arXiv subjects

Aalok Gangopadhyay

Publications and source records attributed to Aalok Gangopadhyay.

9 recordsLinked to original sources

arXiv2026

On the randomized Horn problem and the surface tension of hives

Given two nonincreasing $n$-tuples of real numbers $λ_n$, $μ_n$, the Horn problem asks for a description of all nonincreasing $n$-tuples of real numbers $ν_n$ such that there exist Hermitian matrices $X_n$, $Y_n$ and $Z_n$ respectively with these spectra such that $X_n + Y_n = Z_n$. There is also a randomized version of this problem where $X_n$ and $Y_n$ are sampled uniformly at random from orbits of Hermitian matrices arising from the conjugacy action by elements of the unitary group. One then asks for a description of the probability measure of the spectrum of the sum $Z_n$. Both the original Horn problem and its randomized version have solutions using the hives introduced by Knutson and Tao. In an asymptotic sense, as $n \rightarrow \infty$, large deviations for the randomized Horn problem were given by Narayanan and Sheffield in terms of the surface tension of hives. In this paper, we provide upper and lower bounds on this surface tension function. We also obtain a closed-form expression for the total entropy of a surface tension minimizing continuum hive with boundary conditions arising from GUE eigenspectra. Finally, we give several empirical results for random hives and lozenge tilings arising from an application of the octahedron recurrence for large $n$ and a numerical approximation of the surface tension function.

math.PR↗

arXiv2026

Denoising data using convex relaxations

We study the problem of denoising observations $Y_i=X_i+Z_i$, where the latent variables $X_i$ are sampled from a low-dimensional manifold in $\mathbb{R}^n$ and the noise variables $Z_i$ are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

stat.ME↗

arXiv2025

Flow Symmetrization for Parameterized Constrained Diffeomorphisms

Diffeomorphisms play a crucial role while searching for shapes with fixed topological properties, allowing for smooth deformation of template shapes. Several approaches use diffeomorphism for shape search. However, these approaches employ only unconstrained diffeomorphisms. In this work, we develop Flow Symmetrization - a method to represent a parametric family of constrained diffeomorphisms that contain additional symmetry constraints such as periodicity, rotation equivariance, and transflection equivariance. Our representation is differentiable in nature, making it suitable for gradient-based optimization approaches for shape search. As these symmetry constraints naturally arise in tiling classes, our method is ideal for representing tile shapes belonging to any tiling class. To demonstrate the efficacy of our method, we design two frameworks for addressing the challenging problems of Escherization and Density Estimation. The first framework is dedicated to the Escherization problem, where we parameterize tile shapes belonging to different isohedral classes. Given a target shape, the template tile is deformed using gradient-based optimization to resemble the target shape. The second framework focuses on density estimation in identification spaces. By leveraging the inherent link between tiling theory and identification topology, we design constrained diffeomorphisms for the plane that result in unconstrained diffeomorphisms on the identification spaces. Specifically, we perform density estimation on identification spaces such as torus, sphere, Klein bottle, and projective plane. Through results and experiments, we demonstrate that our method obtains impressive results for Escherization on the Euclidean plane and density estimation on non-Euclidean identification spaces. Code and results: https://dwipddalal.github.io/FlowSymmetry/

cs.GR↗

arXiv2025

DMD-Net: Deep Mesh Denoising Network

We present Deep Mesh Denoising Network (DMD-Net), an end-to-end deep learning framework, for solving the mesh denoising problem. DMD-Net consists of a Graph Convolutional Neural Network in which aggregation is performed in both the primal as well as the dual graph. This is realized in the form of an asymmetric two-stream network, which contains a primal-dual fusion block that enables communication between the primal-stream and the dual-stream. We develop a Feature Guided Transformer (FGT) paradigm, which consists of a feature extractor, a transformer, and a denoiser. The feature extractor estimates the local features, that guide the transformer to compute a transformation, which is applied to the noisy input mesh to obtain a useful intermediate representation. This is further processed by the denoiser to obtain the denoised mesh. Our network is trained on a large scale dataset of 3D objects. We perform exhaustive ablation studies to demonstrate that each component in our network is essential for obtaining the best performance. We show that our method obtains competitive or better results when compared with the state-of-the-art mesh denoising algorithms. We demonstrate that our method is robust to various kinds of noise. We observe that even in the presence of extremely high noise, our method achieves excellent performance.

cs.CV↗

arXiv2025

Hand Shadow Art: A Differentiable Rendering Perspective

Shadow art is an exciting form of sculptural art that produces captivating artistic effects through the 2D shadows cast by 3D shapes. Hand shadows, also known as shadow puppetry or shadowgraphy, involve creating various shapes and figures using your hands and fingers to cast meaningful shadows on a wall. In this work, we propose a differentiable rendering-based approach to deform hand models such that they cast a shadow consistent with a desired target image and the associated lighting configuration. We showcase the results of shadows cast by a pair of two hands and the interpolation of hand poses between two desired shadow images. We believe that this work will be a useful tool for the graphics community.

cs.GR↗

arXiv2023

Search Me Knot, Render Me Knot: Embedding Search and Differentiable Rendering of Knots in 3D

We introduce the problem of knot-based inverse perceptual art. Given multiple target images and their corresponding viewing configurations, the objective is to find a 3D knot-based tubular structure whose appearance resembles the target images when viewed from the specified viewing configurations. To solve this problem, we first design a differentiable rendering algorithm for rendering tubular knots embedded in 3D for arbitrary perspective camera configurations. Utilizing this differentiable rendering algorithm, we search over the space of knot configurations to find the ideal knot embedding. We represent the knot embeddings via homeomorphisms of the desired template knot, where the homeomorphisms are parametrized by the weights of an invertible neural network. Our approach is fully differentiable, making it possible to find the ideal 3D tubular structure for the desired perceptual art using gradient-based optimization. We propose several loss functions that impose additional physical constraints, enforcing that the tube is free of self-intersection, lies within a predefined region in space, satisfies the physical bending limits of the tube material and the material cost is within a specified budget. We demonstrate through results that our knot representation is highly expressive and gives impressive results even for challenging target images in both single view as well as multiple view constraints. Through extensive ablation study we show that each of the proposed loss function is effective in ensuring physical realizability. We construct a real world 3D-printed object to demonstrate the practical utility of our approach. To the best of our knowledge, we are the first to propose a fully differentiable optimization framework for knot-based inverse perceptual art.

cs.GR↗

arXiv2023

A Graph Neural Network Approach for Temporal Mesh Blending and Correspondence

We have proposed a self-supervised deep learning framework for solving the mesh blending problem in scenarios where the meshes are not in correspondence. To solve this problem, we have developed Red-Blue MPNN, a novel graph neural network that processes an augmented graph to estimate the correspondence. We have designed a novel conditional refinement scheme to find the exact correspondence when certain conditions are satisfied. We further develop a graph neural network that takes the aligned meshes and the time value as input and fuses this information to process further and generate the desired result. Using motion capture datasets and human mesh designing software, we create a large-scale synthetic dataset consisting of temporal sequences of human meshes in motion. Our results demonstrate that our approach generates realistic deformation of body parts given complex inputs.

cs.CV↗

arXiv2021

APEX-Net: Automatic Plot Extractor Network

Automatic extraction of raw data from 2D line plot images is a problem of great importance having many real-world applications. Several algorithms have been proposed for solving this problem. However, these algorithms involve a significant amount of human intervention. To minimize this intervention, we propose APEX-Net, a deep learning based framework with novel loss functions for solving the plot extraction problem. We introduce APEX-1M, a new large scale dataset which contains both the plot images and the raw data. We demonstrate the performance of APEX-Net on the APEX-1M test set and show that it obtains impressive accuracy. We also show visual results of our network on unseen plot images and demonstrate that it extracts the shape of the plots to a great extent. Finally, we develop a GUI based software for plot extraction that can benefit the community at large. For dataset and more information visit https://sites.google.com/view/apexnetpaper/.

cs.CV↗

arXiv2015

SA-CNN: Dynamic Scene Classification using Convolutional Neural Networks

The task of classifying videos of natural dynamic scenes into appropriate classes has gained lot of attention in recent years. The problem especially becomes challenging when the camera used to capture the video is dynamic. In this paper, we analyse the performance of statistical aggregation (SA) techniques on various pre-trained convolutional neural network(CNN) models to address this problem. The proposed approach works by extracting CNN activation features for a number of frames in a video and then uses an aggregation scheme in order to obtain a robust feature descriptor for the video. We show through results that the proposed approach performs better than the-state-of-the arts for the Maryland and YUPenn dataset. The final descriptor obtained is powerful enough to distinguish among dynamic scenes and is even capable of addressing the scenario where the camera motion is dominant and the scene dynamics are complex. Further, this paper shows an extensive study on the performance of various aggregation methods and their combinations. We compare the proposed approach with other dynamic scene classification algorithms on two publicly available datasets - Maryland and YUPenn to demonstrate the superior performance of the proposed approach.

cs.CV↗