arXiv subjects

Frederic Schoppert

Publications and source records attributed to Frederic Schoppert.

4 recordsLinked to original sources

arXiv2026

Polynomial curvelets on higher-dimensional spheres

In this article, we introduce and investigate polynomial curvelets on spheres, which form a class of Parseval frames for $L^2(\mathbb{S}^{d-1})$, $d \geq 3$. The proposed construction offers a directionally sensitive multiscale decomposition and provides a sparse representation of spherical data. As a main result, we derive a sharp pointwise localization bound which shows that the frame elements decay rapidly away from their center of mass, making them a powerful tool for position-based analyses. In contrast to previous constructions, polynomial curvelets are not limited in their directional resolution. Consequently, the frames established in this article are particularly powerful when it comes to the analysis of localized anisotropic features, such as edges. To illustrate this point, we show that, given a suitable test signal that exhibits (higher-order) discontinuities at the boundary $\partial A$ of a spherical cap $A\subset \mathbb{S}^{d-1}$, the corresponding curvelet coefficients peak precisely when the analysis function matches some segment of the boundary $\partial A$, both in terms of position and orientation. Otherwise, the coefficients decay rapidly.

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arXiv2026

Directional polynomial frames on spheres

We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $\Psi^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $\Psi^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames.

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arXiv2025

Directional polynomial wavelets on spheres

In this article, we construct discrete tight frames for $L^2(\mathbb{S}^{d-1})$, $d\geq3$, which consist of localized polynomial wavelets with adjustable degrees of directionality. In contrast to the well studied isotropic case, these systems are well suited for the direction sensitive analysis of anisotropic features such as edges. The price paid for this is the fact that at each scale the wavelet transform lives on the rotation group $SO(d)$, and not on $\mathbb{S}^{d-1}$ as in the zonal setting. Thus, the standard approach of building discrete frames by sampling the continuous wavelet transform requires a significantly larger amount of sample points. However, by keeping the directionality limited, this number can be greatly reduced to the point where it is comparable to the number of samples needed in the isotropic case. Moreover, the limited directionality is reflected in the wavelets being steerable and their great localization in space leads to a fast convergence of the wavelet expansion in the spaces $L^p(\mathbb{S}^{d-1})$, $1\leq p \leq \infty$.

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arXiv2025

Edge detection with polynomial frames on the sphere

In a recent article, we have shown that a variety of localized polynomial frames, including isotropic as well as directional systems, are suitable for detecting jump discontinuities along circles on the sphere. More precisely, such edges can be identified in terms of their position and orientation by the asymptotic decay of the frame coefficients in an arbitrary small neighborhood. In this paper, we will extend these results to discontinuities which lie along general smooth curves. In particular, we prove upper and lower estimates for the frame coefficients when the analysis function is concentrated in the vicinity of such a singularity. The estimates are given in an asymptotic sense, with respect to some dilation parameter, and they hold uniformly in a neighborhood of the smooth curve segment under consideration.

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