Spherical Wavelets: Texture Processing

Schröder, Peter; Sweldens, Wim

doi:10.1007/978-3-7091-9430-0_24

Cited by 59 publications

(54 citation statements)

References 15 publications

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“…We also note that characterizations of function spaces using wavelets on stratified Lie groups had been considered in [L]. (The author would like to thank an anonymous referee very much for kindly pointing out these references ([SS1], [SS2], [KL], [L]) to him.) For more recent work on spherical frames, we refer to the paper [NPW] by F. J. Narcowich, P. Petrushev and J. D. Ward, and also the nice survey paper [MP] by H. N. Mhaskar and J. Prestin.…”

Section: B(x R)|mentioning

confidence: 99%

Characterizations of function spaces on the sphere using frames

Dai¹

2006

Trans. Amer. Math. Soc.

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Abstract. In this paper we introduce a polynomial frame on the unit sphere S d−1 of R d , for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere S d−1 , such as L p , H p and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case R d . We also study a related nonlinear m-term approximation problem on S d−1 . In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov ("Compression of wavelet decompositions", Amer. J. Math. 114 (1992), no. 4, 737-785).

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Section: B(x R)|mentioning

confidence: 99%

Characterizations of function spaces on the sphere using frames

Dai¹

2006

Trans. Amer. Math. Soc.

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“…We believe that many applications can benefit from these wavelet bases. For example, using their localization properties a number of spherical image processing algorithms, such as local smoothing and enhancement, can be realized in a straightforward and efficient way [25].…”

Section: Discussionmentioning

confidence: 99%

Spherical wavelets: Efficiently representing functions on a sphere

Schröder¹,

Sweldens²

Wavelets in the Geosciences

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265

409

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Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

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“…Other researchers proposed the use of the lifting scheme to decompose the surfaces. Schroder and Swelden have proposed an extension of the lifting scheme to decompose spherical surfaces [32,33].…”

Section: -Multi-scales Analysis On Surfaces Represented By Triangulamentioning

confidence: 99%

Multi-scale freeform surface texture filtering using a mesh relaxation scheme

Jiang

Abdul-Rahman

Scott

2013

Meas. Sci. Technol.

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Surface filtering algorithms using Fourier, Gaussian, wavelets … etc., are well-established for simple Euclidean geometries. However, these filtration techniques cannot be applied to today's complex freeform surfaces, which have non-Euclidean geometries, without distortion of the results. This paper proposes a new multi-scale filtering algorithm for freeform surfaces that are represented by triangular meshes based on a mesh relaxation scheme. The proposed algorithm is capable of decomposing a freeform surface into different scales and to separate surface roughness, waviness and form from each other as will be demonstrated throughout the paper. Results of applying the proposed algorithm to computer-generated as well as real surfaces are represented and compared with a lifting wavelet filtering algorithm.

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Spherical Wavelets: Texture Processing

Cited by 59 publications

References 15 publications

Characterizations of function spaces on the sphere using frames

Characterizations of function spaces on the sphere using frames

Spherical wavelets: Efficiently representing functions on a sphere

Multi-scale freeform surface texture filtering using a mesh relaxation scheme

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