Harmonic wavelet transform

About: Harmonic wavelet transform is a research topic. Over the lifetime, 9602 publications have been published within this topic receiving 247336 citations.

Phase Retrieval for the Cauchy Wavelet Transform

Abstract: We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the uniqueness and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in \(L^2\)-norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.

A Space-Vector Discrete Fourier Transform for Unbalanced and Distorted Three-Phase Signals

Abstract: In this paper, a space-vector discrete-time Fourier transform is proposed for fast and precise detection of the fundamental-frequency and harmonic positive- and negative-sequence vector components of three-phase input signals. The discrete Fourier transform is applied to the three-phase signals represented by Clarke's αβ vector. It is shown that the complex numbers output from the Fourier transform are the instantaneous values of the positive- and negative-sequence harmonic component vectors of the input three-phase signals. The method allows the computation of any desired positive- or negative-sequence fundamental-frequency or harmonic vector component of the input signal. A recursive algorithm for low-effort online implementation is also presented. The detection performance for variable-frequency and interharmonic input signals is discussed. The proposed and other usual method performances are compared through simulations and experiments.

A Storage Efficient Way to Implement the Discrete Cosine Transform

Abstract: Ahmed has shown that a discrete cosine transform can be implemented by doing one double length fast Fourier transform (FFT). In this correspondence, we show that the amount of work can be cut to doing two single length FFT's.

Rotation, scale, and translation resilient public watermarking for images using a log-polar fourier transform

Abstract: A method for detecting a watermark signal in digital image data. The detecting method includes the steps of: computing a log-polar Fourier transform of the image data to obtain a log-polar Fourier spectrum; projecting the log-polar Fourier spectrum down to a lower dimensional space to obtain an extracted signal; comparing the extracted signal to a target watermark signal; and declaring the presence or absence of the target watermark signal in the image data based on the comparison. Also provided is a method for inserting a watermark signal in digital image data to obtain a watermarked image. The inserting method includes the steps of: computing a log-polar Fourier transform of the image data to obtain a log-polar Fourier spectrum; projecting the log-polar Fourier spectrum down to a lower dimensional space to obtain an extracted signal; modifying the extracted signal such that it is similar to a target watermark; performing a one-to-many mapping of the modified signal back to log-polar Fourier transform space to obtain a set of watermarked coefficients; and performing an inverse log-polar Fourier transform on the set of watermarked coefficients to obtain a watermarked image.

On the restriction of the Fourier transform to a conical surface

Abstract: Let r be the su.rface of a circular cone in R3. We show that if 1 < p < 4/3, 1/q = 3(1-1/p) and f E LP(R3), then the Fourier transform of f belongs to Lq(r, da) for a certain natural measure a on r. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint p = 4/3, with logarithmic growth of the bound as the thickness of the annulus tends to zero.