Rao Sheng Tang Local Regularity Properties of Continuous Wavelet Transforms in N Dimensions

Wavelets: Applications

M. Yamada , in Encyclopedia of Mathematical Physics, 2006

Multidimensional CWT

The CWT can be formulated in an abstract way. We can regard G = {(a, b)|a(≠0), b ∈ R ) as an affine group on R with the group operation of (a, b) (a′, b′) = (aa′, ab′ + b) associated with the invariant measure dμ = da db/a ². The group G has its unitary representation in the Hilbert space H = L ²( R ):

$(U(a,b)f)(x)=\frac{1}{\sqrt{\left|a\right|}}f\left(\frac{x-b}{a}\right)$

and then we can consider the CWT can be constructed as a linear map W from L ²( R ) to L ²(G; da db/a ²):

$W:f(x)\mapsto T_{\psi }(a,b)=\frac{1}{\sqrt{C_{\psi }}}〈U(a,b)\psi ,f〉$

where 〈·, ·〉 is the inner product of L ²( R ) with the complex conjugate taken at the first element, and ψ(x) is a unit vector (analyzing wavelet) satisfying the abstract admissibility condition

$C_{\psi }={\int }_G|〈U(a,b)\psi ,\psi 〉{|}^2\text{d}\mu <\infty$

This formulation is applicable also to a locally compact group G and its unitary and square integrable representation in a Hilbert space H. Note that even the canonical coherent states are included in this framework by taking the Weyl–Heisenberg group and L ²( R ) for G and H, respectively. This abstract formulation allows us to extend the CWT to higher-dimensional Euclidean spaces and other manifolds: for example, 2D sphere S ² for geophysical application and 4D manifold of spacetime taking the Poincaré group into consideration.

In R ⁿ , the CWT of f( x ) ∈ L ²( R ⁿ ) and its inverse transform are given by

$\begin{array}{l}{T}_{\psi }(a,r,\mathit{b})=\frac{1}{\sqrt{C_{\psi }}}{\int }_{{\mathit{R}}^n}\overline{{\psi }^{(a,r,\mathit{b})}(\mathit{x})}f(\mathit{x})\text{d}\mathit{x}\\ f(\mathit{x})=\frac{1}{\sqrt{C_{\psi }}}{\int }_{G}T(a,r,\mathit{b}){\psi }^{(a,r,\mathit{b})}(\mathit{x})\frac{\text{d}a\text{d}r\text{d}\mathit{b}}{a^{n+1}}\end{array}$

where r ∈ SO(n), b ∈ R ⁿ ,dr is the normalized invariant measure of G = SO(n), and the wavelets are defined as ψ ^{(a, r, b )} ( x ) = (1/a ^n/2)ψ(r ⁻¹( x − b )/a), with the analyzing wavelet satisfying the admissibility condition

$C_{\psi }={\int }_{{\mathit{R}}^n}\frac{|\hat{\psi }(\mathit{\omega })|}{|(\mathit{\omega }){|}^n}\text{d}\mathit{\omega }<\infty$

Note that these wavelets are constructed not only by dilation and translation but also by rotation which therefore gives the possibility for directional pattern detection in a data function. In the case of 2D sphere S ², on the other hand, the dilation operation should be reinterpreted in such a way that at the North Pole, for example, it is the normal dilation in the tangent plane followed by lifting it to S ² by the stereographic projection from the South Pole.

Generally, the abstract map W thus defined is injective and therefore reversal, but not surjective in contrast with the Fourier case. Actually in the case of 1D CWT, T _ψ (a, b) is subject to an integral condition:

$\begin{array}{c}{T}_{\psi }(a,b)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{\text{d}a\text{d}b}{a^2}K(a,b;a\prime ,b\prime )T_{\psi }(a\prime ,b\prime )\\ K(a,b;a\prime ,b\prime )={\int }_{-\infty }^{\infty }\overline{{\psi }^{(a,b)}(x)}{\psi }^{(a\prime ,b\prime )}(x)\text{d}x\end{array}$

which defines the range of the CWT, a subspace of L ²( R ). Therefore, if one wants to modify T _ψ (a, b) by, for example, assigning its value as zero in some parameter region just as in a filter process, care should be taken for the resultant T _ψ (a, b) to be in the image of the CWT. The reason may be understood intuitively by noticing that the wavelets ψ ^{(a, b)}(x) are linearly dependent on each other. The expression of a data function by a linear combination of the wavelets is therefore not unique, and thus is redundant. The CWT gives only T _ψ (a, b) of the least norm in L ²( R ²; da db/a ²). In physical interpretations of the CWT, however, this nonuniqueness is often ignored.

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Wavelets: Mathematical Theory

K. Schneider , M. Farge , in Encyclopedia of Mathematical Physics, 2006

Higher Dimensions

The continuous wavelet transform can be extended to higher dimensions in $L^2\left({\mathbb{R}}^n\right)$ in different ways. Either we define spherically symmetric wavelets by setting $\psi \left(x\right)={\psi }^{1d}\left(\right|x\left|\right)$ for $x\in {\mathbb{R}}^n$ or we introduce in addition to dilations $a\in {\mathbb{R}}^{+}$ and translations $b\in {\mathbb{R}}^n$ also rotations to define wavelets with a directional sensitivity. In the two-dimensional case, we obtain for example,

[16] ${\psi }_{a,b,\theta }\left(x\right)=\frac{1}{a}\psi \left(R_{\theta }^{-1}\left(\frac{x-b}{a}\right)\right)$

where $a\in {\mathbb{R}}^{+},b\in {\mathbb{R}}^2$ , and where R _θ is the rotation matrix

[17] $\left(\begin{array}{cc}\text{cos}\theta & -\text{sin}\theta \\ \text{sin}\theta & \text{cos}\theta \end{array}\right)$

The analysis formula [8] then becomes

[18] $\tilde{f}\left(a,b,\theta \right)={\int }_{{\mathbb{R}}^2}f\left(x\right){\psi }_{a,b,\theta }^{*}\left(x\right)\text{d}x$

and for the corresponding inverse wavelet transform [11] we obtain

[19] $f\left(x\right)=\frac{1}{C_{\psi }}{\int }_0^{\infty }{\int }_{R^2}{\int }_0^{2\pi }\tilde{f}\left(a,b,\theta \right){\psi }_{a,b,\theta }\left(x\right)\frac{\text{d}a\text{d}b\text{d}\theta }{a^3}$

Similar constructions can be made in dimensions larger than 2 using n  −   1 angles of rotation.

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Wavelets, Advanced

Su-Yun Huang , Zhidong Bai , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV Discrete Wavelet Transforms

In the continuous wavelet transform, we consider the family of wavelets

${\psi }_{\text{a},\text{b}}(\text{x})=\frac{1}{\sqrt{|\text{a}|}}\psi \left(\frac{\text{x}-\text{b}}{\text{a}}\right)$

where a, b  ∈ R, a ≠ 0 and ψ(x) is admissible. In the multiresolution analysis, there exist such functions ϕ(x) and ψ(x) that {ϕ _{j, k} (x):k  ∈ Z} constitutes an orthonormal basis for V _j and {ψ _{j, k} (x):k  ∈ Z} constitutes an orthonormal basis for W _j . Thus, sampling information of f(x) on the lattice points,

$\begin{array}{llll}\text{a}=2^{\text{j}},\hfill & \text{b}\in 2^{\text{j}}\text{Z}\hfill & \text{for}\hfill & \text{j}\in \text{Z}\hfill \end{array}$

is enough to gain full knowledge of f. In the discretization, we restrict a and b to the above lattice points. For every f(x)   ∈ L ₂(R), f(x) can be represented in terms of orthonormal wavelet series

$\text{f}(\text{x})={\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{c}}_{\text{j},\text{k}}{\phi }_{\text{j},\text{k}}(\text{x})+{\displaystyle \sum _{\mathcal{l}\ge \text{j}}}{\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{d}}_{\mathcal{l},\text{k}}{\psi }_{\mathcal{l},\text{k}}(\text{x})$

or

$\text{f}(\text{x})={\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\displaystyle \sum _{\text{k}\in \text{Z}}}{\text{d}}_{\mathcal{l},\text{k}}{\psi }_{\mathcal{l},\text{k}}(\text{x})$

where

$\begin{array}{ll}{\text{c}}_{\text{j},\text{k}}=〈\text{f},{\phi }_{\text{j},\text{k}}〉,\hfill & {\text{d}}_{\mathcal{l},\text{k}}=〈\text{f},{\psi }_{\mathcal{l},\text{k}}〉\hfill \end{array}$

with 〈·,   ·〉 the inner product in L ₂. By the two-scale equations (4) and (6), we have

(9) ${\phi }_{\text{j}-1,\text{k}}(\text{x})={\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{h}}_{\mathcal{l}-2\text{k}}{\phi }_{\text{j},\mathcal{l}}(\text{x})$

and

(10) ${\psi }_{\text{j}-1,\text{k}}(\text{x})={\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{g}}_{\mathcal{l}-2\text{k}}{\phi }_{\text{j},\mathcal{l}}(\text{x})$

By (9), (10), and the orthogonality property of wavelets, we obtain

$\begin{array}{ll}{\text{c}}_{\text{j}-1,\text{k}}\hfill & =〈\text{f},{\phi }_{\text{j}-1,\text{k}}〉=〈\text{f},{\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{h}}_{\mathcal{l}-2\text{k}}{\phi }_{\text{j},\mathcal{l}}〉\hfill \\ \hfill & ={\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{h}}_{\mathcal{l}-2\text{k}}{\text{c}}_{\text{j},\mathcal{l}}\hfill \end{array}$

Similarly,

$\begin{array}{ll}{\text{d}}_{\text{j}-1,\text{k}}\hfill & =〈\text{f},{\psi }_{\text{j}-1,\text{k}}〉=〈\text{f},{\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{g}}_{\mathcal{l}-2\text{k}}{\phi }_{\text{j},\mathcal{l}}〉\hfill \\ \hfill & ={\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{g}}_{\mathcal{l}-2\text{k}}{\text{c}}_{\text{j},\mathcal{l}}\hfill \end{array}$

In the reverse direction, coefficients in the finer resolution scale can be obtained from coefficients in the coarser resolution scale:

$\begin{array}{ll}{\text{c}}_{\text{j},\text{k}}\hfill & =〈\text{f},{\phi }_{\text{j},\text{k}}〉={\displaystyle \sum _{\mathcal{l}}}{\text{c}}_{\text{j}-1,\mathcal{l}}〈{\phi }_{\text{j}-1,\mathcal{l}},{\phi }_{\text{j},\text{k}}〉\hfill \\ \hfill & +{\displaystyle \sum _{\mathcal{l}}}{\text{d}}_{\text{j}-1,\mathcal{l}}〈{\psi }_{\text{j}-1,\mathcal{l}},{\phi }_{\text{j},\text{k}}〉\hfill \\ \hfill & ={\displaystyle \sum _{\mathcal{l}}}{\text{c}}_{\text{j}-1,\mathcal{l}}{\text{h}}_{\text{k}-2\mathcal{l}}+{\displaystyle \sum _{\mathcal{l}}}{\text{d}}_{\text{j}-1,\mathcal{l}}{\text{g}}_{\text{k}-2\mathcal{l}}\hfill \end{array}$

In summary, we have the following fast wavelet algorithm.

$\begin{array}{ll}\text{Wavelet decomposition}:\hfill & \{\begin{array}{l}{\text{c}}_{\text{j}-1,\text{k}}={\sum }_{\mathcal{l}\in \text{Z}}{\text{h}}_{\mathcal{l}-2\text{k}}{\text{c}}_{\text{j},\mathcal{l}},\hfill \\ {\text{d}}_{\text{j}-1,\text{k}}={\sum }_{\mathcal{l}\in \text{Z}}{\text{g}}_{\mathcal{l}-2\text{k}}{\text{c}}_{\text{j},\mathcal{l}},\hfill \end{array}\hfill \\ \text{Wavelet reconstruction}:\hfill & {\text{c}}_{\text{j},\text{k}}={\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{h}}_{\text{k}-2\mathcal{l}}{\text{c}}_{\text{j}-1,\mathcal{l}}+{\displaystyle \sum _{\mathcal{l}\in \text{Z}}}{\text{g}}_{\text{k}-2\mathcal{l}}{\text{d}}_{\text{j}-1,\mathcal{l}}\hfill \end{array}$

The preceding formulas give the so-called cascade algorithm. The algorithm gives a constructive and efficient way to compute approximate values of the scaling function ϕ(x) with arbitrarily high precision, provided that ϕ(x) has compact support. Use the notation c _j   =   (…, c _j,−1,c _j,0,c _j,1, …) to denote the low-pass sequence and d _j   =   (…, d _j,−1,d _j,0,d _j,1, …) to denote the high-pass sequence. Following is the cascade algorithm for approximate values of the scaling function.

1.

We start with a low-pass sequence c ₀, with c _0,0  =   1 and c _0,k   =   0 for k ≠ 0, and a high-pass sequence d ₀  =   0.

2.

Set j  = j  +   1. Compute c _j based on c _{j  −   1} and d _{j  −   1}  =   0 using the reconstruction formula. At each step of this cascade algorithm, twice as many values are computed. Values at "even points" c _j,2k are refined from the previous iteration

${\text{c}}_{\text{j},2\text{k}}=\sum _{\ell \in \text{Z}}{\text{h}}_{2(\text{k}-\ell )}{\text{c}}_{\text{j}-1,\ell }$

and values at "odd points" c _j,2k+1 are computed by

${\text{c}}_{\text{j},2\text{k}+1}=\sum _{\ell \in \text{Z}}{\text{h}}_{2(\text{k}-\ell )+1}{\text{c}}_{\text{j}-1,\ell }$

3.

Go to step 2, if j  < J; otherwise, terminate the iteration and go to step 4 for approximate values of ϕ(x).

4.

Set η _J (2^−J k)   = c _J,k . Linearly interpolate the values η _J (2^−J k) to obtain η _J (x) for nondyadic x. This function η _J (x) is used to approximate ϕ(x).

The error order of this approximation is given by

$\Vert \phi -{\eta }_{\text{J}}{\Vert }_{\infty }=\text{O}(2^{-\text{J}\alpha })$

where ϕ(x) is assumed to be Hölder continuous with exponent α. Thus, by letting the number of iterations J  →   ∞, the approximation to ϕ(x) can be made with arbitrarily high accuracy. An attractive feature of the cascade algorithm is that it allows one to zoom in on particular features of ϕ(x).

For a discrete signal of finite extent f  =   (f ₁, …, f _n ), the discrete wavelet transform calculates the coefficients of its wavelet transform approximation. This transformation maps the vector f to a vector of n wavelet coefficients. The discrete wavelet transform can be obtained as w  = Wf, where W is an orthogonal matrix corresponding to the discrete wavelet transform. To get the wavelet coefficients w, we do not actually perform the matrix multiplication. Instead we use the fast algorithm with complexity O(n).

Functions in the space L ₂(R) can be represented by orthonormal wavelet series. We shall notice that wavelet series are just as effective in the analysis of other spaces, such as L _p (R) for 1   < p  <   ∞, Sobolev spaces, Hölder spaces, Hardy spaces, and Besov spaces.

Suppose that the mother wavelet ψ(x) arising from a multiresolution analysis of L ₂(R) has regularity of degree r  ≥   1. Then the set {ψ _{j, k} (x):j, k  ∈ Z} constitutes an unconditional basis for L _p (R), 1   < p  <   ∞. It clearly does not apply to the spaces L ₁(R) and L _∞(R), which have no unconditional bases. Moreover, a wavelet series f(x)   =   ∑ _{j, k∈Z} 〈f,ψ _{j, k} 〉ψ _{j, k} (x) belongs to L _p (R) if and only if

${\left({\displaystyle \sum _{\text{j},\text{k}\in \text{Z}}}{〈\text{f},{\psi }_{\text{j},\text{k}}〉}^2{\psi }_{\text{j},\text{k}}^2(\text{x})\right)}^{1/2}\in {\text{L}}_{\text{p}}(\text{R})$

if and only if

${\left({\displaystyle \sum _{\text{j},\text{k}\in \text{Z}}}{〈\text{f},{\psi }_{\text{j},\text{k}}〉}^2\text{I}(2^{-\text{j}}\text{k}\le \text{x}\le 2^{-\text{j}}(\text{k}+1))\right)}^{1/2}\in {\text{L}}_{\text{p}}(\text{R})$

where I  is   the   indicator   function.

Similarly, wavelets provide unconditional bases and characterizations for Sobolev spaces W ₂ ^s (R), s  ≥   0. The Sobolev spaces W ₂ ^s (R) are defined by

${\text{W}}_2^{\text{s}}(\text{R})=\left\{\text{f}\in {\text{L}}_{\text{p}}(\text{R}):{\int }_{-\infty }^{\infty }{(1+|\omega {|}^2)}^{\text{s}}|\hat{\text{f}}(\omega ){|}^2\text{d}\omega <\infty \right\}$

The set {ψ _{j, k} }(x): j, k  ∈ Z} constitutes an unconditional basis for W ₂ ^s (R) for 0   ≤ s  < r. A wavelet series f(x)   =   ∑ _{j, k∈Z} 〈f, ψ _{j, k} 〉ψ _{j, k} (x) belongs to W ₂ ^s (R) if and only if

${\displaystyle \sum _{\text{j},\text{k}\in \text{Z}}}{〈\text{f},{\psi }_{\text{j},\text{k}}〉}^2(1+2^{2\text{j}\text{s}})<\infty$

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Wavelets: Application to Turbulence

M. Farge , K. Schneider , in Encyclopedia of Mathematical Physics, 2006

Wavelet space

To study turbulent signals one uses the continuous wavelet transform for analysis, and the orthogonal wavelet transform for compression and computation. To perform a continuous wavelet transform, one can choose:

•

either a real-valued wavelet, such as the Marr wavelet, also called "Mexican hat," which is the second derivative of a Gaussian,

•

or a complex-valued wavelet, such as the Morlet wavelet,

with the wavenumber k _ψ denoting the barycenter of the wavelet support in Fourier space computed as

[1] $\psi (x)=(1-x^2)exp\left(\frac{-x^2}{2}\right)$

[2] $\{\begin{array}{ll}\hat{\psi }(k)=\frac{1}{2\pi }exp\left(-\frac{{(k-k_{\psi })}^2}{2}\right)& \text{for}k>0\\ \hat{\psi }(k)=0& \text{for}k\le 0\end{array}$

[3] $k_{\psi }=\frac{{\int }_0^{\infty }k|\hat{\psi }(k)|\text{d}k}{{\int }_0^{\infty }|\hat{\psi }(k)|\text{d}k}$

For the orthogonal wavelet transform, there is a large collection of possible wavelets and the choice depends on which properties are preferred, for instance: compact support, symmetry, smoothness, number of cancelations, computational efficiency.

From our own experience, we tend to prefer the Coifman wavelet 12, which is compactly supported, has four vanishing moments, is quasisymmetric, and is defined with a filter of length 12, which leads to a computational cost for the fast wavelet transform in 24N operations, since two filters are used.

As stated above, we recommend the complex-valued continuous wavelet transform for analysis. In this case, one plots the modulus and the phase of the wavelet coefficients in wavelet space, with a linear horizontal axis for the position b, and a logarithmic vertical axis for the scale a, with the largest scale at the bottom and the smallest scale at the top.

In Figure 1a we show the wavelet analysis of a turbulent signal, corresponding to the time evolution of the velocity fluctuations of two successive vortex breakdowns, measured by hot-wire anemometry at N = 32768 = 2¹⁵ instants (Cuypers et al. 2003). The modulus of the wavelet coefficients ( Figure 1b ) shows that during the vortex breakdown, which is due to strong nonlinear flow instability, energy is spread over a wide range of scales. The phase of the wavelet coefficients ( Figure 1c ) is plotted only where the modulus is non-negligible, otherwise the phase information would be meaningless. In Figure 1c , one observes that the lines of constant phase point towards the instants where the signal is less regular, that is, during vortex breakdowns.

Figure 1. Example of a one-dimensional continuous wavelet analysis. (a) the signal to be analyzed, (b) the modulus of its wavelet coefficients, (c) the phase of its wavelet coefficients.

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Signal and Image Representation in Combined Spaces

John J. Benedetto , in Wavelet Analysis and Its Applications, 1998

§3 WAM noise suppression algorithm

This algorithm, constructed with Anthony Teolis, consists of the following components:

1.

A continuous wavelet transform W_g with adaptive analyzing impulse response g;

2.

Signal dependent irregular sampling of W_g ;

3.

Modification of sampled coefficients by nonlinear and thresh-olding operations to achieve compression and noise reduction, respectively;

4.

Iterative irregular sampling reconstruction method.

3.1 Continuous wavelet transform

The continuous wavelet transform W _g of a signal f for a given analyzing impulse response g is the function of two variables

$W_{g}f\left(t,s\right)=\left(f\ast D_s\mathsf{g}\right)\left(t\right)\text{,}$

where t is time, s is scale, $D_s\mathsf{g}\left(t\right)=s^{\frac{1}{2}}\mathsf{g}\left(ts\right)$ and * is convolution. The design of the filter $\widehat{\mathsf{g}}$ (Fourier transform of g) is important, and the con-tinuous wavelet transform is a reasonable mathematical model in the case of overlapping filter banks such as arise in auditory and visual systems. In auditory systems, the impulse response is causal, and the magnitude of the corresponding filter is shark-fin shaped. Consequently, by a calculation involving the Hilbert transform and the Paley-Wiener logarithmic integral theorem, the impulse response has the form of Figure 1.

Figure 1.

3.2 System output

Suppose we are dealing with a system of overlapping dilation related filters, with a mechanical to electrical transference, and a sigmoidal nonlinearity to achieve compression. An example is a mammalian auditory system. An analysis of the derivative of the nonlinearity, and a lateral scale operation, produce sampling sets X   =   {t _m,n : n   =   1, …} for each scale s_m . Our irregular sampling theorem allows us to reconstruct the original signal from the discrete set Lf of sampled values on X and to obtain compression in the process.

It turns out that L can be thought of as an operator with adjoint operator L*, and that S  = L*L is a frame operator of the type described in Section 2. This factorization is the basis of one of the ideas behind our irregular sampling formulas.

3.3 Thresholding

It is natural to introduce thresholding methods, in conjunction with wavelet theory, to achieve noise reduction, e.g. [11,15]. As indicated in Section 2, we have generalized the original idea of WAM beyond the original auditory applications. It can now be used as a noise suppressant in the following way.

Suppose we are given a class of signals f which are to be transmitted in a noisy environment caused by additive noise w. The main theoretical problem is to design a filter $\widehat{\mathsf{g}}$ so that w is incoherent with regard to the "wavelets" {ψ _m,n} determined by the impulse response g. Using the notation L from the section on System Output, this means that Lω  =   {⟨ω,ψ _m,n⟩}and each |⟨ ω,ψ _m,n⟩|   ≤   δ.

In our experiments with TIMIT data, we obtained symmetric envelopes as in Figure 2. For each scale s_m , M_m is defined as max_n{⟨f  + ω,ψ _m,n⟩}, and the maximum is taken over a specified portion of the aforementioned irregular sampling pattern. The noise threshold δ is then subtracted from each nonnegative element of the data set {⟨f  + ω,ψ _m,n⟩} The resulting data set is used in conjunction with our irregular sampling theory to form a signal $\tilde{f}$ which, mathematically, is a reasonable representation of f, e.g. [3]. These representations are effective, both graphically and in audio-taped versions, in the case of speech signals f and white noise ω.

Figure 2.

3.4 Discretization, noise reduction, and iterative reconstruction

In terms of the L operator, the description in the section on Thresholding amounts to the following. In the case of a signal space H and of frames which are not bases, L is not subjective onto the space of finite energy sequences. Thus, the filter g can be designed so that Lω ∉ L(H).In this setting we construct the frame correlation R  = LL* : L(H) ⟶L(H)so that the discrete set R ^{−   1} can be stored off-line.

Consequently,

(3.4.1) $f=L^{*}{R}^{-1}Lf\text{,}$

e.g. f can be reconstructed in terms of the sequence Lf .Therefore, since Lω   ∉ L(H) we expect L ^⁎ R ^{−   1}L(f  +   ω) to give a reasonable representation of f.

This approach makes thresholding viable for noise reduction, and ensures that such reduction is not due to low-pass filtering.

Equation (3.4.1) can be implemented by an iterative procedure with exponentially decreasing error. The examples in Figures 3–5 were generated in this way.

Figure 3.

Figure 4.

Figure 5.

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Logarithmic Wavelets

Laurent Navarro , ... Michel Jourlin , in Advances in Imaging and Electron Physics, 2014

2.2.1 Continuous Wavelet Transform

The wavelet transform is a method to decompose a real signal f into a set of elementary waveforms that provide a way to analyze the signal by examining the wavelet coefficients (Graps 1995 ). The continuous wavelet transform ( CWT) of f is defined by (Grossmann, Kronland-Martinet, & Morlet 1990), Debnath (2003), Grossmann and Morlet (1984), Duval-Destin, Muschietti, & Torresani 1993):

(7) $CW{T}_f\left(a,b\right)={\int }_{-\infty }^{\infty }f\left(x\right)\overline{{\zeta }_{a,b}\left(x\right)}dx,a>0,b\in \mathrm{ℝ}$

where the mother wavelet is

(8) ${\zeta }_{a,b}\left(x\right)=\frac{1}{\sqrt{a}}\zeta \left(\frac{x-b}{a}\right)$

and $\overline{\zeta }$ is the complex conjugate of $\zeta$ , a is the scaling factor (dilation or compression), and b is the translation factor (time shift). In addition $\zeta \in L^2\left(\mathrm{ℝ}\right)$ .

The continuous wavelet transform has the property to be invertible if the condition of admissibility is respected:

(9) $C_{\zeta }={\int }_{-\infty }^{\infty }\frac{\left|\hat{\zeta }\left(\nu \right)\right|}{\left|\nu \right|}d\nu <\infty ,$

where $\hat{\zeta }$ is the Fourier transform of $\zeta$ and

(10) ${\int }_{-\infty }^{\infty }\zeta \left(x\right)dx=0.$

Under this admissibility condition, the inverse wavelet transform can be calculated (Meyer 1992):

(11) $f\left(x\right)=\frac{1}{C_{\zeta }}{\int }_{a=0}^{\infty }{\int }_{b=-\infty }^{\infty }\frac{1}{{\left|a\right|}^2}CWT\left(a,b\right){\zeta }_{a,b}\left(x\right)dadb.$

A famous continuous wavelet has been built by Jean Morlet with the objective of creating a tool like a constant-Q filter bank (Grossmann and Morlet 1984; Strang and Nguyen 1996). A representation of the Morlet wavelet Eq. (12) is shown for three values of parameter a (Figure 6):

Figure 6. The real part of the Morlet wavelet at three different scales: (a) contracted wavelet, (b) mother wavelet, (c) dilated wavelet.

(12) $\zeta \left(x\right)=e^{-\pi x^2}{e}^{10.i\pi x}.$

Concerning the graphical representation of the wavelet coefficients of the CWT, we must consider the fact that the wavelet transform is defined on the half-plane (b,a). If the function $\zeta$ (Mother wavelet) is a complex valued function, the mean to highlight the useful information of the signal is to provide a representation of the magnitude onto the full plane $\left(b,-\log \left(a\right)\right)$ . This is a relevant strategy if we want to display information within a wide range of scale parameters. The horizontal axis represents the time and the vertical axis the scale, with small scales at the top of the representation. In the same plane, the phase can be represented in a grayscale picture, where 0 corresponds to white and $2\pi$ to black. When the phase reaches $2\pi$ , the phase is wrapped around 0.

Another alternative consists in representing the energy of the signal computed using the square modulus of the continuous wavelet transform, named the scalogram (Mallat 2008), which is equivalent to the spectrogram for wavelet (Figure 7). The scale axis is expressed in frequency, using the relation between scale and frequency:

Figure 7. Scalogram (bottom) of the signal (top) composed of four sinusoidal functions at four different frequencies. (See the color plate.)

(13) $a=\frac{v_0}{v},$

where $v_0$ represents the reference frequency linked to the considered Morlet wavelet.

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Signal and Image Representation in Combined Spaces

Boris Rubin , in Wavelet Analysis and Its Applications, 1998

Introduction

During the last decade, a modern wavelet analysis has been extensively developed in the series of books and papers by C. Chui, R. Coifman, I. Daubechies, A. Grossmann, R G. Lemarié, S. Mallat, Y. Meyer and many other authors who contributed greatly to this field. A theory of continuous wavelet transforms is one of the basic tools of this analysis (see, e.g. [3, 9, 18, 21, 24]). In mentioned publications, this theory has been built on the basis of Calderón's reproducing formula, or by making use of square integrable group representations ([18, 19]). Concerning Calderón's reproducing formula, see, e.g. [2, 3, 9, 12, 11, 20, 22, 24, 26]. In the following we present an alternative approach which may be helpful in the cases when methods mentioned above fail or their realization is difficult. Such a situation arises e.g. on a spherical surface. Here the classical Calderón's reproducing formula (and related wavelet transforms) may not be used in principle for geometrical reasons. On the other hand it is not clear how to use representations of the rotation group for setting up a spherical analogue of the Calderón's reproducing formula. A list of such examples may be continued.

Our approach is based on the following observation. Any reproducing formula may be interpreted as a representation of the identity operator I. Consider an analytic operator family {K ^α}, α ϵ ₵, for which K ⁰  = I (for instance, one can take any analytic semigroup of operators). Then a reproducing formula is nothing else but the trivial relation

$\phi =\left(a.c.\right)K^{\alpha }\phi |{}_{\alpha =0}$

Where (a.c) K^α stands for some representation of analytic continuation of K ^α. Now the problem is to choose a suitable operator family {K^α} and to write out (a.c.) Kα in appropriate form. If we want to get a reproducing formula, say, on a set A, it is natural to choose operators K ^α acting just on this set. A desired construction of analytic continuation may be obtained by using a generalization of A. Marchaud's idea ([23, 28]) that was originally used for constructing fractional derivatives (see Section 1 for the details). This construction gives rise to natural continuous wavelet transforms on $\mathcal{A}$ and to the corresponding reproducing formula. Note that the classical Calderón's reproducing formula on IR ⁿ can be obtained in the framework of our approach if we take K ^α to be a semigroup of Riesz potential for which $\left(\widehat{K^{\alpha }\phi }\right)\left(\xi \right)=\left|\xi \right|{}^{-\alpha }\widehat{\phi }\left(\xi \right)$ in the Fourier terms.

In Sections 2 and 3 a new method is illustrated for the case when is an n-dimensional unit sphere ∑ _n   ⊂   IR ^{n  +   1} (one should mention a paper of Segman and Zeevi [33] in which the term "spherical wavelet transform" has another meaning). Various topics of the harmonic analysis on a sphere are presented in [1, 7], [4–6], [16, 17, 25, 30, 34–36]. This list of references may be essentially extended.

The content of this article is to a large extent originated by papers [27, 28].

Notation

${\Sigma }_n=\left\{x\in {\mathrm{I}\mathrm{R}}^{n+1}:|x=1|\right\},\begin{array}{cc}\hfill \hfill & \hfill {\sigma }_n=\left|{\Sigma }_n\right|\hfill \end{array}=2{\pi }^{\left(n+1\right)/2}/\Gamma \left(\left(n+1\right)/2\right)\text{.}$

For x, y ∈ ∑ _n the notation xy is used for the scalar product of x and y; dx denotes a Lebesgue measure on ${\sum }_n$ ; y(∑ _n )   =   {Y_k,j (x)} stands for the complete orthonormal system of spherical harmonics on ∑ _n ; k  =   0,1,…; j  =   1,2,…, d_n (k), d_n (k) being the dimension of the subspace of harmonics of the order k, $d_n\left(k\right)=\left(n+2k-1\right)\frac{\left(n+k-2\right)!}{k!\left(n-1\right)!}$ (see [35]).

$f\left(x,r\right)=\frac{1}{{\sigma }_n}{\displaystyle {\int }_{{\sum }_n}\frac{1-r^2}{{\left|y-rx\right|}^{n+1}}f\left(y\right)}\mathit{dy},\begin{array}{cc}\hfill \hfill & \hfill 0<r<1\hfill \end{array}\text{,}$

is a Poisson integral of f.

Below C(∑ _n ) is a linear space of continuous functions on. ∑ _n The notation L^p (∑ _n ) is standard for $1\le p<\infty \text{.}$ For our convenience, a similar notation L $L^{\infty }\left({\sum }_n\right)$ is used for the space $C\left({\sum }_n\right)$ .

We use the symbol "≤   ≈   ˮ instead of "≤ˮ when corresponding relations hold up to a nonessential constant factor.

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Multirate and Wavelet Signal Processing

In Wavelet Analysis and Its Applications, 1998

5.2 Wavelet transform

The name wavelet comes from the requirement that a function should integrate to zero, waving above and below the axis. The diminutive connotation of wavelet suggests the function has to be well localized. At this point, some people might ask, why not use traditional Fourier methods? Fourier basis functions are localized in frequency, but not in time. So, Fourier analysis is the ideal tool for the efficient representation of very smooth, stationary signals. Wavelet basis functions are localized in time and frequency. So, wavelet analysis is an ideal tool for representing signals that contain discontinuities (in the signal or its derivatives) or for signals that are not stationary.

5.2.1 Definition and properties

Definition 5.2.1.1

The wavelet transform of g(t) with respect to wavelet ψ(t) is defined by

$\text{WT{}g\text{;}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }g\text{(}t\text{)}{\psi }^{*}\left(\frac{t-b}{a}\right)dt$

where, a ≠ 0 and b are called the scale and translation parameters, respectively. Furthermore, the Fourier transform of wavelet ψ(t), denoted ψ(f), must satisfy the following admissibility condition:

$C_{\psi }\text{=}{{\int }^{}}_{-\infty }^{\infty }\frac{\text{|Ψ(}f{\text{)|}}^{\text{2}}}{f}df\text{<}\infty ,$

which shows that ψ(t) has to oscillate and decay. The inverse of the continuous wavelet transform is given by

$g\text{(t)=}\frac{\text{1}}{C_{\psi }}{{\int }^{}}_{-\infty }^{\infty }{{\int }^{}}_{-\infty }^{\infty }\text{WT{}g\text{;}a\text{,}b\text{}}\psi \left(\frac{t-b}{a}\right)\frac{dadb}{a^{\text{2}}}.$

It is often useful to think of functions and their wavelet transforms as occupying two domains. Then, the following properties show the correspondence between operations performed in one domain with operations in the other.

5.2.1.1 Linearity property

Theorem 5.2.1.1

If the wavelet transforms of g and h exist and α and β are scalars, then

$\text{WT{}\alpha g\text{+}\beta h\text{;}a\text{,}b\text{}=}\alpha \text{WT{}g\text{;}a\text{,}b\text{}+}\beta \text{WT{}h\text{;}a\text{,}b\text{}}\text{.}$

Proof

By definition,

$\text{WT{}\alpha g\text{+}\beta h\text{;}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }\text{{}\alpha g\text{(t)+}\beta h\text{(}t\text{)}}{\psi }^{*}\left(\frac{t-b}{a}\right)dt.$

By the linearity of integration,

$\begin{array}{ll}\text{WT{}\alpha g\text{+}\beta h\text{;}a\text{,}b\text{}=}\hfill & \frac{\alpha }{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }g\text{(t)}{\psi }^{*}\left(\frac{t-b}{a}\right)dt\hfill \\ \hfill & +\frac{\beta }{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }h\text{(t)}{\psi }^{*}\left(\frac{t-b}{a}\right)dt.\hfill \end{array}$

Hence,

$\text{WT{}\alpha g\text{+}\beta h\text{;}a\text{,}b\text{}=}\alpha \text{WT{}g\text{;}a\text{,}b\text{}+}\beta \text{WT{}h;a\text{,}b\text{}}$

5.2.1.2 Similarity theorem

Theorem 5.2.1.2

If the wavelet transform of f exists, then the wavelet transform of f(α•), for a constant α, is given by

$\text{WT{}f\text{(}\alpha \text{•);}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}\alpha \text{|}}}\text{WT{}f\text{;}\alpha a\text{,}\alpha b\text{}}\text{.}$

Proof

By definition,

$\text{WT{}f\text{(}\alpha \text{•);}a\text{,}b\text{}±}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(}\alpha t\text{)}{\psi }^{*}\left(\frac{t-b}{a}\right)dt.$

Let u = αt. Then,

$\text{WT{}f\text{(}\alpha \text{•);}a\text{,}b\text{}=}\frac{\text{1}}{\text{|}\alpha \text{|}\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(u)}{\psi }^{*}\text{(}\frac{\frac{u}{\alpha }-b}{a}\text{)}d\text{u,}$

or equivalently,

$\text{WT{}f\text{(}\alpha \text{•);}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}\alpha \text{|}}\sqrt{\text{|}\alpha a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(u)}{\psi }^{*}\text{(}\frac{\text{u-}\alpha b}{\alpha a}\text{)}d\text{u}.$

Hence,

$\text{WT{}f\text{(}\alpha \text{•);}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}\alpha \text{|}}}\text{WT{}f\text{;}\alpha a\text{,}\alpha b\text{}}\text{.}$

5.2.1.3 Shift theorem

Theorem 5.2.1.3

If the wavelet transform of f exists, then the wavelet transform of f(• - α), for a constant α, is given by

$\text{WT{}f\text{(•-}\alpha \text{);}a\text{,}b\text{}=WT{}f\text{;}a\text{,}b-\alpha \text{}}\text{.}$

Proof

By definition,

$\text{WT{}f\text{(•-}\alpha \text{);}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(}t-\alpha \text{)}{\psi }^{*}\left(\frac{t-b}{a}\right)dt.$

Let u = t - α. Then,

$\text{WT{}f\text{(•-}\alpha \text{);}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(u)}{\psi }^{*}\left(\frac{u\text{−(}b-\alpha \text{)}}{a}\right)du,$

or equivalently,

$\text{WT{}f\text{(•-}\alpha \text{);}a\text{,}b\text{}=WT{}f\text{;}a\text{,}b-\alpha \text{}}\text{.}$

5.2.1.4 Differentiation theorem

Theorem 5.2.1.4

If the wavelet transform of f exists and if f' exists, then the wavelet transform of f′ is given by

$\text{WT{}f\prime \text{;}a\text{,}b\text{}=}\frac{\partial }{\partial b}\text{WT{}f\text{;}a\text{,}b\text{}}\text{.}$

Proof

By definition,

$\text{WT{}f\prime \text{;}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\prime \text{(}t\text{)}{\psi }^{*}\left(\frac{t-b}{a}\right)dt.$

But for every real t,

$f\prime \text{(}t\text{)=}\underset{\epsilon \to \text{0}}{\lim }\frac{f\text{(}t\text{+}\epsilon \text{)-}f\text{(t)}}{\epsilon }.$

Then, using Lebesque's Dominated Convergence Theorem, we obtain

$\text{WT{}f\prime \text{;}a\text{,}b\text{}=}\underset{\epsilon \to 0}{\lim }\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }\left(\frac{f\text{(}t\text{+}\epsilon \text{)-}f\text{(}t\text{)}}{\epsilon }\right){\psi }^{*}\left(\frac{t-b}{a}\right)d\text{t}$

Using the Linearity Theorem, we obtain

$\text{WT{}f\prime \text{;}a\text{,}b\text{}=}{\lim }_{\epsilon \to \text{0}}(\frac{1}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(}t\text{+}\epsilon \text{)}{\psi }^{*}\left(\frac{\text{t-}b}{a}\right)dt-\frac{1}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }f\text{(t)}{\psi }^{*}\left(\frac{t\text{−b}}{a}\right)dt)$

Using the Shift Theorem, we obtain

$\text{WT{}f\prime \text{;}a\text{,}b\text{}=}{\lim }_{\epsilon \to \text{0}}\frac{\text{WT{}f\text{;}a\text{,}b\text{+}\epsilon \text{}-WT{}f\text{;}a\text{,}b\text{}}}{\epsilon }.$

Therefore,

$\text{WT{}f\prime \text{;}a\text{,}b\text{}=}\frac{\partial }{\partial b}\text{WT{}f;a\text{,}b\text{}}\text{.}$

5.2.1.5 Convolution theorem

Theorem 5.2.1.5

If the wavelet transform of g exists and if f * g exists, then the wavelet transform of f * g is given by

$\text{WT{}f\star g\text{;}a\text{,}b\text{}=}f\stackrel{b}{\star }\text{WT{}g\text{;}a\text{,}b\text{},}$

where ${}_{*}^{\text{b}}$ denotes convolution with respect to the b variable.

Proof

By definition,

$\text{WT{}f\star g\text{;}a\text{,}b\text{}=}\frac{\text{1}}{\sqrt{\text{|}a\text{|}}}{{\int }^{}}_{-\infty }^{\infty }\left[{{\int }^{}}_{-\infty }^{\infty }f\text{(}u\text{)}g\text{(}t\text{−u)}d\text{u}\right]{\psi }^{*}\left(\frac{t-b}{a}\right)d\text{t,}$

or equivalently,

$\text{WT{}f\star g\text{;}a\text{,}b\text{}=}{{\int }^{}}_{-\infty }^{\infty }f\left(u\right)\left[{{\int }^{}}_{-\infty }^{\infty }g\text{(}t\text{−u)}d\text{u}{\psi }^{*}\left(\frac{t-b}{a}\right)d\text{t}\right]du\text{.}$

Using the Shift Theorem we obtain,

$WT\{f*g;a,b\}={\displaystyle {\int }_{-\infty }^{\infty }f(u)WT\{g;a,b-u\}du,}$

or equivalently,

$\text{WT{}f\star g\text{;}a\text{,}b\text{}=}f\stackrel{b}{\star }\text{{}g\text{;}a\text{,}b\text{}}\text{.}$

The following subsection illustrates the role that wavelets naturally play in radar signal processing.

5.2.2 Radar signal processing and wavelets

The problem consists of estimating the location and velocity of some target in a radar application. The estimation procedure can be described by the following. Suppose x(t) is a known emitting signal. In the presence of a target, this signal x(t) will return to the source as the received signal h(t) with a delay τ, due to the target's location and a Doppler effect distortion, due to the target's velocity. If x(t) is a narrow-band signal, then the Doppler effect amounts to a single frequency shift f ₀. The characteristics of the target can be determined by maximizing the cross-correlation function. This estimator is called the Narrow-Band Ambiguity Function, that is,

${{\int }^{}}_{-\infty }^{\infty }x\text{(}t\text{)}h\text{(}t-\tau \text{)}\exp \text{(-j2}\pi f_{\text{0}}t\text{)}dt$

which is simply the Short Time Fourier Transform (STFT) with shift τ about frequency f ₀.

However, for wide-band signals, the Doppler frequency shift varies the signal spectrum, causing a stretching or compression of the emitted signal. The resulting estimator is called the Wide-Band Ambiguity Function, that is

${\left(\text{|}a\text{|}\right)}^{\text{−1/2}}{{\int }^{}}_{-\infty }^{\infty }x\text{(}t\text{)}h\left(\frac{t-\tau }{a}\right)d\text{t}$

which is continuous wavelet transform (CWT) about a point τ at a scale given by a. Thus, the wavelet transform is an operation that determines the similarity or cross correlation between the emitted signal x(t) and the received signal, the wavelet h(t-τ/a) at scale a and shift τ.

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Numerical Analysis of Wavelet Methods

Albert Cohen , in Studies in Mathematics and Its Applications, 2003

2.15 Historical notes

Although the concepts and constructions that we have described throughout this chapter have mostly been developed since the end of the 1980's, they should be viewed as the result of the merging of several independent developments that took place in different areas of research:

1.

Time-frequency and time-scale analysis: since the 1950's, several ideas had been proposed to "localize" Fourier analysis in the context of signal processing. For example, Gabor proposed computing inner products Gf(a, b)   =   〈f, g_a,b 〉 of the analyzed function f against localized waves,

(2.15.1) $g_{a,b}\left(x\right)=e^{iax}g\left(x-b\right),a,b\in ℝ,$

where g is typically a gaussian function. In the early 1980's, the geophysicist Morlet who was looking for an analysis with arbitrarily high resolution in space (limited in (2.15.1) by the fixed support of g), proposed using

(2.15.2) ${\psi }_{a,b}\left(x\right)=a^{-1}\psi \left(\frac{x-b}{a}\right),a>0,b\in ℝ,$

where ψ is a function which is well localized both in space and frequency. The function ψ should also satisfy the condition

(2.15.3) $2\pi {\displaystyle \int {\left|\omega \right|}^{-1}\left|\widehat{\psi }{\left(\omega \right)}^2\right|d\omega =}C<+\infty ,$

which implies ∫ ψ  =   0. Moreover, he suggested that a function f can be reconstructed from its continuous wavelet transform Wf(a, b)   =   〈f, ψ_a,b 〉 by the formula

(2.15.4) $f=C^{-1}{\displaystyle \int {\displaystyle \int Wf\left(a,b\right){\psi }_{a,b}\left(x\right)\frac{dadb}{a}}}.$

This formula was proved to be true by Grossmann in 1983. From a more numerical point of view, Daubechies proved in 1984 that the sampling

(2.15.5) ${\psi }_{m,n}\left(x\right)=a_0^{m/2}\psi \left(a_0^{m}x-b_{0,n}\right),m,n\in \mathbb{Z},$

generates a frame of L ²(ℝ) if a ₀ > 1 and b ₀ > 0 are chosen small enough. The question of the existence of an orthonormal basis with the above structure and ψ well localized both in space and frequency (in contrast to the Haar system), was left open until a construction by Meyer in 1985, in which ψ belongs to the Schwartz class $S\left(ℝ\right)$ . A detailed and comprehensive survey of these developments will be found in Daubechies [1992].

2.

Harmonic analysis: Since the 1930's, Littlewood and Paley had proposed a systematic way of decomposing a function f into "almost orthogonal blocks". One fixes a positive C ^∞ function r(ω) which has compact support in [–2, 2] and equals 1 on [–––––1,1]. Any tempered distribution f can then be decomposed according to

(2.15.6) $f=S_{0}f+{\displaystyle \sum _{j\ge 0}{\text{Δ}}_{j}f},$

with $ℱS_{j}f\left(\omega \right)=\widehat{f}\left(\omega \right)r\left(2^{-j}\omega \right)$ and ${\text{Δ}}_{j}f=S_{j+1}f-S_{j}f$ . The block Δ _j f thus represents the frequency content of f between 2 ^j and 2 ^j+1. Such a decomposition has proved to be a powerful tool for the study of linear and multilinear operators, as well as for characterizing various function spaces, in particular smoothness classes, from the size properties of the blocks Δ _j (see Frazier, Jawerth and Weiss [1991] for a review on Littlewood-Paley theory and its applications). Clearly, wavelet decompositions can be viewed as a variant of (2.15.6) which is more adapted for numerical computations, since the Δ _j f are replaced by combinations of local functions.

3.

Approximation theory: the study of spline functions began with the work of Schonberg in the 1940's ( Schonberg [1946]), who is also at the origin of the Strang-Fix conditions described in §2.8. There is a huge amount of existing literature on splines, as well as on finite elements (we already mentioned some general references in this chapter). Let us say that an important step in the understanding of spline approximation was the introduction of local quasi-interpolant operators (which have the same structure as our local projectors P_j , see Remark 2.3.2) in the 1970's by de Boor and Fix. The general relations between spline functions and subdivision algorithms were analyzed in the 1980's ( Dahmen and Micchelli [1984]).

4.

Multiresolution image processing: many algorithms developed in image processing, as well as in computer-aided geometric design, since the 1960's, are based on a multiresolution processing of the visual information. Beside the subdivision algorithms (that have been discussed in §2.4) for fast curve or surface generation, and the iterated filterbanks (see Kovacevic and Vetterli [1995] for a detailed review) that are principally used for image compression and coding, one should also mention the laplacian pyramids introduced in Burt and Adelson [1983] which first made the connexion between techniques of multiscale refinement and image coding ideas, and can be viewed as a discrete implementation of the quasi-interpolant operators that were evoked in Remark 2.3.2.

The introduction of the concept of multiresolution approximation by Mallat in 1986, and the resulting constructions of compactly supported wavelets by Daubechies in 1988, can thus be viewed as a beautiful combination of these ideas.

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Automatic landmark registration of 1D curves

Jérémie Bigot , in Recent Advances and Trends in Nonparametric Statistics, 2003

4.1 Wavelets and structure of a signal

In many practical experiments, typical examples of landmarks are extrema and inflections points of a smooth function (see e.g. [20], [12], [16]). More generally, the landmarks considered in this paper are the points y ₀ where a function f is r-times (r >   1) continuously differentiable in a neighborhood of y ₀ and such that the rth derivative of f has a zero at y ₀ To characterize such landmarks, we propose to use the framework considered in [2 ] which consists in following the propagation at fine scales of the zero-crossings and the wavelet maxima lines of the continuous wavelet transform of a signal. In [ 2], it is shown how to detect the singularities of a function via its wavelet maxima lines, but in this paper, we restrict our study to the estimation of the zeros of the rth derivative of a smooth signal.

4.1.1 The continuous wavelet transform

We assume that we are working with an admissible real-valued wavelet ψ with r vanishing moments (r ∈ IN). We will suppose that the wavelet ψ has a fast decay and has no more than r vanishing moments which implies (see Theorem 6.2 of Mallat [15]) that there exists θ with a fast decay such that: $\psi \left(u\right)={\left(-1\right)}^r\frac{d^r\theta \left(u\right)}{d{t}^r}$ and ${\displaystyle {\int }_{-\infty }^{+\infty }\theta }\left(u\right)du\ne 0$ . Moreover, we will assume that the wavelet ψ is normalized to one i.e.: ${\displaystyle {\int }_{-\infty }^{+\infty }{\left(\psi \left(u\right)\right)}^2}du=1$ . Then, by definition, the continuous wavelet transform of a function f ∈ L ²(ℝ) at a given scale s is: $W_s\left(f\right)\left(x\right)={\displaystyle {\int }_{-\infty }^{+\infty }f}\left(u\right){\psi }_s\left(u-x\right)du,$ , where ${\psi }_s\left(u\right)=\frac{1}{\sqrt{S}}\psi \left(\frac{u}{s}\right)$ .

4.1.2 Zero-crossings of the wavelet transform

Suppose that a function f ∈ L ²(ℝ) is r-times continuously differentiable in an interval [a, b]. By commuting the convolution and differentiation operators, and given that θ has a fast decay, we have that, for all x ∈]a, b[,

(1) $\underset{s\to 0}{lim}\frac{W_s\left(f\right)\left(x\right)}{s^{r+1/2}}=\underset{s\to 0}{lim}{f}^{\left(r\right)}\star \frac{1}{s}\theta \left(\frac{-x}{s}\right)=K{f}^{\left(r\right)}\left(x\right),\mathrm{Where}K={\displaystyle {\int }_{-\infty }^{+\infty }\theta \left(u\right)du\ne 0}$

The term zero-crossings will be used to describe any point (z ₀, s ₀) in the time-scale space such that $z\mapsto \left|W_{s_0}\left(f\right)\left(z\right)\right|$ has exactly one zero at z  = z ₀ in a neighborhood of z ₀. Equation (1) proves that the zero-crossings of the wavelet transform converge to the zeros of the rth derivative of f that are located in [a, b] when the scale s  →   0. Hence, one can find the location of the extrema (resp. the points of inflexion) of a function by following the propagation of the zero-crossings of its wavelet transform when the scale s decreases and when the mother wavelet has r =   1 (resp. r =   2) vanishing moments.

4.1.3 Some properties of the zero-crossings lines

We will call zero-crossings line any connected curve z(s) in the time-scale plane (x, s) along which all points are zero-crossings. Prom equation (1), one might think that it is sufficient to follow any zero-crossings line to detect the landmarks of a function. However, we are not guaranteed that, for any wavelet ψ  =   (−   1) ^rθ ^(r), r ≥   0, a zero-crossings located at (z ₁, s ₁) belongs to a zero-crossings line that propagates towards finer scales. A zero-crossings line might be interrupted when the scale decreases. However, if θ is a Gaussian (i.e $\theta \left(x\right)=K\frac{1}{\sqrt{2\pi \beta }}{e}^{-\frac{x^2}{2{\beta }^2}}$ for some non-negative reals K and β), then for any f ∈ L ²(ℝ), the zero-crossings of W_s (f)(x) belong to connected curves that are never interrupted when the scale decreases (see e.g. Proposition 6.1 of Mallat, [15]). In Figure 1, we plotted the zero-crossings of the Time Shifted Sine f_T =   0.3 sin 3π[g(g(g(g(x))))   + x] +   0.5, x ∈ [0,1], where g(x)   =   (1   −   cos(πx))/2, for a Gaussian wavelet with r =   1 and β =   0.5. It. can be seen in Figure 1b that all the zero-crossings lines propagate to fine scales. In [2], the properties of the zero-crossings lines are investigated in detail by using the notion of causality arising from the scale-space literature (see Lindeberg [13]). However, in this paper, we will not investigate the properties of these lines for any wavelet but will restrict our study to wavelets that are derivatives of a Gaussian. Then, in this case, one obtain that the limits of the zero-crossings lines correspond the landmarks of a signal (see Proposition 3.3 in [2]).

Figure 1. (a) Time function f_T . (b) Zero-crossings lines of f_T computed with a Gaussian wavelet (β =   0.5) with r =   1 vanishing moments, (c) Structural intensity of the zero-crossings. (d) Noisy signal f_T with n =   512. (e) Estimated zero-crossings (thick lines) and true zero-crossings (thin lines) of f_T for r =   1. (f) Structural intensity of the estimated zero-crossing of f_T .

4.1.4 Structural intensity of the zero-crossings lines

As pointed out in [2], we only have a visual representation of the zero-crossings lines in the time-scale plane that indicates "where" the landmarks are located, and the analytical expression of these lines is generally unknown. To overcome this drawback, a new tool called "structural intensity" has been introduced in [2] (see Proposition 3.4) to identify the limits of the zero-crossings lines computed for a scale-space representation derived from B-Spline wavelets. The following proposition extends the concept of structural intensity to wavelets that are not compactly supported:

Proposition 1

Let f ∈ L² (ℝ) and ψ  =   (−   l) ^r θ ^(r), r ≥   1 where θ is a Gaussian. Suppose that there exists p zero-crossings lines Z_i (s) that respectively converge to y_i ∈ ℝ as s tends to zero and that are C ¹ in a neighborhood, of y_i ,i =   1,…,p. For x ∈ ℝ, define the structural intensity of the zero-crossings G_z (x) as:

$G_z\left(x\right)={\displaystyle \sum _{i=1}^p{\displaystyle {\int }_0^{s_{z_i}}\frac{1}{s}\theta \left(\frac{x-z_i\left(s\right)}{s}\right)ds\text{,}}}$

where $\left[0,s_{z_i}\right]$ is the support of the zero-crossings line z_i (.). Then, G_z is continuously differentiable on ℝ \ {y ₁,…,y_p } and such that G_z (x)   →   +∞ as x  → y_i , i =   1,…,p.

Proof. Let f ∈ L ²(ℝ) satisfying the above assumptions. Define:

$g_{z_i}\left(x\right)={\displaystyle {\int }_0^{s_{z_i}}\frac{1}{s}\theta \left(\frac{x-z_i\left(s\right)}{s}\right)ds\mathrm{f}\mathrm{o}\mathrm{r}x\in \mathrm{ℝ}\mathrm{and}1\le i\le p\text{.}}$

Given that θ has a fast decay, there exists a constant C such that for all x ∈ ℝ and all $s\in \left[0,s_{z_i}\right]$ , ${\theta }^{\prime }\left(\frac{x-z_i\left(s\right)}{s}\right)\le \frac{C}{1+{\left(\frac{x-z_i\left(s\right)}{s}\right)}^2}$ . Let a  < y _i, and x ∈]   −   ∞, a[. Given that z_i (s)   → y_i as s  →   0, there exists a constant M and a scale s ₀  >   0 such that for all x ∈]   −   ∞, a[ and all s  ≤ s ₀, |x  − z_i (s)|   ≥ M which imlies that:

(2) ${\theta }^{\prime }\left(\frac{x-z_i\left(s\right)}{s}\right)\le \frac{C{s}^2}{M^2}\text{,}$

for all x ∈]   −   ∞,a,[ and all s  ≤ s ₀. Now, note that for all $s\in ]0,s_{m_i}]$ , $x\mapsto \frac{1}{s}\theta \left(\frac{x-z_i\left(s\right)}{s}\right)$ is continuously differentiable on ]   −   ∞ a[. From equation (2), we have that $s\mapsto \frac{1}{s^2}{\theta }^{\prime }\left(\frac{x-z_i\left(s\right)}{s}\right)$ is bounded on $\left[0,s_{m_i}\right]$ and the differentiation theorem under the integral sign finally implies that $g_{z_i}$ is continuously differentiable on ]   −   ∞ a[. We can similarly show that $g_{z_i}$ is continuously differentiable on ]a,+∞[ for a  > y_i .

Now, since z_i (.) is C ¹ in a neighborhood of y _i, we can define $A={max}_{\mathrm{s}\in \left[0,s_{m_i}\right]}\left\{\frac{\left|y_i-z_i\left(s\right)\right|}{s}\right\}$ . Given that θ is a Gaussian, we obtain that for all $s\in ]0,s_{m_i}],\theta \left(\frac{y_i-z_i\left(s\right)}{s}\right)\ge \theta \left(A\right)$ which implies that $s\mapsto \frac{1}{s}\theta \left(\frac{y_i-z_i\left(s\right)}{s}\right)$ is not integrable on $\left[0,s_{m_i}\right]$ and finally shows that $g_{z_i}\left(x\right)\to +\infty$ as x  → y_i . Since $G_z\left(x\right)={\sum }_{i=1}^p{g}_{z_i}\left(x\right)$ , the result immediately follows. □

Roughly speaking, calculating the structural intensity consists in computing the "density" of the zero-crossings along various scales. Then, Proposition 1 shows that in practice, the modes of the resulting "density" are located at the landmarks of the signal f as it can be seen in Figures 1c where the local maxima of G_z (.) correspond to the limits of the zero-crossings lines of the signal f_T . Note that in practice, the structural intensities are normalized to be probability density functions.

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