Wavelets on Graphs via Spectral Graph Theory

We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $Ł$. Given a wavelet generating kernel $g$ and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(tŁ)$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $Ł$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.

Wavelets on Graphs via Spectral Graph Theory

We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $Ł$. Given a wavelet generating kernel $g$ and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(tŁ)$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $Ł$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.

Haar Wavelet Transform, In-place

Haar Wavelet Transform, In-place. GitHub Gist: instantly share code, notes, and snippets.

MotionWavelet: Human Motion Prediction via Wavelet Manifold Learning

Modeling temporal characteristics and the non-stationary dynamics of body movement plays a significant role in predicting human future motions. However, it is challenging to capture these features due to the subtle transitions involved in the complex human motions. This paper introduces MotionWavelet, a human motion prediction framework that utilizes Wavelet Transformation and studies human motion patterns in the spatial-frequency domain. In MotionWavelet, a Wavelet Diffusion Model (WDM) learns a Wavelet Manifold by applying Wavelet Transformation on the motion data therefore encoding the intricate spatial and temporal motion patterns. Once the Wavelet Manifold is built, WDM trains a diffusion model to generate human motions from Wavelet latent vectors. In addition to the WDM, MotionWavelet also presents a Wavelet Space Shaping Guidance mechanism to refine the denoising process to improve conformity with the manifold structure. WDM also develops Temporal Attention-Based Guidance to enhance prediction accuracy. Extensive experiments validate the effectiveness of MotionWavelet, demonstrating improved prediction accuracy and enhanced generalization across various benchmarks. Our code and models will be released upon acceptance.

ptwt - The PyTorch Wavelet Toolbox

Two methods for baseline correction of spectral data • NIRPY Research

Worked examples of two methods for baseline correction of spectra applied to Raman and XRF data.

𝗝𝗼𝘂𝗿𝗻𝗮𝗹 𝗼𝗳 𝗘𝗻𝘃𝗶𝗿𝗼𝗻𝗺𝗲𝗻𝘁𝗮𝗹 & 𝗘𝗮𝗿𝘁𝗵 𝗦𝗰𝗶𝗲𝗻𝗰𝗲𝘀 | 𝗩𝗼𝗹𝘂𝗺𝗲 𝟬𝟰 | 𝗜𝘀𝘀𝘂𝗲 𝟬𝟮 | 𝗢𝗰𝘁𝗼𝗯𝗲𝗿 𝟮𝟬𝟮𝟮

𝗝𝗼𝘂𝗿𝗻𝗮𝗹 𝗼𝗳 𝗘𝗻𝘃𝗶𝗿𝗼𝗻𝗺𝗲𝗻𝘁𝗮𝗹 & 𝗘𝗮𝗿𝘁𝗵 𝗦𝗰𝗶𝗲𝗻𝗰𝗲𝘀 | 𝗩𝗼𝗹𝘂𝗺𝗲 𝟬𝟰 | 𝗜𝘀𝘀𝘂𝗲 𝟬𝟮 | 𝗢𝗰𝘁𝗼𝗯𝗲𝗿 𝟮𝟬𝟮𝟮 🌏 Quantum Biophysics of the Atmosphere: Asymmetric Wavelets of the Average Annual Air Temperature of Irkutsk for …