Abstract
The pseudo-spectral (PS) method on the basis of the Fourier transform is a numerical method for estimating derivatives. Generally, the discrete Fourier transform (DFT) is used when implementing the PS method. However, when the values on both sides of the sequences differ significantly, oscillatory approximations around both sides appear due to the periodicity resulting from the DFT. To address this problem, we propose a new PS method based on symmetric extension. We mathematically derive the proposed method using the discrete cosine transform (DCT) in the forward transform from the relation between DFT and DCT. DCT allows a sequence to function as a symmetrically extended sequence and estimates derivatives in the transformed domain. The superior performance of the proposed method is demonstrated through image interpolation. Potential applications of the proposed method are numerical simulations using the Fourier based PS method in many fields such as fluid dynamics, meteorology, and geophysics.
Highlights
The discrete cosine transform (DCT) and discrete sine transform (DST) have been extensively studied, and they have played a crucial role in science and engineering for decades
We study PS methods based on symmetric extension to address the problem of the Gibbs phenomenon induced by PS-discrete Fourier transform (DFT) in image interpolation
We proposed PS methods based on symmetric extension to attenuate the oscillatory approximation that occurs with DFT
Summary
The discrete cosine transform (DCT) and discrete sine transform (DST) have been extensively studied, and they have played a crucial role in science and engineering for decades. DCT and DST are closely related to the discrete Fourier transform (DFT) [1,2,3]. The PS method using Fourier series as expansion functions is employed for estimating derivatives at spaced points, which is common in fluid dynamics [12,13], meteorology, and geophysics, e.g., a direct numerical simulation for a turbulent flow [14], wave prediction [15], multibody modeling of wave energy converters [16], and seismogram. Image interpolation by Hermite polynomials is a good example to understand the accuracy of derivatives. We study PS methods based on symmetric extension to address the problem of the Gibbs phenomenon induced by PS-DFT in image interpolation. We are motivated by the seminal work [21] combining Hermite polynomials with PS-DFT for image interpolation. We use an asterisk to denote the complex conjugate, i.e., X ∗ (k ) is the complex conjugate of X (k)
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