Zenodo (CERN European Organization for Nuclear Research), Sep 28, 2006
A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or ge... more A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or generally by wavelets). The orthogonal projection to the subspaces generates a decomposition (multiresolution) of a signal. Regarding the rate of convergence and the number of operations, this kind of decomposition is much more favorable then the conventional Fourier expansion. In this paper, starting from Haar-like systems we will introduce a new type of multiresolution. The transition to higher levels in this case, instead of dilation will be realized by a twofold map. Starting from a convenient scaling function and twofold map, we will introduce a large class of Haar-like systems. Besides others, the original Haar system and Haar-like systems of trigonometric polynomials, and rational functions can be constructed in this way. We will show that the restriction of Haar-like systems to an appropriate set can be identified by the original Haar-system. Haar-like rational functions are used for the approximation of rational transfer functions which play an important role in signal processing [
Measurement and mathematical description of the corneal surface and the optical properties of the... more Measurement and mathematical description of the corneal surface and the optical properties of the human eye is an active field in biomedical engineering and ophthalmology. One particular problem is to correct certain types of corneal shape measurement errors, e.g. ones that arise due to unintended eye-movements and spontaneous rotations of the eye-ball. In this paper we present the mathematical background of our recent approach, which is based on constructions of systems of orthonormal functions based on the well-known Zernike polynomials. For this, argument transformation by suitable Blaschke functions are used, as they correspond to the congruent transformations in the Poincare disk model of the Bolyai–Lobachevsky hyperbolic geometry, making them a practical choice. The problem of discretization, and computation of the translated Zernike coefficients is also discussed.
Zenodo (CERN European Organization for Nuclear Research), Sep 28, 2006
A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or ge... more A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or generally by wavelets). The orthogonal projection to the subspaces generates a decomposition (multiresolution) of a signal. Regarding the rate of convergence and the number of operations, this kind of decomposition is much more favorable then the conventional Fourier expansion. In this paper, starting from Haar-like systems we will introduce a new type of multiresolution. The transition to higher levels in this case, instead of dilation will be realized by a twofold map. Starting from a convenient scaling function and twofold map, we will introduce a large class of Haar-like systems. Besides others, the original Haar system and Haar-like systems of trigonometric polynomials, and rational functions can be constructed in this way. We will show that the restriction of Haar-like systems to an appropriate set can be identified by the original Haar-system. Haar-like rational functions are used for the approximation of rational transfer functions which play an important role in signal processing [
Measurement and mathematical description of the corneal surface and the optical properties of the... more Measurement and mathematical description of the corneal surface and the optical properties of the human eye is an active field in biomedical engineering and ophthalmology. One particular problem is to correct certain types of corneal shape measurement errors, e.g. ones that arise due to unintended eye-movements and spontaneous rotations of the eye-ball. In this paper we present the mathematical background of our recent approach, which is based on constructions of systems of orthonormal functions based on the well-known Zernike polynomials. For this, argument transformation by suitable Blaschke functions are used, as they correspond to the congruent transformations in the Poincare disk model of the Bolyai–Lobachevsky hyperbolic geometry, making them a practical choice. The problem of discretization, and computation of the translated Zernike coefficients is also discussed.
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Papers by Ferenc Schipp