Clifford-wavelet Transform and the uncertainty principle

TL;DR

This paper establishes a Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform, expanding the theoretical framework of Clifford analysis and wavelet theory.

Contribution

It introduces a novel uncertainty principle specific to Clifford wavelets, integrating Clifford algebra with wavelet analysis.

Findings

Derived a Heisenberg-type uncertainty principle for Clifford wavelet transform.

Reviewed Clifford algebra, wavelet transform, and Clifford-Fourier transform properties.

Applied concepts to develop an uncertainty principle based on Clifford wavelets.

Abstract

In this paper we derive a Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform. A brief review of Clifford algebra/analysis, wavelet transform on $\mathbb{R}$ and Clifford-Fourier transform and their proprieties has been conducted. Next, such concepts have been applied to develop an uncertainty principle based on Clifford wavelets.

Equations

$${\mathcal{A}}_{\psi }={\int }_{\mathbb{R}}\frac{{\left|\hat{\psi }(\xi )\right|}^2}{\left|\xi \right|}d\xi <+\infty ,$$

$$e_i{e}_j+e_j{e}_i=-2{\delta }_{ij},$$

$${\mathbb{R}}_n={\mathbb{R}}_n^0\oplus {\mathbb{R}}_n^1\oplus \cdots \oplus {\mathbb{R}}_n^n,$$

$${\mathbb{R}}_n^k=spa{n}_{\mathbb{R}}\left\{e_A=e_{i_1{i}_2\dots i_k},1\le i_1<i_2<\cdots <i_k\le n\right\},$$

$${\mathbb{R}}_n={\mathbb{R}}_n^{+}\oplus {\mathbb{R}}_n^{-}=\underset{k\text{even}}{⨁}{\mathbb{R}}_n^k\oplus \underset{k\text{odd}}{⨁}{\mathbb{R}}_n^k$$

$$a=\sum _A{a}_A{e}_A,a_A\in \mathbb{R},$$

$$a=\sum _{k=0}^n\sum _{|A|=k}{a}_A{e}_A,$$

$$\underline{x}\underline{y}=\underline{x}\cdot \underline{y}+\underline{x}\wedge \underline{y},$$

$$\underline{x}\cdot \underline{y}=-<\underline{x},\underline{y}>=-\sum _{j=1}^n{x}_j{y}_j,$$

$$\underline{x}\cdot \underline{y}=\frac{1}{2}(\underline{x}\underline{y}+\underline{y}\underline{x})and\underline{x}\wedge \underline{y}=\frac{1}{2}(\underline{x}\underline{y}-\underline{y}\underline{x}).$$

$${\underline{x}}^2=-{\left|\underline{x}\right|}^2=-\sum _{j=1}^n|x_j{|}^2.$$

$$T(\underline{x})=\left[R_{{\underline{\omega }}_1}\circ R_{{\underline{\omega }}_2}\circ ......\circ R_{{\underline{\omega }}_{2l}}\right](\underline{x})={\underline{\omega }}_1{\underline{\omega }}_2,......{\underline{\omega }}_{2l}\underline{x}{\underline{\omega }}_{2l}{\underline{\omega }}_{2l-1},......{\underline{\omega }}_2{\underline{\omega }}_1.$$

$$Spin(n)=\left\{s\in {\mathbb{R}}_n;s=\prod _{j=1}^{2l}{\underline{\omega }}_j,{\underline{\omega }}_j^2=-1,1\le j\le 2l\right\}.$$

$$f(\underline{x})=\sum _A{f}_A(\underline{x})e_A,$$

$$<f,g{>}_{L^2({\mathbb{R}}_n,dV(\underline{x}))}={\int }_{{\mathbb{R}}^n}{\left[f(\underline{x})\right]}^{†}g(\underline{x})dV(\underline{x})$$

$$\Vert f{\Vert }_{L^2({\mathbb{R}}_n,dV(\underline{x}))}=<f,f{>}_{L^2({\mathbb{R}}_n,dV(\underline{x}))}^{\frac{1}{2}}.$$

$${\Vert f\Vert }_{L^1({\mathbb{R}}_n,dV(\underline{x}))}={\int }_{{\mathbb{R}}^n}|f(\underline{x})|dV(\underline{x})$$

$$\left|<f,g{>}_{L^2({\mathbb{R}}_n,dV(\underline{x}))}\right|\le {\Vert f\Vert }_{L^2({\mathbb{R}}_n,dV(\underline{x}))}{\Vert g\Vert }_{L^2({\mathbb{R}}_n,dV(\underline{x}))}$$

$$0<{\mathcal{A}}_{\psi }={\int }_{\mathbb{R}}\frac{(\hat{\psi }(\xi ){)}^2}{\left|\xi \right|}d\xi <\infty ,$$

$$\exists N\in \mathbb{N},\underset{\mathbb{R}}{\int }{x}^n\psi (x)dx=0,\forall n\le N.$$

$$T_{\psi }[f](a,b)=\underset{\mathbb{R}}{\int }\overline{{\psi }^{a,b}(x)}f(x)dx=(\widetilde{{\psi }_a}\ast f)(b),$$

$${\int }_{{\mathbb{R}}^{\ast }}{\int }_{\mathbb{R}}{\left|T_{\psi }[f](a,b)\right|}^{2}db\frac{da}{a^2}<\infty .$$

$$\left[T_{\psi }\left[f\right],T_{\psi }\left[g\right]\right]=\frac{1}{{\mathcal{A}}_{\psi }}\underset{\mathbb{R}}{\int }\underset{\mathbb{R}}{\int }\overline{T_{\psi }\left[f\right](a,b)}{T}_{\psi }\left[g\right](a,b)\frac{da}{a^2}db,$$

$$\left[T_{\psi }\left[f\right],T_{\psi }\left[g\right]\right]=<f,g{>}_{L^2(\mathbb{R},dx)}.$$

$$f(x)=\frac{1}{{\mathcal{A}}_{\psi }}\underset{\mathbb{R}}{\int }(\underset{{\mathbb{R}}^{+}}{\int }{\psi }^{a,b}(x)T_{\psi }\left[f\right](a,b)\frac{da}{a^2})db.$$

$${\Vert Af\Vert }_2{\Vert Bf\Vert }_2\ge \frac{1}{2}\left|<\left[A,B\right]f,f>\right|,\forall f\in \mathcal{D}(\left[A,B\right]).$$

$$\mathcal{F}\left[f\right](\underline{\xi })=\hat{f}(\underline{\xi })=\frac{1}{(2\pi {)}^{\frac{n}{2}}}{\int }_{{\mathbb{R}}^n}{e}^{-i<\underline{x},\underline{\xi }>}f(\underline{x})dV(\underline{x}).$$

$${\mathcal{F}}^{-1}\left[f\right](\underline{x})=\frac{1}{(2\pi {)}^{\frac{n}{2}}}{\int }_{{\mathbb{R}}^n}{e}^{i<\underline{x},\underline{\xi }>}\hat{f}(\underline{\xi })dV(\underline{\xi }).$$

$$A_{k}f(\underline{x})=x_{k}f(\underline{x})and{B}_{k}f(\underline{x})={\partial }_{x_k}f(\underline{x}),k=1,2,\cdots ,n.$$

$${\Vert A_{k}f\Vert }_{L^2({\mathbb{R}}_n,dV(\underline{x}))}{\Vert B_{k}f\Vert }_{L^2({\mathbb{R}}_n,dV(\underline{x}))}\ge \frac{1}{2}\left|<\left[A_k,B_k\right]f,f>\right|,$$

Full text

1. Introduction

Transformations such as the Fourier one are powerful methods for signals representations and features detection in signals. The signals are transformed from the original domain to the spectral or frequency one. In the frequency domain many characteristics of a signal are seen more clearly. Contrarily to the Fourier modes, wavelet basis functions are localized in both spatial and frequency domains and thus yield very sparse and well-structured representations of signals, important facts from a signal processing point of view. The first work on wavelet analysis has been done by Morlet in [18] to study seismic waves. He also, with Grossman, gave a mathematical study of continuous wavelet transform (see [19]). In [20], Meyer recognised the link between harmonic analysis and Morlet’s theory and gave a mathematical foundation to the continuous wavelet theory. The continuous-wavelet analysis of a square integrable function $f$ begins by a convolution with copies of a given “mother wavelet” $\psi$ translated and dilated respectively by $b\in\mathbb{R}$ and $a>0$. Such a function $\psi$ has to fulfil an admissibility condition which states that

[TABLE]

where $\widehat{\psi}$ is the classical Fourier transform of $\psi$. More information on real wavelet can be found in [15] and [16] and a generalization to Sobolev spaces $\mathcal{H}^{s}(\mathbb{R})$ for an arbitrary real number $s$ in [17].

On the other hand, Clifford analysis leads to the generalization of real and harmonic analysis to higher dimensions. Clifford algebra accurately treats geometric entities depending on their dimension such as scalars, vectors, bivectors and volume elements, etc. The distinction of axial and polar vectors in physics, e.g. is resolved in the form of vectors and bivectors. For example, the quaternion description of rotations is fully incorporated in the form of rotors. With respect to the geometric product of vectors, division by non-zero vectors is defined. Clifford analysis/algebra has started to take place especially in signal and image processing (See for instance [2], [4], [5], [6], [8], [9], [13]).

The present paper lies in the same topic of Clifford algebra/analysis applications. We aim to develop an uncertainty principle proof in the Clifford analysis framework based on Clifford wavelets.

The paper is organized as follows: In section 2 we give a brief review of the Clifford analysis, introduce the notion of wavelet transform in $\mathbb{R}$ and the uncertainty principle. The third section is devoted to some results and properties of the Clifford-Fourier transform. In section 4, Clifford-wavelet transform has been investigated. In section 5, the uncertainty principle for the Clifford wavelet transform is established.

2. Preliminaries

In this section, we aim to recall the basic properties of the Clifford algebras (See [8], [13] and the references therein). Next, a brief review of continuous wavelet transform in $\mathbb{R}$ and the Heisenberg uncertainty principle are developed.

2.1. Clifford Algebras

The Clifford algebra $\mathbb{R}_{n}$ associated to $\mathbb{R}^{n}$ is an associative algebra generated by an orthonormal (the canonical) basis $\left\{e_{1},e_{2},\dots,e_{n}\right\}$ by means of a non commutative product

[TABLE]

where $\delta$ is the Kronecker symbol. This yields a finite $2^{n}$-dimensional algebra known as the Clifford algebra $\mathbb{R}_{n}$. It is decomposed as a direct sum

[TABLE]

where $\mathbb{R}_{n}^{k}$ are the spaces of $k$-multi vectors defined by

[TABLE]

where $e_{A}=e_{i_{1}i_{2}\dots i_{k}}=e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}$. We may also decompose $\mathbb{R}_{n}$ as two sub-algebras

[TABLE]

called respectively the even and odd sub-algebras. Consequently, any Clifford number $a\in\mathbb{R}_{n}$ has a representation of the form

[TABLE]

where $e_{A}=e_{i_{1}i_{2}\cdots i_{k}}=e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}$, with $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n$, $e_{\emptyset}=1$ and $A=(i_{1},i_{2},\dots,i_{k})$. Denoting $|A|$ the length or the cardinality for the multi-index $A$, the element $a$ may be written as

[TABLE]

On the algebra $\mathbb{R}_{n}$, we may introduce some involutive operators such as

• •

Main-involution: $\widetilde{e_{j}}=-e_{j}$, $\forall j$, which yiels that $\widetilde{e_{A}}=(-1)^{\left|A\right|}e_{A}$ and consequently, $\widetilde{ab}=\widetilde{a}\widetilde{b}$, $\forall a,b\in\mathbb{R}_{n}$.

• •

Reversion: $e_{j}^{\ast}=e_{j}$, $\forall j$, which in turns yields that $e_{A}^{\ast}=(-1)^{\frac{\left|A\right|(\left|A\right|-1)}{2}}e_{A}$ and thus $(ab)^{\ast}=b^{\ast}a^{\ast}$, $\forall a,b\in\mathbb{R}_{n}$.

• •

Conjugation: $\overline{e_{j}}=-e_{j}$, $\forall j$, yielding that $\overline{e_{A}}=(-1)^{\frac{|A|(|A|+1)}{2}}e_{A}$ and consequently, $\overline{ab}=\overline{b}\overline{a}$, $\forall a,b\in\mathbb{R}_{n}$.

The concept of real Clifford algebra can be extended to the complex Clifford algebra $\mathbb{C}_{n}=\mathbb{R}_{n}+i\mathbb{R}_{n}$. An element $\lambda\in\mathbb{C}_{n}$ may be written on the form $\lambda=\sum_{A}\lambda_{A}e_{A},\,\lambda_{A}\in\mathbb{C}$ and thus possesses the decomposition $\lambda=a+ib$, $a,b\in\mathbb{R}_{n}$. This induces the

• •

Hermitian conjugation $\lambda^{\dagger}=\overline{a}-i\overline{b}$.

In this context, a vector $x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}$ may be identified to the Clifford element in $\mathbb{R}_{n}$, $\underline{x}={\displaystyle\sum_{j=1}^{n}x_{j}e_{j}}$. This permits to define the Clifford product of two vectors by

[TABLE]

where the $\cdot$ product is

[TABLE]

the classical inner product on $\mathbb{R}^{n}$ and where the $\wedge$ product is the outer product $\underline{x}\wedge\underline{y}={\displaystyle\sum_{j<k}e_{j}e_{k}(x_{j}y_{k}-x_{k}y_{j})}$. This yields that

[TABLE]

In particular we have

[TABLE]

Any vector $\underline{x}$ is decomposed as $\underline{x}=\underline{x}_{\parallel\underline{\omega}}+\underline{x}_{\perp\underline{\omega}}$ for a $\underline{\omega}\in\mathcal{S}^{n-1}$ with $\underline{\omega}\cdot\underline{x}_{\perp\underline{\omega}}=0$ and $\underline{\omega}\wedge\underline{x}_{\parallel\underline{\omega}}=0$, which in turns induces that $\underline{x}_{\parallel\underline{\omega}}=\left\langle\underline{x},\underline{\omega}\right\rangle\underline{\omega}$ and $\underline{x}_{\perp\underline{\omega}}=\underline{\omega}\left(\underline{x}\wedge\underline{\omega}\right)$. This permits next to characterize the reflection $R_{\underline{\omega}}$ with respect to the hyperplane $\underline{\omega}^{\perp}$ as $R_{\underline{\omega}}(\underline{x})=\underline{\omega}\underline{x}\underline{\omega}$. Cartan-Dieudonné Theorem ([3]) relates the reflection to the so-called spinors. It states that there exists elements $\underline{\omega}_{1},\underline{\omega}_{2},\dots,\underline{\omega}_{2l}\in\mathcal{S}^{n-1}$ with $\underline{\omega}_{j}^{2}=-1,1\leq j\leq 2l$ and a rotation $T\in SO(n)$ such that

[TABLE]

Denoting next $s=\underline{\omega}_{1}\underline{\omega}_{2},......\underline{\omega}_{2l}$ and $\overline{s}=\underline{\omega}_{2l}\underline{\omega}_{2l-1},......\underline{\omega}_{2}\underline{\omega}_{1}$, we have $T(\underline{x})=s\underline{x}\overline{s}$. The element $s$ is called a spinor. Generally speaking, the spin group of order $n$ is

[TABLE]

To finish with this brief overview on Clifford algebra/analysis, it remains to recall the functional framework. Let $f:\mathbb{R}^{n}\longrightarrow\mathbb{C}_{n}$. It may be expressed as

[TABLE]

where $f_{A}$ are generally $\mathbb{C}$-valued functions and $A\subset\left\{1,2,\cdots,n\right\}$. The inner product of two functions, $f=\displaystyle\sum_{A}f_{A}(\underline{x})e_{A}$ and $g=\displaystyle\sum_{B}g_{B}(\underline{x})e_{B}$ is defined by

[TABLE]

and the associated norm by

[TABLE]

We denote also

[TABLE]

where $dV(\underline{x})$ stands for the Lebesgue measure. The inner product (2.1) satisfies the Cauchy-Schwartz inequality

[TABLE]

2.2. Wavelets on $\mathbb{R}$

Wavelet analysis of functions is based as the Fourier one on some type of transform known as wavelet transform which consists in some product and/or projection of the function on suitable windows issued from one source analyzing function called mother wavelet and which plays the role of the Fourier mode $e^{ix}$. Denote $\psi$ such function. It should satisfy

• •

A finite energy or space/time localization assumption: $\psi\in L^{2}(\mathbb{R},dx)$.

• •

An admissibility assumption stating that

[TABLE]

where $\widehat{\psi}(\xi)=\displaystyle\int_{\mathbb{R}}\psi(x)e^{-ix\xi}dx$ is the classical Fourier transform of $\psi$.

• •

Vanishing moments:

[TABLE]

Definition 1**.**

(Continuous-wavelet transform) Let $f\in{L}^{2}(\mathbb{R},dx)$. Its continuous wavelet transform is defined as

[TABLE]

where $\psi^{a,b}(x)=\frac{1}{\sqrt{a}}\psi(\frac{x-b}{a})$, $(a>0,\,b\in\mathbb{R})$ and $\widetilde{\psi_{a}}(x)=\frac{1}{\sqrt{a}}\overline{\psi(\frac{-x}{a})}.$

Using the admissibility assumption above, we immediately affirm that

[TABLE]

More precisely, an inner product may be defined for the wavelet transforms as

[TABLE]

which in turns may be related to the inner product of the analyzed functions $f$ and $g$ by means of a Fourier-Plancherel type formula as

[TABLE]

Moreover, a reconstruction formula may also be induced yielding that

[TABLE]

2.3. Uncertainty Principle

The uncertainty principle also known as Heisenberg’s uncertainty principle discovered in 1927 by Heisenberg in [10] is certainly one of the most famous and important concepts of quantum mechanics. It plays an important role in the development and understanding of quantum physics. The physical origin of uncertainty principle is related to quantum systems and states that: the determination of positions by performing measurement on the system disturbs it sufficiently to make the determination of momentum imprecise and vice-versa.

Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wave function in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). Hence, a non-zero function and its Fourier transform cannot both be sharply localized. The next theorem formally summarizes the Heisenberg’s Uncertainty Principle

Theorem 2**.**

(Uncertainty Principle [14]) Let $A$ and $B$ be two self-adjoint operators on a Hilbert space $\mathcal{X}$ with domains $\mathcal{D}(A)$ and $\mathcal{D}(B)$ respectively and denote finally $\left[A,B\right]=AB-BA$ their commutator. Then

[TABLE]

3. Clifford-Fourier Transform

In this section we propose to review some basic concepts of the Clifford-Fourier transform. Fore more details we may refer to [7] and [12]. The Clifford-Fourier transform of a Clifford-valued function $f\in L^{1}\cap L^{2}(\mathbb{R}_{n},dV(\underline{x}))$ is

[TABLE]

It is an invertible transform and its inverse is

[TABLE]

In the sequel, we shall use the two operators

[TABLE]

Using Theorem 2, we obtain

[TABLE]

which reads otherwise as

[TABLE]

4. Clifford-Wavelet Transform

In this section we introduce the concept of the Clifford-wavelet transform and some of its important properties to be used later. In this context, a function $\psi\in L^{1}\cap L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ will be considered as a Clifford mother wavelet. To join the admissibility assumptions in the real case, here-also we assume that

• •

$\widehat{\psi}(\underline{\xi})\left[\widehat{\psi}(\underline{\xi})\right]^{\dagger}$ is scalar.

• •

$\mathcal{A}_{\psi}=\displaystyle(2\pi)^{n}\int_{\mathbb{R}^{n}}\frac{\widehat{\psi}(\underline{\xi})\left[\widehat{\psi}(\underline{\xi})\right]^{\dagger}}{|\underline{\xi}|^{n}}dV(\underline{\xi})<\infty.$

For $(a,\underline{b},s)\in\mathbb{R}^{+}\times\mathbb{R}^{n}\times Spin(n)$, we denote

[TABLE]

It holds in fact that these copies are also admissible and that

[TABLE]

Proposition 3**.**

The set $\left\{\psi^{a,\underline{b},s},a>0,\underline{b}\in\mathbb{R}^{n},\,s\in Spin(n)\right\}$ is dense in the space $L^{2}(\mathbb{R}^{n},dV(\underline{x}))$.

Proof.

Let $f$ be an analyzed function such that

[TABLE]

We shall prove that $f=0$. Due to the Parseval identity of the Clifford-Fourier transform, we obtain

[TABLE]

Since,

[TABLE]

then,

[TABLE]

Recall now that for a fixed $\underline{\xi}\not=0$ in $\mathbb{R}^{n}$,

[TABLE]

It results that

[TABLE]

∎

Definition 4**.**

(Clifford Wavelet Transform) The Clifford-wavelet transform of an analyzed function $f$ with respect to the mother wavelet $\psi$ is

[TABLE]

Definition 5**.**

(Inner product relation) Let $\mathcal{H}_{\psi}=\left\{T_{\psi}\left[f\right],\;f\in L^{2}(\mathbb{R}^{n},dV(\underline{x}))\right\}$ be the image of $L^{2}(\mathbb{R}_{n},dV(\underline{x}))$ relatively to the operator $T_{\psi}$. We define the inner product by

[TABLE]

where $ds$ stands for the Haar measure on $Spin(n)$.

Proposition 6**.**

$T_{\psi}:L^{2}(\mathbb{R}^{n},dV(\underline{x}))\longrightarrow\mathcal{H}_{\psi}$* is an isometry.*

Proof.

We have to show that

[TABLE]

Put

[TABLE]

Hence,

[TABLE]

Applying Parseval formula, we get

[TABLE]

Next,

[TABLE]

Observing now that

[TABLE]

we get immediately

[TABLE]

where we denoted $\Gamma(t,\nu)=\widehat{\psi}(t\underline{\nu})\left[\widehat{\psi}(t\underline{\nu})\right]^{\dagger}$. Otherwise, by taking $\underline{u}=t\underline{\nu}$, we obtain

[TABLE]

As a result we have

[TABLE]

∎

As a result of the last Proposition and as in the real case, we have here a Clifford-wavelet reconstruction formula.

Proposition 7**.**

For all $f\in L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ we have

[TABLE]

5. Clifford wavelet uncertainty principle

In this section, we established the Heisenberg uncertainty principle for the Clifford wavelet transform. Backgrounds may be found in [1].

Theorem 8**.**

Let $\psi\in L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ an admissible Clifford mother wavelet. Then for $f\in L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ the following inequality holds

[TABLE]

where $k=1,2,\cdots,n$.

To prove this result we need the following lemma :

Lemma 9**.**

[TABLE]

Proof.

As $\widehat{\psi^{a,\underline{b},s}}(\underline{\xi})=a^{\frac{n}{2}}e^{-i<\underline{b},\underline{\xi}>}s\widehat{\psi}(a\overline{s}\underline{\xi}s)\overline{s}$, we get

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

Using of (5.4) we obtain

[TABLE]

According to (4.2), we get in fin

[TABLE]

∎

Proof.

of Theorem 8. Using (3.1) and setting $\underline{x}=\underline{b}\in\mathbb{R}^{n}$, we deduce that

[TABLE]

Therefore

[TABLE]

[TABLE]

According to the Cauchy-Schwartz inequality (2.2), it follows that

[TABLE]

[TABLE]

Now, using Lemma 9 and the fact that the wavelet-transform is an isometry, we get by (4.4)

[TABLE]

The inequality (5.5) becomes

[TABLE]

Hence, we obtain

[TABLE]

∎

6. Conclusion

In this paper, a new uncertainty principle associated with the continuous wavelet transform in the Clifford algebra’s settings has been formulated and proved. Starting from the definition of real Clifford algebra and the real continuous wavelet transform, we defined a continuous Clifford-Wavelet Transform, presented its proprieties and formulated an uncertainty relation based on the uncertainty principle for the Clifford-Fourier transform.