arXiv subjects

Paola Causin

Publications and source records attributed to Paola Causin.

10 recordsLinked to original sources

arXiv2026

A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds

Speciation in generative diffusion models denotes the emergence of distinct stable branches during denoising, through which initially undifferentiated trajectories progressively commit to different data classes. In this work we develop an intrinsic theory of speciation for diffusion models supported on compact Riemannian manifolds: the aim is to go beyond existing theoretical descriptions, which usually identify speciation with a symmetric pitchfork bifurcation and assume to work in a large-dimensional space. We characterize speciation by bifurcations of the critical points of the evolving probability density. A spectral heat-kernel representation makes explicit the role of the manifold geometry, while Poincaré-Hopf and Morse theory impose global constraints on the number and type of score equilibria and reveal topologically-imposed geometrical modes. For mixtures of heat kernels, we prove that generic speciation events have a one-dimensional critical kernel and admit an A2 fold normal form; pitchforks and simultaneous multidirectional transitions arise from nongeneric symmetric configurations. We derive geometry-dependent estimates of speciation times for bimodal mixtures and Riemannian regular simplices. We further establish structural stability of nondegenerate folds under score perturbations and show that the first-order time shift is determined solely by the component of the score error along the critical direction. The theory is illustrated on the sphere using mixtures of von Mises-Fisher distributions, where pitchfork and saddle-node bifurcations, topological modes, and hierarchical multiple speciations are observed. Finally, a chart-based intrinsic score-learning scheme based on neural networks contrasts the theoretically predicted transitions on prototypal and more complex datasets.

cs.LG↗

arXiv2025

Optimization of gridding algorithms for FFT by vector optimization

The Fast Fourier Transform (FFT) is widely used in applications such as MRI, CT, and interferometry; however, because of its dependence on uniformly sampled data, it requires the use of gridding techniques for practical implementation. The performance of these algorithms strongly depends on the choice of the gridding kernel, with the first prolate spheroidal wave function (PSWF) regarded as optimal. This work redefines kernel optimality through the lens of vector optimization (VO), introducing a rigorous framework that characterizes optimal kernels as Pareto-efficient solutions of an error shape operator. We establish the continuity of such operator, study the existence of solutions, and propose a novel methodology to construct kernels tailored to a desired target error function. The approach is implemented numerically via interior-point optimization. Comparative experiments demonstrate that the proposed kernels outperform both the PSWF and the state-of-the-art methods (MIRT-NUFFT) in specific regions of interest, achieving orders-of-magnitude improvements in mean absolute errors. These results confirm the potential of VO-based kernel design to provide customized accuracy profiles aligned with application-specific requirements. Future research will extend this framework to multidimensional cases and relative error minimization, with potential integration of machine learning for adaptive target error selection.

math.NA↗

arXiv2025

Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems

High-dimensional datasets often exhibit low-dimensional geometric structures, as suggested by the manifold hypothesis, which implies that data lie on a smooth manifold embedded in a higher-dimensional ambient space. While this insight underpins many advances in machine learning and inverse problems, fully leveraging it requires to deal with three key tasks: estimating the intrinsic dimension (ID) of the manifold, constructing appropriate local coordinates, and learning mappings between ambient and manifold spaces. In this work, we propose a framework that addresses all these challenges using a Mixture of Variational Autoencoders (VAEs) and tools from Riemannian geometry. We specifically focus on estimating the ID of datasets by analyzing the numerical rank of the VAE decoder pullback metric. The estimated ID guides the construction of an atlas of local charts using a mixture of invertible VAEs, enabling accurate manifold parameterization and efficient inference. We how this approach enhances solutions to ill-posed inverse problems, particularly in biomedical imaging, by enforcing that reconstructions lie on the learned manifold. Lastly, we explore the impact of network pruning on manifold geometry and reconstruction quality, showing that the intrinsic dimension serves as an effective proxy for monitoring model capacity.

cs.LG↗

arXiv2024

A Modular Deep Learning-based Approach for Diffuse Optical Tomography Reconstruction

Medical imaging is nowadays a pillar in diagnostics and therapeutic follow-up. Current research tries to integrate established - but ionizing - tomographic techniques with technologies offering reduced radiation exposure. Diffuse Optical Tomography (DOT) uses non-ionizing light in the Near-Infrared (NIR) window to reconstruct optical coefficients in living beings, providing functional indications about the composition of the investigated organ/tissue. Due to predominant light scattering at NIR wavelengths, DOT reconstruction is, however, a severely ill-conditioned inverse problem. Conventional reconstruction approaches show severe weaknesses when dealing also with mildly complex cases and/or are computationally very intensive. In this work we explore deep learning techniques for DOT inversion. Namely, we propose a fully data-driven approach based on a modularity concept: first data and originating signal are separately processed via autoencoders, then the corresponding low-dimensional latent spaces are connected via a bridging network which acts at the same time as a learned regularizer.

math.NA↗

arXiv2022

A Learned-SVD approach for Regularization in Diffuse Optical Tomography

Diffuse Optical Tomography (DOT) is an emerging technology in medical imaging which employs light in the NIR spectrum to estimate the distribution of optical coefficients in biological tissues for diagnostic and monitoring purposes. DOT reconstruction implies the solution of a severely ill-posed inverse problem, for which regularization techniques are mandatory in order to achieve reasonable results. Traditionally, regularization techniques put a variance prior on the desired solution/gradient via regularization parameters, whose choice requires a fine tuning, specific for each case. In this work we explore deep learning techniques in a fully data-driven approach, able of reconstructing the generating signal (optical absorption coefficient) in an automated way. We base our approach on the so-called Learned Singular Value Decomposition, which has been proposed for general inverse problems, and we tailor it to the DOT application. We perform tests with increasing levels of noise on the measure, and compare it with standard variational approaches.

math.NA↗

arXiv2021

Unsupervised deep learning techniques for powdery mildew recognition based on multispectral imaging

Objectives. Sustainable management of plant diseases is an open challenge which has relevant economic and environmental impact. Optimal strategies rely on human expertise for field scouting under favourable conditions to assess the current presence and extent of disease symptoms. This labor-intensive task is complicated by the large field area to be scouted, combined with the millimeter-scale size of the early symptoms to be detected. In view of this, image-based detection of early disease symptoms is an attractive approach to automate this process, enabling a potential high throughput monitoring at sustainable costs. Methods. Deep learning has been successfully applied in various domains to obtain an automatic selection of the relevant image features by learning filters via a training procedure. Deep learning has recently entered also the domain of plant disease detection: following this idea, in this work we present a deep learning approach to automatically recognize powdery mildew on cucumber leaves. We focus on unsupervised deep learning techniques applied to multispectral imaging data and we propose the use of autoencoder architectures to investigate two strategies for disease detection: i) clusterization of features in a compressed space; ii) anomaly detection. Results. The two proposed approaches have been assessed by quantitative indices. The clusterization approach is not fully capable by itself to provide accurate predictions but it does cater relevant information. Anomaly detection has instead a significant potential of resolution which could be further exploited as a prior for supervised architectures with a very limited number of labeled samples.

cs.CV↗

arXiv2021

A Deep Learning Generative Model Approach for Image Synthesis of Plant Leaves

Objectives. We generate via advanced Deep Learning (DL) techniques artificial leaf images in an automatized way. We aim to dispose of a source of training samples for AI applications for modern crop management. Such applications require large amounts of data and, while leaf images are not truly scarce, image collection and annotation remains a very time--consuming process. Data scarcity can be addressed by augmentation techniques consisting in simple transformations of samples belonging to a small dataset, but the richness of the augmented data is limited: this motivates the search for alternative approaches. Methods. Pursuing an approach based on DL generative models, we propose a Leaf-to-Leaf Translation (L2L) procedure structured in two steps: first, a residual variational autoencoder architecture generates synthetic leaf skeletons (leaf profile and veins) starting from companions binarized skeletons of real images. In a second step, we perform translation via a Pix2pix framework, which uses conditional generator adversarial networks to reproduce the colorization of leaf blades, preserving the shape and the venation pattern. Results. The L2L procedure generates synthetic images of leaves with a realistic appearance. We address the performance measurement both in a qualitative and a quantitative way; for this latter evaluation, we employ a DL anomaly detection strategy which quantifies the degree of anomaly of synthetic leaves with respect to real samples. Conclusions. Generative DL approaches have the potential to be a new paradigm to provide low-cost meaningful synthetic samples for computer-aided applications. The present L2L approach represents a step towards this goal, being able to generate synthetic samples with a relevant qualitative and quantitative resemblance to real leaves.

cs.CV↗

arXiv2016

Mathematical modeling of local perfusion in large distensible microvascular networks

Microvessels -blood vessels with diameter less than 200 microns- form large, intricate networks organized into arterioles, capillaries and venules. In these networks, the distribution of flow and pressure drop is a highly interlaced function of single vessel resistances and mutual vessel interactions. In this paper we propose a mathematical and computational model to study the behavior of microcirculatory networks subjected to different conditions. The network geometry is composed of a graph of connected straight cylinders, each one representing a vessel. The blood flow and pressure drop across the single vessel, further split into smaller elements, are related through a generalized Ohm's law featuring a conductivity parameter, function of the vessel cross section area and geometry, which undergo deformations under pressure loads. The membrane theory is used to describe the deformation of vessel lumina, tailored to the structure of thick-walled arterioles and thin-walled venules. In addition, since venules can possibly experience negative transmural pressures, a buckling model is also included to represent vessel collapse. The complete model including arterioles, capillaries and venules represents a nonlinear system of PDEs, which is approached numerically by finite element discretization and linearization techniques. We use the model to simulate flow in the microcirculation of the human eye retina, a terminal system with a single inlet and outlet. After a phase of validation against experimental measurements, we simulate the network response to different interstitial pressure values. Such a study is carried out both for global and localized variations of the interstitial pressure. In both cases, significant redistributions of the blood flow in the network arise, highlighting the importance of considering the single vessel behavior along with its position and connectivity in the network.

q-bio.TO↗

arXiv2015

A Poroelastic Mixture Model of Mechanobiological Processes in Tissue Engineering. Part I: Mathematical Formulation

An adequate control of cell response in tissue engineering applications is of utmost importance to obtain products suitable to clinical practice. This paper is the first part of a series of two connected publications in which we study via mathematical tools the cultivation in bioreactors of articular chondrocytes. The proposed model combines poroelastic theory of mixtures and cellular population models into a framework including stress state and oxygen tension as main determinants of engineered culture evolution. The special mechanosensitivity of articular chondrocytes to the surrounding environment is accounted for in the model through the novel concept of "force isotropy" acting on the cell which is assumed as the promoting factor of the production of new cells or extracellular matrix.

q-bio.TO↗

arXiv2015

A Poroelastic Mixture Model of Mechanobiological Processes in Tissue Engineering. Part II: Numerical Simulations

In Part I of this article we have developed a novel mechanobiological model of a Tissue Engineering process that accounts for the mechanisms through which an isotropic or anisotropic adherence condition regulates the active functions of the cells in the construct. The model expresses mass balance and force equilibrium balance for a multi-phase mixture in a 3D computational domain and in time dependent conditions. In the present Part II, we study the mechanobiological model in a simplified 1D geometrical setting with the purpose of highlighting the ability of the formulation to represent the influence of force isotropy and nutrient availability on the growth of the tissue construct. In particular, an example of isotropy estimator is proposed and coded within a fixed-point solution map that is used at each discrete time level for system linearization and subsequent finite element approximation of the linearized equations. Extensively conducted simulations show that: 1) the spatial and temporal evolution of the cellular populations are in good agrement with the local growth/production conditions predicted by the mechanobiological stress-dependent model; and 2) the isotropy indicator and all model variables are strongly influenced by both maximum cell specific growth rate and mechanical boundary conditions enforced at the interface between the biomass construct and the interstitial fluid.

math.NA↗