In the present work, we are concerned with the development of a new uncertainty principle based on the wavelet transform in the Clifford analysis/algebra framework. We precisely derive a sharp Heisenberg-type uncertainty principle for the continuous Clifford wavelet transform.

1.
B.
Amri
and
L. T.
Rachdi
, “
Beckner logarithmic uncertainty principle for the Riemann-Liouville operator
,”
Int. J. Math.
24
(
09
),
1350070
(
2013
).
2.
B.
Amri
and
L. T.
Rachdi
, “
Uncertainty principle in terms of entropy for the Riemann–Liouville operator
,”
Bull. Malaysian Math. Sci. Soc.
39
(
1
),
457
481
(
2015
).
3.
S.
Arfaoui
,
I.
Rezgui
, and
A.
Ben Mabrouk
,
Wavelet Analysis on the Sphere: Spheroidal Wavelets
(
Walter de Gruyter
,
2017
), ISBN: 10: 311048109X, ISBN: 13: 978-3110481099.
4.
S.
Arfaoui
and
A.
Ben Mabrouk
, “
Some ultraspheroidal monogenic Clifford Gegenbauer Jacobi polynomials and associated wavelets
,”
Adv. Appl. Clifford Algebras
27
,
2287
2306
(
2017
).
5.
S.
Arfaoui
and
A.
Ben Mabrouk
, “
Some old orthogonal polynomials revisited and associated wavelets: Two-parameters Clifford-Jacobi polynomials and associated spheroidal wavelets
,”
Acta Appl. Math.
155
,
177
195
(
2018
).
6.
S.
Arfaoui
,
A.
Ben Mabrouk
, and
C.
Cattani
, “
New type of Gegenbauer–Hermite monogenic polynomials and associated Clifford wavelets
,”
J. Math. Imaging Vis.
62
,
73
97
(
2020
).
7.
S.
Arfaoui
,
A.
Ben Mabrouk
, and
C.
Cattani
, “
New type of Gegenbauer-Jacobi-Hermite monogenic polynomials and associated continuous Clifford wavelet transform
,”
Acta Appl. Math.
(
2020
).
8.
H.
Banouh
,
A.
Ben Mabrouk
, and
M.
Kesri
, “
Clifford wavelet transform and the uncertainty principle
,”
Adv. Appl. Clifford Algebras
29
,
106
(
2019
).
9.
F.
Brackx
,
J. S. R.
Chisholm
, and
V.
Soucek
,
Clifford Analysis and Its Applications
, NATO Science Series, Series II: Mathematics, Physics and Chemistry Vol. 25 (
Springer
,
2000
).
10.
F.
Brackx
,
R.
Delanghe
, and
F.
Sommen
,
Clifford Analysis
(
Pitman Publication
,
1982
).
11.
F.
Brackx
,
N.
De Schepper
, and
F.
Sommen
, “
The two-dimensional Clifford-Fourier transform
,”
J. Math. Imaging
26
,
5
18
(
2006
).
12.
F.
Brackx
,
N.
De Schepper
, and
F.
Sommen
, “
The Fourier transform in Clifford analysis
,”
Adv. Imaging Electron Phys.
156
,
55
201
(
2009
).
13.
F.
Brackx
,
N.
De Schepper
, and
F.
Sommen
, “
Clifford-Hermite and two-dimensional Clifford-Gabor filters for early vision
,” in
(digital) Proceedings 17th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering
, edited by
K.
Gürlebeck
and
C.
Könke
(
Bauhaus-Universität Weimar
,
2006
).
14.
F.
Brackx
,
E.
Hitzer
, and
S. J.
Sangwine
, “
History of quaternion and Clifford-Fourier transforms and wavelets
,” in
Quaternion and Clifford-Fourier Transforms and Wavelets
, Trends in Mathematics Vol. 27 (
Springer, Basel
,
2013
), pp.
XI
XXVII
.
15.
F.
Brackx
and
F.
Sommen
, “
The continuous wavelet transform in Clifford analysis
,” in
Clifford Analysis and Its Applications
(
Springer Netherlands
,
2001
), pp.
9
26
.
16.
R.
Bujack
,
H.
De Bie
,
N.
De Schepper
, and
G.
Scheuermann
, “
Convolution products for hypercomplex Fourier transforms
,”
J. Math. Imaging Vis.
48
(
3
),
606
624
(
2013
).
17.
R.
Bujack
,
G.
Scheuermann
, and
E.
Hitzer
, “
A general geometric Fourier transform
,” in
Quaternion and Clifford Fourier Transforms and Wavelets
, Trends in Mathematics Vol. 27, edited by
E.
Hitzer
and
S. J.
Sangwine
(
Birkhauser
,
Basel
,
2013
), pp.
155
176
.
18.
R.
Bujack
,
G.
Scheuermann
, and
E.
Hitzer
, “
A general geometric Fourier transform convolution theorem
,”
Adv. Appl. Clifford Alg.
23
(
1
),
15
38
(
2013
).
19.
W. K.
Clifford
, “
On the classification of geometric algebras
,” in
Mathematical Papers
(
MacMillan, London
,
1882
), pp.
397
401
.
20.
S.
Dahkle
,
G.
Kutyniok
,
P.
Maass
,
C.
Sagiv
,
H.-G.
Stark
, and
G.
Teschke
, “
The uncertainty principle associated with the continuous shearlet transform
,”
Int. J. Wavelets, Multiresol. Inform. Process.
6
(
2
),
157
181
(
2008
).
21.
R.
Delanghe
, “
Clifford analysis: History and perspective
,”
Comput. Methods Funct. Theory
1
(
1
),
107
153
(
2001
).
22.
H.
De Bie
, “
Clifford algebras, Fourier transforms and quantum mechanics
,”
Math. Method. Appl. Sci.
35
(
18
),
2198
2228
(
2012
).
23.
H.
De Bie
and
Y.
Xu
, “
On the Clifford-Fourier transform
,”
Int. Math. Res.
22
,
5123
5163
(
2011
).
24.
N.
De Schepper
, “
Multi-dimensional continuous wavelet transforms and generalized Fourier transforms in Clifford analysis
,” Ph.D. thesis,
Ghent University
,
2006
.
25.
N.
Dian Tunjung
,
A.
Zainal Arifin
, and
R.
Soelaiman
, “
Medical image segmentation using generalized Gradient vector flow and Clifford geometric algebra
,” in
International Conference on Biomedical Engineering, Surabaya, Indonesia
(
2008
), p.
5
.
26.
P. A. M.
Dirac
, “
The quantum theory of the electron
,”
Proc. R. Soc. London, Ser. A
117
(
778
),
610
624
(
1928
).
27.
Y.
El Haoui
and
S.
Fahlaoui
, “
The continuous quaternion algebra-valued wavelet transform and the associated uncertainty principle
,” arXiv:1902.08461 (
2019
).
28.
Y.
El Haoui
,
S.
Fahlaoui
, and
E.
Hitzer
, “
Generalized uncertainty principles associated with the quaternionic offset linear canonical transform
,” arXiv:1807.04068v2 [math.CA] (
2019
).
29.
Y.
El Haoui
and
S.
Fahlaoui
, “
Donoho-Stark’s uncertainty principles in real Clifford algebras
,”
Adv. Appl. Clifford Al.
29
,
94
(
2019
).
30.
H. G.
Feichtinger
and
K.
Gröchenig
, “
Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view
,”
Wavelets
2
,
359
398
(
1992
).
31.
A.
Grossmann
,
J.
Morlet
, and
T.
Paul
, “
Transforms associated to square integrable group representations. II: Examples
,”
Ann. l’IHP Phys. Théor.
45
(
3
),
293
309
(
1986
).
32.
A.
Grossmann
and
J.
Morlet
, “
Decomposition of hardy functions into square integrable wavelets of constant shape
,”
SIAM J. Math. Anal.
15
,
723
736
(
1984
).
33.
W. R.
Hamilton
, “
On a new species of imaginary quantities connected with a theory of quaternions
,” in
Proc. R. Irish Acad.
2
,
424
434
(
1844
).
34.
W. R.
Hamilton
,
Elements of Quaternions
(
Longmans, Green, & Company
,
1866
).
35.
W.
Heisenberg
, “
Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik
,”
Z. Phys.
43
,
172
198
(
1927
).
36.
W.
Heisenberg
, “
Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik
,” in
Original Scientific Papers Wissenschaftliche Originalarbeiten
(
Springer Berlin Heidelberg
,
1985
), pp.
478
504
.
37.
E.
Hitzer
, “
New developments in Clifford Fourier transforms
,” in
Advances in Applied and Pure Mathematics
[Proceedings of the 2014 International Conference on Pure Mathematics, Applied Mathematics, Computational Methods (PMAMCM 2014), Santorini Island, Greece, July 17–21, 2014] Mathematics and Computers in Science and Engineering Series Vol. 29, edited by
N. E.
Mastorakis
,
P. M.
Pardalos
,
R. P.
Agarwal
, and
L.
Kocinac
(
Dekker
,
2014
), pp.
19
25
.
38.
E.
Hitzer
, “
Clifford (geometric) algebra wavelet transform
,” in
Proceedings of the GraVisMa 2009, Plzen, Czech Republic, 02–04 September 2009
, edited by
V.
Skala
and
D.
Hildenbrand
(
Vaclav Skala-Union Agency
,
2009
), pp.
94
101
.
39.
E.
Hitzer
, “
Directional uncertainty principle for quaternion Fourier transform
,”
Adv. Appl. Clifford Alg.
20
,
271
284
(
2010
).
40.
E.
Hitzer
and
B.
Mawardi
, “
Uncertainty principle for the Clifford-geometric algebra C3,0 based on Clifford Fourier transform
,” in
International Conference on Numerical Analysis and Applied Mathematics 2005
, edited by
T. E.
Simos
,
G.
Sihoyios
, and
C.
Tsitouras
(
Wiley-VCH, Weinheim
,
2005
), pp.
922
925
.
41.
E.
Hitzer
, “
Tutorial on Fourier transformations and wavelet transformations in Clifford geometric algebra
,” in
Lecture Notes of the International Workshop for Computational Science with Geometric Algebra (FCSGA2007)
, edited by
K.
Tachibana
(
Nagoya University
,
Japan
,
2007
), pp.
65
87
.
42.
P.
Jorgensen
and
F.
Tian
,
Non-Commutative Analysis
(
World Scientific Publishing Company
,
2017
).
43.
K. I.
Kou
,
J.-Y.
Ou
, and
J.
Morais
, “
On uncertainty principle for quaternionic linear canonical transform
,”
Abst. Appl. Anal.
2013
,
725952
.
44.
G.
Ma
and
J.
Zhao
, “
Quaternion ridgelet transform and curvelet transform
,”
Adv. Appl. Clifford Algebras
28
,
80
(
2018
).
45.
B.
Mawardi
, “
Construction of quaternion-valued wavelets
,”
Matematika
26
(
1
),
107
114
(
2010
).
46.
B.
Mawardi
and
A.
Ryuichi
, “
A simplified proof of uncertainty principle for quaternion linear canonical transform
,”
Abst. Appl. Anal.
2016
,
5874930
.
47.
B.
Mawardi
and
R.
Ashino
, “
Logarithmic uncertainty principle for quaternion linear canonical transform
,” in
Proceedings of the 2016 International Conference on Wavelet Analysis and Pattern Recognition, Jeju, South Korea
(
IEEE
,
2016
), p.
6
.
48.
B.
Mawardi
and
R.
Ashino
, “
A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms
,”
Abst. Appl. Anal.
2017
,
3795120
.
49.
B.
Mawardi
and
E.
Hitzer
, “
Clifford algebra Cl(3, 0)-valued wavelets and uncertainty inequality for Clifford Gabor wavelet transformation
,” in
Preprints of Meeting of the Japan Society for Industrial and Applied Mathematics
(
Tsukuba University
,
Tsukuba, Japan
,
2006
), pp.
64
65
.
50.
B.
Mawardi
and
E.
Hitzer
, “
Clifford algebra Cl(3, 0)-valued wavelet transformation, Clifford wavelet uncertainty inequality and Clifford Gabor wavelets
,”
Int. J. Wavelets, Multiresol. Inform. Process.
5
(
6
),
997
1019
(
2007
).
51.
B.
Mawardi
and
E.
Hitzer
, “
Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra Cl3,0
,”
Adv. Appl. Clifford Alg.
16
,
41
61
(
2006
).
52.
B.
Mawardi
and
E.
Hitzer
, “
Clifford Fourier transform on multivector Fields and uncertainty principles for dimensions n = 2(mod 4) and n = 3(mod 4)
,”
Adv. Appl. Clifford Alg.
18
,
715
736
(
2008
).
53.
B.
Mawardi
,
E.
Hitzer
,
A.
Hayashi
, and
R.
Ashino
, “
An uncertainty principle for quaternion Fourier transform
,”
Comput. Math. Appl.
56
,
2398
2410
(
2008
).
54.
B.
Mawardi
,
R.
Ashino
, and
R.
Vaillancourt
, “
Two-dimensional quaternion wavelet transform
,”
Appl. Math. Comput.
218
,
10
21
(
2011
).
55.
K.
Nagata
and
T.
Nakamura
, “
Violation of Heisenberg’s uncertainty principle
,”
Open Access Library J.
2
,
e1797
(
2015
).
56.
L. T.
Rachdi
and
F.
Meherzi
, “
Continuous wavelet transform and uncertainty principle related to the spherical mean operator
,”
Mediterr. J. Math.
14
(
1
),
11
(
2016
).
57.
L. T.
Rachdi
,
B.
Amri
, and
A.
Hammami
, “
Uncertainty principles and time frequency analysis related to the Riemann–Liouville operator
,”
Ann. Univ. Ferrara
65
,
139
(
2018
).
58.
L. T.
Rachdi
and
H.
Herch
, “
Uncertainty principles for continuous wavelet transforms related to the Riemann–Liouville operator
,”
Ric. Mat.
66
(
2
),
553
578
(
2017
).
59.
D.
Rizo-Rodríguez
,
H.
Méndez-Vázquez
, and
E.
García-Reyes
, “
Illumination invariant face recognition using quaternion-based correlation Filters
,”
J. Math. Imaging Vis.
45
,
164
175
(
2013
).
60.
K.
Sau
,
R. K.
Basak
, and
A.
Chanda
, “
Image compression based on block truncation coding using Clifford algebra
,”
Proc. Technol.
10
,
699
706
(
2013
).
61.
D.
Sen
, “
The uncertainty relations in quantum mechanics
,”
Current Science
107
(
2
),
203
218
(
2018
).
62.
F.
Sommen
and
H.
De Schepper
,
Introductory Clifford Analysis
(
Springer
,
Basel
,
2015
), pp.
1
27
.
63.
R.
Soulard
and
P.
Carré
, “
Characterization of color images with multiscale monogenic maxima
,”
IEEE Trans. Pattern Anal. Mach. Intell.
40
(
10
),
2289
2302
(
2018
).
64.
P. A.
Stabnikov
, “
Geometric interpretation of the uncertainty principle
,”
Nat. Sci.
11
(
5
),
146
148
(
2019
).
65.
L.
Wietzke
and
G.
Sommer
, “
The signal multi-vector
,”
J. Math. Imaging Vis.
37
,
132
150
(
2010
).
66.
H.
Weyl
,
The Theory of Groups and Quantum Mechanics
, 2nd ed. (
Dover
,
New York
,
1950
).
67.
Y.
Yang
,
P.
Dang
, and
T.
Qian
, “
Stronger uncertainty principles for hypercomplex signals
,”
Complex Var. Elliptic Equations
60
(
12
),
1696
1711
(
2015
).
68.
Y.
Yang
and
K.
Ian Kou
, “
Uncertainty principles for hypercomplex signals in the linear canonical transformdomains
,”
Signal Process.
95
,
67
75
(
2014
).
69.
C.
Zou
and
K. I.
Kou
, “
Hypercomplex signal energy concentration in the spatial and quaternionic linear canonical Frequency domains
,” arXiv:1609.00890 (
2016
).
You do not currently have access to this content.