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1
Content available Continuous Head-related Transfer Function Representation Based on Hyperspherical Harmonics
Szwajcowski Adam
Archives of Acoustics
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2023
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Vol. 48, No. 1
127--139
EN
Expressing head-related transfer functions (HRTFs) in the spherical harmonic (SH) domain has been thoroughly studied as a method of obtaining continuity over space. However, HRTFs are functions not only of direction but also of frequency. This paper presents an extension of the SH-based method, utilizing hyperspherical harmonics (HSHs) to obtain an HRTF representation that is continuous over both space and frequency. The application of the HSH approximation results in a relatively small set of coefficients which can be decoded into HRTF values at any direction and frequency. The paper discusses results obtained by applying the method to magnitude spectra extracted from exemplary HRTF measurements. The HRTF representations based on SHs and HSHs exhibit similar reproduction accuracy, with the latter one featuring continuity over both space and frequency and requiring much lower number of coefficients. The developed HSH-based continuous functional model can serve multiple purposes, such as interpolation, compression or parametrization for machine-learning applications.
2
Content available remote Notes on continuity result for conformable diffusion equation on the sphere: the linear case
Nguyen Van Tien
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
952--962
EN
In this article, we are interested in the linear conformable diffusion equation on the sphere. Our main goal is to establish some results on the continuity problem with respect to fractional order. The main technique is based on several evaluations on the sphere using spherical basis functions. To overcome the difficulty, we also need to use some calculations to control the generalized integrals.
3
Content available Measures of growth and approximation of entire harmonic functions in n-dimensional space in some Banach spaces
Kumar Devendra
Journal of Mathematics and Applications
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2021
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Vol. 44
57--70
EN
The relationship between the classical order and type of an entire harmonic function in space Rn, n≥3, and the rate of its best harmonic polynomial approximation for some Banach spaces of functions harmonic in the ball of radius R has been studied.
4
Content available Error Analysis of Sound Source Directivity Interpolation Based on Spherical Harmonics
Szwajcowski Adam, Krause Daniel, Snakowska Anna
Archives of Acoustics
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2021
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Vol. 46, No. 1
95--104
EN
Precise measurement of the sound source directivity not only requires special equipment, but also is time-consuming. Alternatively, one can reduce the number of measurement points and apply spatial interpolation to retrieve a high-resolution approximation of directivity function. This paper discusses the interpolation error for different algorithms with emphasis on the one based on spherical harmonics. The analysis is performed on raw directivity data for two loudspeaker systems. The directivity was measured using sampling schemes of different densities and point distributions (equiangular and equiareal). Then, the results were interpolated and compared with these obtained on the standard 5° regular grid. The application of the spherical harmonic approximation to sparse measurement data yields a mean error of less than 1 dB with the number of measurement points being reduced by 89%. The impact of the sparse grid type on the retrieval error is also discussed. The presented results facilitate optimal sampling grid choice for low-resolution directivity measurements.
5
Content available remote Local Ionospheric Modeling Using the Localized Global Ionospheric Map and Terrestrial GPS
Sharifi M. A., Farzaneh S.
Acta Geophysica
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2016
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Vol. 64, no. 1
237--252
EN
Global ionosphere maps are generated on a daily basis at CODE using data from about 200 GPS/GLONASS sites of the IGS and other institutions. The vertical total electron content is modeled in a solargeomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. The spherical Slepian basis is a set of bandlimited functions which have the majority of their energy concentrated by optimization inside an arbitrarily defined region, yet remain orthogonal within the spatial region of interest. Hence, they are suitable for decomposing the spherical harmonic models into the portions that have significant strength only in the selected areas. In this study, the converted spherical harmonics to the Slepian bases were updated by the terrestrial GPS observations by use of the least-squares estimation with weighted parameters for local ionospheric modeling. Validations show that the approach adopted in this study is highly capable of yielding reliable results.
6
Content available remote Implementation of the spherical harmonics in glare sources simulation
Kadłubowski M., Sawicki D.
Przegląd Elektrotechniczny
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2015
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R. 91, nr 4
163--166
EN
The UGR index is practically used measure of a discomfort glare for the interior working environment. The basic parameters of the index formula are the angular dependences which define the visibility of light sources. In the paper the new method for light sources description has been presented. The mathematical model using spherical harmonics is discussed. Spherical harmonics allows for convenient and efficient description of the light sources positions as well as their shapes in the field of view.
PL
Wskaźnik UGR jest praktycznie stosowaną miarą olśnienia przykrego dla stanowisk pracy we wnętrzach. Podstawowymi parametrami wpływającymi na wartość wskaźnika są zależności kątowe w jakich widoczne są źródła światła. W artykule została przedstawiona nowa metoda opisu źródeł światła z wykorzystaniem harmonicznych sferycznych. Pozwalają one na wygodny i skuteczny opis położenia źródeł światła, jak również ich kształtu w polu widzenia.
7
Content available remote Features extraction of 3D objects by using Laplace series
Nowak K., Starzyński J.
Przegląd Elektrotechniczny
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2010
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R. 86, nr 1
172-174
EN
This paper deals with methods of features extraction of 3D objects focusing on their usage in a classification and recognition area. A brief review of possible approaches is given. An algorithm based on the spherical harmonics transform is presented with an appropriate theoretical background, implementation and tests results.
PL
Niniejszy artykuł dotyczy metod ekstrakcji cech obiektów trójwymiarowych w celu klasyfikacji i rozpoznawania. Po krótkiej klasyfikacji dostępnych metod skoncentrowano się na dokładnym omówieniu podejścia wykorzystującego szereg Laplace’a (harmoniki sferyczne). Przedstawiono podstawy teoretyczne, implementację i przykład zastosowania tej metody.
8
Content available remote Contribution of 1st -3rd order terms of a binomial expansion of topographic heights in topographic and atmospheric effects on satellite gravity gradiometric data
Eshagh M.
Artificial Satellites : Journal of Planetary Geodesy
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2009
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Vol. 44, No. 1
21--31
EN
In mathematical modeling of the topographic and atmospheric potentials in spherical harmonics, the topographic heights can binomially be expanded a certain order, usually to the third order. Some studies have been done on the effect of each order on geoid and gravity anomaly. However similar study on the satellite gravity gradiometric data is missed yet. This paper will investigate this matter globally. It presents that the contribution of the second- and third-order topographic terms is within 0.08 E and 2 mE, respectively on satellite gravity gradiometric data at 250 km level. Also the contribution of these terms is within 0.5 mE and 0.08 mE for the atmospheric effect.
9
Content available remote Generalized growth of entire harmonic functions
Srivastava G. S.
Fasciculi Mathematici
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2008
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Nr 40
79-89
EN
Let H(x), x = (x1, x2, ... , xn), be an entire harmonic function in Rn.Fryant and Shankar [1] had obtained growth properties of H explicitly in terms of its Fourier coeffcients. In this paper, we obtain the characterizations of generalized order and type and introduce the generalized lower order for H. Special case of functions of slow growth has also been considered. Our results generalize some of the results obtained in [1].
10
Content available remote Frequency-based representation of 3D point-based surfaces using spherical harmonics
Mousa M., Chaine R., Akkouche S.
Machine Graphics and Vision
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2006
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Vol. 15, No. 3/4
537-546
EN
In this paper, we propose a precise frequency-based representation for oriented point-based surfaces using spherical harmonics. The representation can be useful in many applications, such as filtering, progressive transmission and coding of 3D surfaces. The basic computation in our approach is the spherical harmonics transform of local spherical radial functions induced by a set of points. An important feature of our approach is that the calculations are performed directly on local 2D triangulations of the point-based surface without any prior space voxelization. This property ensures that the complexity of our computation of the spherical harmonics transform is linear in the number of triangles in the local patch. We present some experimental results which demonstrate our technique.
11
Orthogonal bases for spaces of complex spherical harmonics
Menegatto V. A., Oliveira C. P.
Journal of Applied Analysis
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2005
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Vol. 11, nr 1
113-132
EN
This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L²(Ω 2q). As a bypass, we study the inner product [f,g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables [wzór], in which f(D) is the differential operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [. , .] reduces to a multiple of the L²(Ω 2q) inner product. Bi-orthogonality in (F(Cq), [. , .] ) is fully investigated.
12
Content available remote Almost everywhere convergence of Cesáro means of spherical harmonic expansions for functions in fractional integral spaces
Ma B.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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1998
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[Z] 38
63-75
EN
In this paper, we mainly investigate the almost everywhere convergence of Cesáro means of spherical harmonic expansions for functions in fractional integral spaces. It is proved that if a function f has certain smoothness which is measured by the degree L them the non-negative summation indices b can chosen, correspondingly.
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