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This work investigates the structure and representation capacity of $sinusoidal$ MLPs, which have recently shown promising results in encoding low-dimensional signals. This success can be attributed to its smoothness and high representation capacity. The first allows the use of the network's derivatives during training, enabling regularization. However, defining the architecture and initializing its parameters to achieve a desired capacity remains an empirical task. This work provides theoretical and experimental results justifying the capacity property of sinusoidal MLPs and offers control mechanisms for their initialization and training. We approach this from a Fourier series perspective and link the training with the model's spectrum. Our analysis is based on a $harmonic$ expansion of the sinusoidal MLP, which says that the composition of sinusoidal layers produces a large number of new frequencies expressed as integer linear combinations of the input frequencies (weights of the input layer). We use this novel $identity$ to initialize the input neurons which work as a sampling in the signal spectrum. We also note that each hidden neuron produces the same frequencies with amplitudes completely determined by the hidden weights. Finally, we give an upper bound for these amplitudes, which results in a $bounding$ scheme for the network's spectrum during training.

In this paper, we design a controller for an interconnected system consisting of a linear Stochastic Differential Equation (SDE) actuated through a linear hyperbolic Partial Differential Equation (PDE). Our approach aims to minimize the variance of the state of the SDE component. We leverage a backstepping technique to transform the original PDE into an uncoupled stochastic PDE. As such, we reformulate our initial problem as the control of a delayed SDE with a non-deterministic drift. Under standard controllability assumptions, we design a controller steering the mean of the states to zero while keeping its covariance bounded. As final step, we address the optimal control of the delayed SDE employing Artstein's transformation and Linear Quadratic stochastic control techniques., Comment: 8 pages, 1 figure. Submitted to L-CSS journal jointly with CDC conference

We explore sinusoidal neural networks to represent periodic tileable textures. Our approach leverages the Fourier series by initializing the first layer of a sinusoidal neural network with integer frequencies with a period $P$. We prove that the compositions of sinusoidal layers generate only integer frequencies with period $P$. As a result, our network learns a continuous representation of a periodic pattern, enabling direct evaluation at any spatial coordinate without the need for interpolation. To enforce the resulting pattern to be tileable, we add a regularization term, based on the Poisson equation, to the loss function. Our proposed neural implicit representation is compact and enables efficient reconstruction of high-resolution textures with high visual fidelity and sharpness across multiple levels of detail. We present applications of our approach in the domain of anti-aliased surface.

Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal-to-primal Hodge star operator. Combining these three `basic operators' we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz-Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

In this paper, we introduce a novel approach to solve the (mean-covariance) steering problem for a fairly general class of linear continuous-time stochastic systems subject to input delays. Specifically, we aim at steering delayed linear stochastic differential equations to a final desired random variable with given mean and covariance. We first establish a controllability result for these control systems, revealing the existence of a lower bound under which the covariance of the control system can not be steered. This structural threshold covariance stems from a unique combined effect due to stochastic diffusions and delays. Next, we propose a numerically cheap approach to reach any neighbor of this threshold covariance in finite time. Via an optimal control-based strategy, we enhance the aforementioned approach to keep the system covariance small at will in the whole control horizon. Under some additional assumptions on the dynamics, we give theoretical guarantees on the efficiency of our method. Finally, numerical simulations are provided to ground our theoretical findings, showcasing the ability of our methods in optimally approaching the covariance threshold.

Eukaryotic swimming cells such as spermatozoa, algae or protozoa use flagella or cilia to move in viscous fluids. The motion of their flexible appendages in the surrounding fluid induces propulsive forces that balance with the viscous drag on the cells and lead to a directed swimming motion. Here, we use our recently built database of cell motility (BOSO-Micro) to investigate the extent to which the shapes of eukaryotic swimming cells may be optimal from a hydrodynamic standpoint. We first examine the morphology of flexible flagella undergoing waving deformation and show that their amplitude-to-wavelength ratio is near the one predicted theoretically to optimise the propulsive efficiency of active filaments. Next, we consider ciliates, for which locomotion is induced by the collective beating of short cilia covering their surface. We show that the aspect ratio of ciliates are close to the one predicted to minimise the viscous drag of the cell body. Both results strongly suggest a key role played by hydrodynamic constraints, in particular viscous drag, in shaping eukaryotic swimming cells., Comment: to be published in J. Fluid Mech

Face morphing is a problem in computer graphics with numerous artistic and forensic applications. It is challenging due to variations in pose, lighting, gender, and ethnicity. This task consists of a warping for feature alignment and a blending for a seamless transition between the warped images. We propose to leverage coord-based neural networks to represent such warpings and blendings of face images. During training, we exploit the smoothness and flexibility of such networks by combining energy functionals employed in classical approaches without discretizations. Additionally, our method is time-dependent, allowing a continuous warping/blending of the images. During morphing inference, we need both direct and inverse transformations of the time-dependent warping. The first (second) is responsible for warping the target (source) image into the source (target) image. Our neural warping stores those maps in a single network dismissing the need for inverting them. The results of our experiments indicate that our method is competitive with both classical and generative models under the lens of image quality and face-morphing detectors. Aesthetically, the resulting images present a seamless blending of diverse faces not yet usual in the literature., Comment: 14 pages, 20 figures, accepted for CVPR 2024

In this paper, we consider continuous-time stochastic optimal control problems where the cost is evaluated through a coherent risk measure. We provide an explicit gradient descent-ascent algorithm which applies to problems subject to non-linear stochastic differential equations. More specifically, we leverage duality properties of coherent risk measures to relax the problem via a smooth min-max reformulation which induces artificial strong concavity in the max subproblem. We then formulate necessary conditions of optimality for this relaxed problem which we leverage to prove convergence of the gradient descent-ascent algorithm to candidate solutions of the original problem. Finally, we showcase the efficiency of our algorithm through numerical simulations involving trajectory tracking problems and highlight the benefit of favoring risk measures over classical expectation.

We present MR-Net, a general architecture for multiresolution neural networks, and a framework for imaging applications based on this architecture. Our coordinate-based networks are continuous both in space and in scale as they are composed of multiple stages that progressively add finer details. Besides that, they are a compact and efficient representation. We show examples of multiresolution image representation and applications to texturemagnification, minification, and antialiasing. This document is the extended version of the paper [PNS+22]. It includes additional material that would not fit the page limitations of the conference track for publication.

This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time $\mathbb{R}^3\times \mathbb{R}$, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually $\mathbb{R}^3 \times \{0\}$. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions. To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches. We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface.

We introduce a novel approach for rendering static and dynamic 3D neural signed distance functions (SDF) in real-time. We rely on nested neighborhoods of zero-level sets of neural SDFs, and mappings between them. This framework supports animations and achieves real-time performance without the use of spatial data-structures. It consists of three uncoupled algorithms representing the rendering steps. The multiscale sphere tracing focuses on minimizing iteration time by using coarse approximations on earlier iterations. The neural normal mapping transfers details from a fine neural SDF to a surface nested on a neighborhood of its zero-level set. It is smooth and it does not depend on surface parametrizations. As a result, it can be used to fetch smooth normals for discrete surfaces such as meshes and to skip later iterations when sphere tracing level sets. Finally, we propose an algorithm for analytic normal calculation for MLPs and describe ways to obtain sequences of neural SDFs to use with the algorithms., Comment: 9 pages, 8 figures

Depth maps captured with commodity sensors often require super-resolution to be used in applications. In this work we study a super-resolution approach based on a variational problem statement with Tikhonov regularization where the regularizer is parametrized with a deep neural network. This approach was previously applied successfully in photoacoustic tomography. We experimentally show that its application to depth map super-resolution is difficult, and provide suggestions about the reasons for that., Comment: 9 pages

The Muon $g-2$ Experiment at Fermilab uses a gaseous straw tracking detector to make detailed measurements of the stored muon beam profile, which are essential for the experiment to achieve its uncertainty goals. Positrons from muon decays spiral inward and pass through the tracking detector before striking an electromagnetic calorimeter. The tracking detector is therefore located inside the vacuum chamber in a region where the magnetic field is large and non-uniform. As such, the tracking detector must have a low leak rate to maintain a high-quality vacuum, must be non-magnetic so as not to perturb the magnetic field and, to minimize energy loss, must have a low radiation length. The performance of the tracking detector has met or surpassed the design requirements, with adequate electronic noise levels, an average straw hit resolution of $(110 \pm 20) \,\mu$m, a detection efficiency of 97% or higher, and no performance degradation or signs of aging. The tracking detector's measurements result in an otherwise unachievable understanding of the muon's beam motion, particularly at early times in the experiment's measurement period when there are a significantly greater number of muons decaying. This is vital to the statistical power of the experiment, as well as facilitating the precise extraction of several systematic corrections and uncertainties. This paper describes the design, construction, testing, commissioning, and performance of the tracking detector., Comment: 37 pages, 27 figures

Unicellular microscopic organisms living in aqueous environments outnumber all other creatures on Earth. A large proportion of them are able to self-propel in fluids with a vast diversity of swimming gaits and motility patterns. In this paper we present a biophysical survey of the available experimental data produced to date on the characteristics of motile behaviour in unicellular microswimmers. We assemble from the available literature empirical data on the motility of four broad categories of organisms: bacteria (and archaea), flagellated eukaryotes, spermatozoa and ciliates. Whenever possible, we gather the following biological, morphological, kinematic and dynamical parameters: species, geometry and size of the organisms, swimming speeds, actuation frequencies, actuation amplitudes, number of flagella and properties of the surrounding fluid. We then organise the data using the established fluid mechanics principles for propulsion at low Reynolds number. Specifically, we use theoretical biophysical models for the locomotion of cells within the same taxonomic groups of organisms as a means of rationalising the raw material we have assembled, while demonstrating the variability for organisms of different species within the same group. The material gathered in our work is an attempt to summarise the available experimental data in the field, providing a convenient and practical reference point for future studies., Comment: Database available in the Center for Open Science (OSF) repository (https://osf.io/4tyx6/). Open source version of the database available on GitHub (https://github.com/marcos-fvr/BOSO-micro)

This chapter describes how to use intersection and closest-hit shaders to implement real-time visualizations of complex fractals using distance functions. The Mandelbulb and Julia Sets are used as examples., Comment: 11 pages, 4 figures