Fundamental Inverse Problem Research Articles

Deformable surface reconstruction via Riemannian metric preservation

Estimating the pose of an object from a monocular image is a fundamental inverse problem in computer vision. Due to its ill-posed nature, solving this problem requires incorporating deformation priors. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a reliable and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach for inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and achieves state-of-the-art performance without the need for offline training. Being a method that performs per-frame optimization, our method can refine its estimates, contrary to those based on performing a single inference step. Despite enforcing differential geometry constraints at each update, our approach is the fastest of all the tested optimization-based methods.

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search space possesses many interesting symmetry properties that were studied in the last decade. In this paper, we present a new algorithm for this subclass of the DGP, which exploits DMDGP symmetries more effectively than its predecessors. Computational results show that the speedup, with respect to the classic BP algorithm, is considerable for sparse DMDGP instances related to protein conformation.

On deep learning techniques to boost monocular depth estimation for autonomous navigation

Inferring the depth of images is a fundamental inverse problem within the field of Computer Vision since depth information is obtained through 2D images, which can be generated from infinite possibilities of observed real scenes. Benefiting from the progress of Convolutional Neural Networks (CNNs) to explore structural features and spatial image information, Single Image Depth Estimation (SIDE) is often highlighted in scopes of scientific and technological innovation, as this concept provides advantages related to its low implementation cost and robustness to environmental conditions. In the context of autonomous vehicles, state-of-the-art CNNs optimize the SIDE task by producing high-quality depth maps, which are essential during the autonomous navigation process in different locations. However, such networks are usually supervised by sparse and noisy depth data, from Light Detection and Ranging (LiDAR) laser scans, and are carried out at high computational cost, requiring high-performance Graphic Processing Units (GPUs). Therefore, we propose a new lightweight and fast supervised CNN architecture combined with novel feature extraction models which are designed for real-world autonomous navigation. We also introduce an efficient surface normals module, jointly with a simple geometric 2.5D loss function, to solve SIDE problems. We also innovate by incorporating multiple Deep Learning techniques, such as the use of densification algorithms and additional semantic, surface normals and depth information to train our framework. The method introduced in this work focuses on robotic applications in indoor and outdoor environments and its results are evaluated on the competitive and publicly available NYU Depth V2 and KITTI Depth datasets.

Remotely-sensed planform morphologies reveal fluvial and tidal nature of meandering channels

Meandering channels extensively dissect fluvial and tidal landscapes, critically controlling their morphodynamic evolution and sedimentary architecture. In spite of an apparently striking dissimilarity of the governing processes, planform dimensions of tidal and fluvial meanders consistently scale to local channel width, and previous studies were unable to identify quantitative planimetric differences between these landforms. Here we use satellite imagery, measurements of meandering patterns, and different statistical analyses applied to about 10,000 tidal and fluvial meanders worldwide to objectively disclose fingerprints of the different physical processes they are shaped by. We find that fluvial and tidal meanders can be distinguished on the exclusive basis of their remotely-sensed planforms. Moreover, we show that tidal meanders are less morphologically complex and display more spatially homogeneous characteristics compared to fluvial meanders. Based on existing theoretical, numerical, and field studies, we suggest that our empirical observations can be explained by the more regular processes carving tidal meanders, as well as by the higher lithological homogeneity of the substrates they typically cut through. Allowing one to effectively infer processes from landforms, a fundamental inverse problem in geomorphology, our results have relevant implications for the conservation and restoration of tidal environments, as well as from planetary exploration perspectives.

Universal nature of different methods of obtaining the exact Kohn–Sham exchange-correlation potential for a given density

Finding the external potential or the Kohn–Sham (KS) potential for a given ground state density is a fundamental inverse problem in density functional theory. Furthermore, it is important in generating the exact exchange-correlation potential for a density which can then serve as a benchmark for testing the accuracy of an exchange-correlation energy functional. Over the years different methods have been proposed to do the density-to-potential inversion and all of these appear to be disjoint. In this paper we show that these methods are connected to and can be derived from one general algorithm that utilizes two forms of the equation for the density; these are the KS equation in terms of orbitals and the Euler equation in terms of the density. We obtain the condition for the convergence of the general method and show that all the methods considered by us satisfy this condition. We demonstrate the method and its flexibility by obtaining the KS potential for some spherical systems in a variety of ways.

Modeling Point Spread Function in Fluorescence Microscopy With a Sparse Gaussian Mixture: Tradeoff Between Accuracy and Efficiency.

Deblurring is a fundamental inverse problem in bioimaging. It requires modeling the point spread function (PSF), which captures the optical distortions entailed by the image formation process. The PSF limits the spatial resolution attainable for a given microscope. However, recent applications require a higher resolution and have prompted the development of super-resolution techniques to achieve sub-pixel accuracy. This requirement restricts the class of suitable PSF models to analog ones. In addition, deblurring is computationally intensive, hence further requiring computationally efficient models. A custom candidate fitting both the requirements is the Gaussian model. However, this model cannot capture the rich tail structures found in both the theoretical and empirical PSFs. In this paper, we aim at improving the reconstruction accuracy beyond the Gaussian model, while preserving its computational efficiency. We introduce a new class of analog PSF models based on the Gaussian mixtures. The number of Gaussian kernels controls both the modeling accuracy and the computational efficiency of the model: the lower the number of kernels, the lower the accuracy and the higher the efficiency. To explore the accuracy-efficiency tradeoff, we propose a variational formulation of the PSF calibration problem, where a convex sparsity-inducing penalty on the number of Gaussian kernels allows trading accuracy for efficiency. We derive an efficient algorithm based on a fully split formulation of alternating split Bregman. We assess our framework on synthetic and real data, and demonstrate a better reconstruction accuracy in both geometry and photometry in point source localization-a fundamental inverse problem in fluorescence microscopy.

Network reconstruction from infection cascades.

Accessing the network through which a propagation dynamics diffuses is essential for understanding and controlling it. In a few cases, such information is available through direct experiments or thanks to the very nature of propagation data. In a majority of cases however, available information about the network is indirect and comes from partial observations of the dynamics, rendering the network reconstruction a fundamental inverse problem. Here we show that it is possible to reconstruct the whole structure of an interaction network and to simultaneously infer the complete time course of activation spreading, relying just on single epoch (i.e. snapshot) or time-scattered observations of a small number of activity cascades. The method that we present is built on a belief propagation approximation, that has shown impressive accuracy in a wide variety of relevant cases, and is able to infer interactions in the presence of incomplete time-series data by providing a detailed modelling of the posterior distribution of trajectories conditioned to the observations. Furthermore, we show by experiments that the information content of full cascades is relatively smaller than that of sparse observations or single snapshots.

Reconstructing signed networks via Ising dynamics.

Revealing unknown network structure from observed data is a fundamental inverse problem in network science. Current reconstruction approaches were mainly proposed to infer the unsigned networks. However, many social relationships, such as friends and foes, can be represented as signed social networks that contain positive and negative links. To the best of our knowledge, the method of reconstructing signed networks has not yet been developed. To this purpose, we develop a statistical inference approach to fully reconstruct the signed network structure (positive links, negative links, and nonexistent links) based on the Ising dynamics. By the theoretical analysis, we show that our approach can transfer the problem of maximum likelihood estimation into the problem of solving linear systems of equations, where the solution of the linear system of equations uncovers the neighbors and the signs of links of each node. The experimental results on both synthetic and empirical networks validate the reliability and efficiency of our method. Our study moves the first step toward reconstructing signed networks.

Performance of Landweber iteration algorithm in tomographic image reconstruction

The rapid growth of computed tomography has been accompanied by equally advancing in image reconstruction algorithm also. The fundamental inverse problem is the reconstruction of a function from finitely many measurements, so pertaining to that function. The measured data are limited, and it cannot serve to determine one single correct solution. The iterative reconstruction image reconstruction algorithms have stable solution to the limited projection. In this paper, Landweber-based iteration image reconstruction is simulated and its performance is compared with different algorithm. Then, the quality of the reconstructed image is expressed in terms of mean absolute error and correlation coefficient as compared to the original image. The entire simulations are performed in Matlab tool.

Performance of Landweber iteration algorithm in tomographic image reconstruction

The rapid growth of computed tomography has been accompanied by equally advancing in image reconstruction algorithm also. The fundamental inverse problem is the reconstruction of a function from finitely many measurements, so pertaining to that function. The measured data are limited, and it cannot serve to determine one single correct solution. The iterative reconstruction image reconstruction algorithms have stable solution to the limited projection. In this paper, Landweber-based iteration image reconstruction is simulated and its performance is compared with different algorithm. Then, the quality of the reconstructed image is expressed in terms of mean absolute error and correlation coefficient as compared to the original image. The entire simulations are performed in Matlab tool.

Solution of three dimensional inverse problem of magnetic induction tomography using Tikhonov regularization method

Magnetic induction tomography, highly promising imaging technique for medical purposes, applies a magnetic field from an exciter to induce eddy currents in the object of low conductivity, and the magnetic field from these is then detected by a sensor. The induced eddy currents are proportional to the conductivity distribution. This paper deals with the solution of the fundamental inverse problem existing in magnetic induction tomography, which consists in the reconstruction of the eddy currents distribution basing on the magnetic field measurements. The problem has been solved by the Tikhonov regularization method. The choice of the regularization parameter and the accuracy of the proposed method have also been discussed.

Direct analytical solution of the inverse gear tooth contact analysis problem

Despite the advances in gear tooth contact analysis and the existence of many competent theories, the fundamental inverse problem of determining the gear profile form that produces a desired kinematical response, or function of transmission errors, remains to be solved. This is because the usually employed form of the equations governing tooth contact is so complex and implicit, that it is impossible to solve inversely. To bypass this handicap, current design methodologies have to rely on indirect calculations, often requiring substantial computational effort. Here, a new more versatile formulation of the fundamental surface contact equations is proposed, leading to a set of meshing equations that allows the direct analytical solution of the inverse problem. The solution itself is in elegant vector-matrix form and it is explicit and fast. Applications of the proposed solution are discussed.