glgl, h˜lh˜l, g˜lg˜l) have only a finite number of nonzero elements, and hence all these matrices are banded.

Example: Haar wavelet transform

In case of the orthogonal Haar transform, H˜j*=Hj*H˜j*=Hj* and is of the form

H ˜ j * = h 0 h 1 h 0 h 1 ... h 0 h 1 H ˜ j * = h 0 h 1 h 0 h 1 ... h 0 h 1
(8)

since only h0h0 and h1h1 are different from zero : h0=h1=1/2h0=h1=1/2. The high-pass filter {gl}{gl} is such that g0=-1/2g0=-1/2 and g1=1/2g1=1/2. The forward transform Equation 2-Equation 3 reduces to

s j , k = 1 2 s j + 1 , 2 k + 1 + 1 2 s j + 1 , 2 k d j , k = 1 2 s j + 1 , 2 k + 1 - 1 2 s j + 1 , 2 k , s j , k = 1 2 s j + 1 , 2 k + 1 + 1 2 s j + 1 , 2 k d j , k = 1 2 s j + 1 , 2 k + 1 - 1 2 s j + 1 , 2 k ,
(9)

and the reconstruction is given by

s j + 1 , 2 k = 1 2 s j , k - 1 2 d j , k s j + 1 , 2 k + 1 = 1 2 s j , k + 1 2 d j , k . s j + 1 , 2 k = 1 2 s j , k - 1 2 d j , k s j + 1 , 2 k + 1 = 1 2 s j , k + 1 2 d j , k .
(10)

Two-dimensional wavelet transform

The wavelet transform has been successfully applied to compress images, which are modelled as functions defined on a regular two-dimensional grid.

Figure 1: Two-dimensional wavelet transform: first the filters are applied on the column of the matrix SJSJ, this produces two matrices. The filters are applied a second time on the columns of these two matrices, resulting in four elements: a matrix of scaling coefficients, and three detail matrices.
Figure 1 (image1.png)

The easiest way to build a two-dimensional MRA is probably to use tensor products of spaces, see [2], [4]. In terms of wavelet transforms, this leads to applying two times a one-dimensional transform: first on the `row' of the signal matrix SJSJ, and second on the `columns' of the resulting two matrices, see Figure 1. In this figure, we see that, at each level of the decomposition, three types of detail coefficients are produced: DjhDjh, DjvDjv and DjdDjd. These superscripts recall that, in an image, horizontal edges will lead to large values of DjhDjh, vertical edges will show up in DjvDjv and DjdDjd will be sensitive to diagonal lines.

However, such a transform is not able to compress efficiently an image that contains curves. More complex bidimensional bases are now proposed in the literature to better model discontinuities along curves, see for example [1], [3], [5], [9].

References

  1. Candès, E.J. (1999). Ridgelets: Estimating with Ridge Functions. Technical report. Department of Statistics, Stanford University.
  2. Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: SIAM.
  3. Jansen, M. and Choi, H. and Lavu, S. and Baraniuk, R. (2001). Multiscale Image Processing Using Normal Triangulated Meshes. In IEEE International Conference on Image Processing.
  4. Mallat, S. (1998). A Wavelet Tour of Signal Processing. (Second). Academic Press.
  5. Starck, J. L. and Candès, E. J. and Donoho, D. L. (2002). The Curvelet Transform for Image Denoising. [To appear in IEEE Transactions on Signal Processing].
  6. Sweldens, W. and Piessens, R. (1993). Wavelet Sampling Techniques. In 1993 Proceedings of the Statistical Computing Section. (pp. 20-29). American Statistical Association.
  7. Sweldens, W. and Piessens, R. (1994). Quadrature Formulae and Asymptotic Error Expansions for wavelet approximations of smooth functions. SIAM J. Numer. Anal., 31(4), 1240-1264.
  8. Sweldens, W. (1994). Construction and Applications of Wavelets in Numerical Analysis. Ph. D. Thesis. Department of Computer Science, Katholieke Universiteit Leuven, Belgium.
  9. Willett, R. and Nowak, R. (2002). A Multiscale Approach for Recovering Edges and Surfaces in Photon Limited Medical Imaging. Technical report. Dept. of Electrical and Computing Engineering, Rice University.

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