Recently, implicit neural representations (INR) have made significant strides in various vision-related domains, providing a novel solution for Multispectral and Hyperspectral Image Fusion (MHIF) tasks.

Existing convolutional neural networks widely adopt spatial down-/up-sampling for multi-scale modeling. However, spatial up-sampling operators (e.g., interpolation, transposed convolution, and un-pooling) heavily depend on local pixel attention, incapably exploring the global dependency. In contrast, the Fourier domain is in accordance with the nature of global modeling according to the spectral convolution theorem. Unlike the spatial domain that easily performs up-sampling with the property of local similarity, up-sampling in the Fourier domain is more challenging as it does not follow such a local property. In this study, we propose a theoretically feasible Deep Fourier Up-Sampling (FourierUp) to solve these issues. We revisit the relationships between spatial and Fourier domains and reveal the transform rules on the features of different resolutions in the Fourier domain, which provide key insights for FourierUp's designs. FourierUp as a generic operator consists of three key components: 2D discrete Fourier transform, Fourier dimension increase rules, and 2D inverse Fourier transform, which can be directly integrated with existing networks. Extensive experiments across multiple computer vision tasks, including object detection, image segmentation, image de-raining, image dehazing, and guided image super-resolution, demonstrate the consistent performance gains obtained by introducing our FourierUp. Code will be publicly available.

Recently, implicit neural representations (INR) have made significant strides in various vision-related domains, providing a novel solution for Multispectral and Hyperspectral Image Fusion (MHIF) tasks.

Neural ODEs (NODEs) have emerged as powerful tools for modeling time series data, offering the flexibility to adapt to varying input scales and capture complex dynamics. However, they face significant challenges: first, their reliance on time-domain representations often limits their ability to capture long-term dependencies and periodic structures; second, the inherent mismatch between their continuous-time formulation and the discrete nature of real-world data can lead to loss of granularity and predictive accuracy. To address these limitations, we propose Fourier Ordinary Differential Equations (FODEs), an approach that embeds the dynamics in the Fourier domain. By transforming time-series data into the frequency domain using the Fast Fourier Transform (FFT), FODEs uncover global patterns and periodic behaviors that remain elusive in the time domain. Additionally, we introduce a learnable element-wise filtering mechanism that aligns continuous model outputs with discrete observations, preserving granularity and enhancing accuracy. Experiments on various time series datasets demonstrate that FODEs outperform existing methods in terms of both accuracy and efficiency. By effectively capturing both long- and short-term patterns, FODEs provide a robust framework for modeling time series dynamics.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This work presents a method to efficiently train object detectors in the presence of geometric transformations that can be represented as vector-matrix multiplications. This has recently been developed for the case of translation transformations ([1,8,14]) but has not been too obvious for other transformations, such as rotations, let alone non-rigid deformations. The authors propose to adapt the Fourier-based method that originally allowed the development of efficient algorithms for the case of translations so that we can now deal with rotations and other `cyclic' signal transformations (e.g. the walking pattern of a pedestrian). The condition for this to hold is that the transformation is norm-preserving, can be represented as a matrix multiplication, x_transformed = Q x, has an inverse Q^{-1} = Q^T and for some s Q^s = I. The starting point for the previous works was the fact that the'data matrix' obtained by'stacking' together all translated versions of a signal is a circulant NxN matrix, where N is the length of the signal, and as such can be diagonalized using the discrete harmonic basis (or, Discrete Fourier Transform-DFT matrix), Eq. 3, and reference [5].

In many datasets, the samples are related by a known image transformation, such as rotation, or a repeatable non-rigid deformation. This applies to both datasets with the same objects under different viewpoints, and datasets augmented with virtual samples. Such datasets possess a high degree of redundancy, because geometrically-induced transformations should preserve intrinsic properties of the objects. Likewise, ensembles of classifiers used for pose estimation should also share many characteristics, since they are related by a geometric transformation. By assuming that this transformation is norm-preserving and cyclic, we propose a closed-form solution in the Fourier domain that can eliminate most redundancies. It can leverage off-the-shelf solvers with no modification (e.g.

Conventional Fourier-domain Optical Coherence Tomography (FD-OCT) systems depend on resampling into wavenumber (k) domain to extract the depth profile. This either necessitates additional hardware resources or amplifies the existing computational complexity. Moreover, the OCT images also suffer from speckle noise, due to systemic reliance on low coherence interferometry. We propose a streamlined and computationally efficient approach based on Deep-Learning (DL) which enables reconstructing speckle-reduced OCT images directly from the wavelength domain. For reconstruction, two encoder-decoder styled networks namely Spatial Domain Convolution Neural Network (SD-CNN) and Fourier Domain CNN (FD-CNN) are used sequentially. The SD-CNN exploits the highly degraded images obtained by Fourier transforming the domain fringes to reconstruct the deteriorated morphological structures along with suppression of unwanted noise. The FD-CNN leverages this output to enhance the image quality further by optimization in Fourier domain (FD). We quantitatively and visually demonstrate the efficacy of the method in obtaining high-quality OCT images. Furthermore, we illustrate the computational complexity reduction by harnessing the power of DL models. We believe that this work lays the framework for further innovations in the realm of OCT image reconstruction.