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The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations, and for formulating invariant visual operations at higher levels. This paper defines and proves a set of joint covariance properties under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations, which make it possible to characterize how different types of image transformations interact with each other and the associated spatio-temporal receptive field responses. In this regard, we also extend the notion of scale-normalized derivatives to affine-normalized derivatives, to be able to obtain true affine-covariant properties of spatial derivatives, that are computed based on spatial smoothing with affine Gaussian kernels. The derived relations show how the parameters of the receptive fields need to be transformed, in order to match the output from spatio-temporal receptive fields under composed spatio-temporal image transformations. As a side effect, the presented proof for the joint covariance property over the integrated combination of the different geometric image transformations also provides specific proofs for the individual transformation properties, which have not previously been fully reported in the literature. The paper also presents an in-depth theoretical analysis of geometric interpretations of the derived covariance properties, as well as outlines a number of biological interpretations of these results.

We present an unsupervised 3D shape co-segmentation method which learns a set of deformable part templates from a shape collection. To accommodate structural variations in the collection, our network composes each shape by a selected subset of template parts which are affine-transformed. To maximize the expressive power of the part templates, we introduce a per-part deformation network to enable the modeling of diverse parts with substantial geometry variations, while imposing constraints on the deformation capacity to ensure fidelity to the originally represented parts. We also propose a training scheme to effectively overcome local minima. Architecturally, our network is a branched autoencoder, with a CNN encoder taking a voxel shape as input and producing per-part transformation matrices, latent codes, and part existence scores, and the decoder outputting point occupancies to define the reconstruction loss. Our network, coined DAE-Net for Deforming Auto-Encoder, can achieve unsupervised 3D shape co-segmentation that yields fine-grained, compact, and meaningful parts that are consistent across diverse shapes. We conduct extensive experiments on the ShapeNet Part dataset, DFAUST, and an animal subset of Objaverse to show superior performance over prior methods. Code and data are available at //github.com/czq142857/DAE-Net.

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity.

The NIR-to-RGB spectral domain translation is a formidable task due to the inherent spectral mapping ambiguities within NIR inputs and RGB outputs. Thus, existing methods fail to reconcile the tension between maintaining texture detail fidelity and achieving diverse color variations. In this paper, we propose a Multi-scale HSV Color Feature Embedding Network (MCFNet) that decomposes the mapping process into three sub-tasks, including NIR texture maintenance, coarse geometry reconstruction, and RGB color prediction. Thus, we propose three key modules for each corresponding sub-task: the Texture Preserving Block (TPB), the HSV Color Feature Embedding Module (HSV-CFEM), and the Geometry Reconstruction Module (GRM). These modules contribute to our MCFNet methodically tackling spectral translation through a series of escalating resolutions, progressively enriching images with color and texture fidelity in a scale-coherent fashion. The proposed MCFNet demonstrates substantial performance gains over the NIR image colorization task. Code is released at: //github.com/AlexYangxx/MCFNet.

Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a new way of deriving coefficients of implicit Runge-Kutta methods. This approach based on repeated integrals yields both new and well-known Butcher's tableaux. We discuss the properties of newly derived methods and compare them with standard collocation implicit Runge-Kutta methods in a series of numerical experiments. In particular, we observe higher accuracy and higher experimental order of convergence of some newly derived methods.

Satellite imaging generally presents a trade-off between the frequency of acquisitions and the spatial resolution of the images. Super-resolution is often advanced as a way to get the best of both worlds. In this work, we investigate multi-image super-resolution of satellite image time series, i.e. how multiple images of the same area acquired at different dates can help reconstruct a higher resolution observation. In particular, we extend state-of-the-art deep single and multi-image super-resolution algorithms, such as SRDiff and HighRes-net, to deal with irregularly sampled Sentinel-2 time series. We introduce BreizhSR, a new dataset for 4x super-resolution of Sentinel-2 time series using very high-resolution SPOT-6 imagery of Brittany, a French region. We show that using multiple images significantly improves super-resolution performance, and that a well-designed temporal positional encoding allows us to perform super-resolution for different times of the series. In addition, we observe a trade-off between spectral fidelity and perceptual quality of the reconstructed HR images, questioning future directions for super-resolution of Earth Observation data.

Experimental particle physics uses machine learning for many of tasks, where one application is to classify signal and background events. The classification can be used to bin an analysis region to enhance the expected significance for a mass resonance search. In natural language processing, one of the leading neural network architectures is the transformer. In this work, an event classifier transformer is proposed to bin an analysis region, in which the network is trained with special techniques. The techniques developed here can enhance the significance and reduce the correlation between the network's output and the reconstructed mass. It is found that this trained network can perform better than boosted decision trees and feed-forward networks.

The implication problem for conditional independence (CI) asks whether the fact that a probability distribution obeys a given finite set of CI relations implies that a further CI statement also holds in this distribution. This problem has a long and fascinating history, cumulating in positive results about implications now known as the semigraphoid axioms as well as impossibility results about a general finite characterization of CI implications. Motivated by violation of faithfulness assumptions in causal discovery, we study the implication problem in the special setting where the CI relations are obtained from a directed acyclic graphical (DAG) model along with one additional CI statement. Focusing on the Gaussian case, we give a complete characterization of when such an implication is graphical by using algebraic techniques. Moreover, prompted by the relevance of strong faithfulness in statistical guarantees for causal discovery algorithms, we give a graphical solution for an approximate CI implication problem, in which we ask whether small values of one additional partial correlation entail small values for yet a further partial correlation.

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalising this idea and, as illustration, construct and analyse straight-line programs for the Bruhat decomposition of $d\times d$ matrices with determinant $1$ over a finite field of order $q$ that have length $O(d^2\log(q))$ and require storing only $O(\log(q))$ matrices during evaluation.

We present self-supervised geometric perception (SGP), the first general framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels (e.g., camera poses, rigid transformations). Our first contribution is to formulate geometric perception as an optimization problem that jointly optimizes the feature descriptor and the geometric models given a large corpus of visual measurements (e.g., images, point clouds). Under this optimization formulation, we show that two important streams of research in vision, namely robust model fitting and deep feature learning, correspond to optimizing one block of the unknown variables while fixing the other block. This analysis naturally leads to our second contribution -- the SGP algorithm that performs alternating minimization to solve the joint optimization. SGP iteratively executes two meta-algorithms: a teacher that performs robust model fitting given learned features to generate geometric pseudo-labels, and a student that performs deep feature learning under noisy supervision of the pseudo-labels. As a third contribution, we apply SGP to two perception problems on large-scale real datasets, namely relative camera pose estimation on MegaDepth and point cloud registration on 3DMatch. We demonstrate that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.