Time-frequency analysis: theory and applications
Time-frequency analysis: theory and applications
Fourier analysis and applications: filtering, numerical computation, wavelets
Fourier analysis and applications: filtering, numerical computation, wavelets
Discrete-time signal processing (2nd ed.)
Discrete-time signal processing (2nd ed.)
Stochastic Complexity in Statistical Inquiry Theory
Stochastic Complexity in Statistical Inquiry Theory
Partial Differential Equations: Analytical and Numerical Methods
Partial Differential Equations: Analytical and Numerical Methods
PHLST5: A Practical and Improved Version of Polyharmonic Local Sine Transform
Journal of Mathematical Imaging and Vision
Improvement of DCT-Based Compression Algorithms Using Poisson's Equation
IEEE Transactions on Image Processing
Hi-index | 7.29 |
In 2006, Naoki Saito proposed a Polyharmonic Local Fourier Transform (PHLFT) to decompose a signal f@?L^2(@W) into the sum of a polyharmonic componentu and a residualv, where @W is a bounded and open domain in R^d. The solution presented in PHLFT in general does not have an error with minimal energy. In resolving this issue, we propose the least squares approximant to a given signal in L^2([-1,1]) using the combination of a set of algebraic polynomials and a set of trigonometric polynomials. The maximum degree of the algebraic polynomials is chosen to be small and fixed. We show in this paper that the least squares approximant converges uniformly for a Holder continuous function. Therefore Gibbs phenomenon will not occur around the boundary for such a function. We also show that the PHLFT converges uniformly and is a near least squares approximation in the sense that it is arbitrarily close to the least squares approximant in L^2 norm as the dimension of the approximation space increases. Our experiments show that the proposed method is robust in approximating a highly oscillating signal. Even when the signal is corrupted by noise, the method is still robust. The experiments also reveal that an optimum degree of the trigonometric polynomial is needed in order to attain the minimal l^2 error of the approximation when there is noise present in the data set. This optimum degree is shown to be determined by the intrinsic frequency of the signal. We also discuss the energy compaction of the solution vector and give an explanation to it.