On the convergence of derivatives of B-splines to derivatives of the Gaussian function

 

 

Ralph Brinks

Philips Research Laboratories, Weisshausstrasse 2, 52066 Aachen, Germany. E-mail: rbrinks@gmx.de

 

 

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ABSTRACT

In 1992 Unser and colleagues proved that the sequence of normalized and scaled B-splines B_m tends to the Gaussian function as the order m increases, [1]. In this article the result of Unser et al. is extended to the derivatives of the B-splines. As a consequence, a certain sequence of wavelets defined by B-splines, tends to the famous Mexican hat wavelet. Another consequence can be observed in the continuous wavelet transform (CWT) of a function analyzed with different B-spline wavelets.
Mathematical subject classification: 43A15, 42C40, 65T60, 65R10.

Key words: B-splines, Gaussian function, Continuous wavelet transform, Mexican hat wavelet, Scalogram.

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1 Introduction

In 1992 Unser and colleagues proved that the sequence of normalized and scaled B-splines B_m tends to the Gaussian function as the order m increases, [1]. In this article the result of Unser et al. is extended to the derivatives of the B-splines. After some definitions and introductions this is done in the second section.

Due to numerical reasons it has become common to use B-splines as analyzing functions (wavelets) in wavelet analysis. Two implications of the convergence result of the second section in wavelet analysis are presented in the third section. First, a certain sequence of wavelets defined by B-splines tends to the famous Mexican hat wavelet. Second, similarities in continuous wavelet transform (CWT) can be observed, when a function is analyzed with different B-spline wavelets.

 

2 B-splines: definition and some properties

Let p Î ^>1 and L^p() as usual denote the set

For p = 2 and f, g Î L²() define the inner product

and the norm

making L²() to a Hilbert space. For f, g Î L²() the function f * g is defined as

f * g is called convolution product of f and g and is in L²().

For f Î L¹() define the Fourier transform f^Ù and the inverse Fourier transform f^Ú as

This definition can be extended to functions f Î L²(), see for example [2].

Now let us define the cardinal B-splines:

Definition 2.1. Define

and let B_m be defined as

B_m : = B₀ * B_m-1, m Î .

Then B_m, m Î ₀, has the compact support and is in C^m-1(), (C^-1() denotes the set of the piecewise constant functions). The B_m are the famous cardinal B-Splines.

B-splines play an important role in computer-aided design (CAD) and signal processing. For an extensive monography see [3].

For all B_m, m Î ₀, it holds B_m Î L¹() Ç L²() and

where sinc denotes the sinus cardinalis function:

After the necessary definitions have been provided, we can come to the main result of this article. Unser and colleagues in [1] have shown that normalized and scaled versions of the B-splines tend to the Gaussian function:

where the limit may be taken pointwise or in L^p(), p Î [2, ¥).

Now it is shown that taking the k-th derivative of both sides of equation (2.1) is possible. It holds:

Theorem 2.2. Let be k Î ₀ then for m > k + 1 the sequence of the k-th derivatives of the cardinal B-splines converges to the k-th derivative of the Gaussian function,

where the limit may be taken pointwise or in L^p(), p Î [2, ¥).

The proof of Theorem 2.2 is based on two lemmata.

Lemma 2.3. Let be k Î ₀ and A : ® be a two times continuously differentiable function with a positive maximum at x₀ such that

1. A(x₀) > 0,

2. A¢(x₀) = 0,

3. A¢¢(x₀) = - a² A(x₀) < 0, a ¹ 0,

then for the sequence

it holds

where the limit is taken pointwise.

Proof. First of all let be k = 0. Define L_n(x) := ln A_n . Then with t_n := it holds

Since it holds A₁(0) = 1 and A¢₁(0) = 0, for the limit n ® ¥, and hence t_n ® 0, we may apply l'Hôpital's rule twice:

This implies the hypothesis for k = 0. For k > 0 the hypothesis follows from the case k = 0 and the limit theorems (multiplying x^k on both sides).

Lemma 2.4. For k Î ₀ and n Î , n > k + 2, there is a constant c_k Î ^>0 such that for

it holds

G_k Î L^p(), " p Î [1, ¥),

and

Proof. The fact that G_k(x) Î L^p(), p > 1, may be concluded from exp(-x²) < x^-2k for x large in modulus. Due to the symmetry we may assume x > 0. For x Î [0, 1] it holds

and hence sinc(p x) < 1 - x².

1. Let x Î [ 0, ].

Then it holds

Now we show that

Let

then it holds

and thus = p(0) = 1.

2. Let x Î [, ¥).

Set

The existence of c_k follows from the convergence of the sequence

Then one follows

By noting that

the hypothesis follows.

Proof of Theorem 2.2. Set A(x) := sinc(), then for A the assumptions of Lemma 2.3 are fulfilled with a = . The lemma states that

tends to x^k exp() if n ® ¥. Furthermore, it holds = sinc^(m+1) (), and B_m Î C^m-1(). The later yields for k < m - 1:

Consequently, one obtains

where the limit in equation (2.2) follows from Lemma 2.3 and is meant to be taken pointwise. The L^p-convergence, p > 1, follows from Lebesgue's Dominated Convergence Theorem, using Lemma 2.4. The hypothesis follows from taking the Fourier transform and the fact that for p Î [1, 2] and = 1 the Fourier transform is a bounded operator from L^p() to L^q(), see [2, Sect. V.1].

 

3 Wavelet analysis

In this section two implications of Theorem 2.2 for wavelet analysis are presented briefly. For a more detailed treatment of wavelet analysis, see for example the books of Daubechies [4] or Chui [5].

Definition 3.1. A function y Î L²()\ {0} is called a wavelet, iff y fulfills the admissibility condition

holds. Henceforth, we assume y to be normalized: ||y|| = 1.

Example 3.2. Here we give two examples of famous wavelets:

1. The Haar wavelet is defined by

2. The second derivative of the Gaussian exp

is another standard wavelet, the Mexican hat wavelet. The normalized version is (t² - 1) exp .

3.1 Convergence to the Mexican hat wavelet

The construction of the Mexican hat wavelet as a derivative of the Gaussian function may be generalized:

Proposition 3.3. Let j Î C^k() Ç L²()\ {0}, k Î , with j^(m)(t) ® 0 as |t| ® 0 for all m Î {0, ..., k - 1} and j^(k) Î L¹() Ç L²(), then

is a normalized wavelet.

Proof. First of all y cannot be constant zero, which would imply that j is a polynomial, what contradicts the assumptions. Without loss of generality assume ||j^(k)|| = 1. From Fourier analysis it follows y^Ù(w) = (i w)^k · j^Ù(w) and hence

The Parseval formula in L²() proves that y is admissible.

With a view to numerical calculations wavelets with compact support are mostly preferred. Thus, the derivatives of the cardinal B-splines B_m, which of course have compact support, are nice candidates for wavelets. For those wavelets, we now can state the first consequence from Theorem 2.2:

Corollary 3.4. For m Î , m > 3, define the normalized wavelets

then the sequence (y_m)_m>3 converges to the normalized Mexican hat wavelet as m tends to infinity.

3.2 Similarity in the continuous wavelet transform

The second consequence of the convergence result of the previous section relates to the continuous wavelet transform (CWT). The CWT is an integral transform, which correlates a given function f to the dilated and translated versions of wavelet y.

Definition 3.5. Let f, y Î L²(), then define

If y is a wavelet, W_y[f] is called continuous wavelet transform (CWT) of f.

Remark 3.6. By Cauchy-Schwarz's inequality W_y[f] is well-defined and bounded.

In the continuous wavelet transform (3.2) the parameter a Î ^¹0 is the scale or dilation parameter, and b Î is the translation parameter, which may be interpreted as time. Since |y(t)| ® 0 as |t| ® 0, the scale parameter a is proportional to the inverse of the frequency which can be detected by correlating the signal f to : the larger the scale a, the wider the window of the wavelet, and hence the smaller the corresponding analyzed frequency. The CWT is maximal if the frequency of the input signal f matches that of the corresponding wavelet and the phases coincide.

Hence, for an one-dimensional input signal the transform W_y[f] is a two-dimensional description of the signal with respect to the time b and the scale a. The (b, a)-space is called parameter plane and a plot of |W_y[f](b, a)|² against the parameter plane is called a scalogram. Since we are using real wavelets only, henceforth a scalogram is a plot of |W_y[f](b, a)| against the parameter plane. Those are often arranged as contour plots where the brightness of each pixel represents the modulus of the CWT, see Figure 1.

 

 

The comparison of the zero-lines of different B-Spline wavelets shows an astonishing similarity for the same order of differentiation. For example the zero-lines in the CWT for normalized versions of , and are just vertically shifted as may be seen in Figure 2.

 

 

If only the bending points of the zero-lines are plotted in the parameter plane, the shift may be seen more impressively by cascades of three points with the same abscissa values, cf. Figure 3.

 

 

The groups of three bending points may be explained by the qualitative similarity of the analyzing wavelets , and , see Figure 4. It is striking that the ratios of the ordinate values of the groups of bending points remain almost constant. Within such a triple group the corresponding ordinates ratios of / and / are around 0.92 and 0.93 respectively. To explain this, let y₁ and y₂ be two analyzing wavelets where y₂ is just a scaled (and normalized) version of y₁. Let s Î ^>0 and

y₂(t) = y₁(s · t).

 

 

Then it holds

which means that for any (b₀, a₀) Î × ^¹0:

Consequently, each zero (b₀, a₀) of W_y2[f] is shifted to the zero (b₀, ) of W_y1[f].

Theorem 2.2 explains the vertical shifts of the scalograms by values around 0.9. For m = 4 and m = 5 respectively the ratios of the coefficients inside the arguments of and are » 0.913 and » 0.926 respectively.

 

4 Summary

In this article it has been shown that a convergence result of the sequence of B-splines of increasing order to the Gaussian function can be generalized to the derivatives. Since derivatives of B-splines are often used as wavelets in the continuous wavelet transform (CWT), this generalization has implications in the field of wavelet analysis. First, the sequence of the second order derivatives of the B-splines tends to the Mexican hat wavelet, and second, the CWTs of a function with different B-spline wavelets have astonishing similarities.

 

REFERENCES

[1] M. Unser, A. Aldroubi and M. Eden, On the Asymptotic Convergence of B-Spline Wavelets to Gabor Functions. IEEE Trans. Inform. Theo., 38(2) (1992).         [ Links ]

[2] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, 1975.         [ Links ]

[3] C. de Boor, A Practical Guide to Splines. Springer-Verlag, New York, 1978.         [ Links ]

[4] I. Daubechies, Ten Lectures on Wavelets. SIAM, Philadelphia, 1992.         [ Links ]

[5] C.K. Chui, An Introduction to Wavelets. Academic Press, Boston, San Diego, 1992.        [ Links ]

 

 

Received: 28/II/07. Accepted: 26/VI/07.

 

 

#710/07.