Minimal Solver Research Articles

Minimal solver for relative pose estimation under planar motion

Minimal solver for relative pose estimation under planar motion

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

Correction: Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

Correction: Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically, we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.

MinBackProp – Backpropagating through Minimal Solvers

MinBackProp – Backpropagating through Minimal Solvers

Single-particle reconstruction in cryo-EM based on three-dimensional weighted nuclear norm minimization

Single-particle reconstruction (SPR) in cryogenic electron microscopy (cryo-EM) aims at aligning and averaging two-dimensional micrographs to reconstruct a three-dimensional particle.How to reconstruct micrographs from heavy noise is a crucial point for achieving better micrograph quality, and thus many methods focus on noise removal. However, new problems such as over-smoothing often occur in their results due to failure in handling heavy noise well. This paper proposes a three-dimensional weighted nuclear norm minimization (3DWNNM) model for SPR in the cryo-EM task to address these issues. Specifically, we design a minimization solver based on the forward-backward splitting algorithm to tackle our model efficiently. Under certain conditions, this solution has an energy-decaying feature and performs exceptionally well in reconstruction. Numerical experiments fully demonstrate the effectiveness and the robustness of the proposed method.

On Making SIFT Features Affine Covariant

An approach is proposed for recovering affine correspondences (ACs) from orientation- and scale-covariant, e.g., SIFT, features exploiting pre-estimated epipolar geometry. The method calculates the affine parameters consistent with the epipolar geometry from the point coordinates and the scales and rotations which the feature detector obtains. The proposed closed-form solver returns a single solution and is extremely fast, i.e., 0.5 upmu seconds on average. Possible applications include estimating the homography from a single upgraded correspondence and, also, estimating the surface normal for each correspondence found in a pre-calibrated image pair (e.g., stereo rig). As the second contribution, we propose a minimal solver that estimates the relative pose of a vehicle-mounted camera from a single SIFT correspondence with the corresponding surface normal obtained from, e.g., upgraded ACs. The proposed algorithms are tested both on synthetic data and on a number of publicly available real-world datasets. Using the upgraded features and the proposed solvers leads to a significant speed-up in the homography, multi-homography and relative pose estimation problems with better or comparable accuracy to the state-of-the-art methods.

A Minimal Solution for Image-Based Sphere Estimation

We propose a novel minimal solver for sphere fitting via its 2D central projection, i.e., a special ellipse. The input of the presented algorithm consists of contour points detected in a camera image. General ellipse fitting problems require five contour points. However, taking advantage of the isotropic spherical target, three points are enough to define the tangent cone parameters of the sphere. This yields the sought ellipse parameters. Similarly, the sphere center can be estimated from the cone if the radius is known. These proposed geometric methods are rapid, numerically stable, and easy to implement. Experimental results—on synthetic, photorealistic, and real images—showcase the superiority of the proposed solutions to the state-of-the-art methods. A real-world LiDAR-camera calibration application justifies the utility of the sphere-based approach resulting in an error below a few centimeters.

An Active Contour Model with Local Variance Force Term and Its Efficient Minimization Solver for Multiphase Image Segmentation

In this paper, we propose a general active contour model with a local variance force (LVF) term that can be applied to multiphase image segmentation problems. With the LVF, the proposed model is very effective in the segmentation of images with noise. A well-targeted minimization algorithm called ICTM-LVF is then designed to solve this model efficiently. This minimization algorithm, developed from the iterative convolution-thresholding method (ICTM), enjoys the energy-decaying property under some conditions and has highly efficient performance in the segmentation. To overcome the initialization issue of active contour models, we generalize the inhomogeneous graph Laplacian initialization method to the multiphase case and then apply it to give the initial contour of the ICTM-LVF solver. Numerical experiments are conducted on synthetic images and real images to demonstrate the capability of our initialization method, and the effectiveness of the LVF for noise robustness in the multiphase image segmentation.

A Novel Spatio-Temporal Synchronization Method of Roadside Asynchronous MMW Radar-Camera for Sensor Fusion

Roadside sensors, such as camera and millimeter-wave (MMW) radar, provide traffic information beyond the visual range of intelligent vehicles in cooperative vehicle-infrastructure systems. Unlike onboard equipment, roadside sensors are affiliated with different systems and lack synchronization in both space and time. In this paper, we propose a novel spatio-temporal synchronization method of asynchronous roadside MMW radar–camera for sensor fusion, which utilizes features of the scenario to extract lane line corner points to pre-calibrate the camera. Based on the consistent time flow rate of the separate sensors, multiple virtual detection lines are set up to match the time headway of successive vehicles and conduct objective matching to track data. Finally, a synchronization optimization model is formulated and a constrained nonlinear minimization solver is applied to tune the parameters. Measure data from Donghai Bridge in Shanghai is applied to verify the feasibility and effectiveness of the method. The results determine that there are 33 frames (33*40 ms) of temporal deviation between the camera and the radar in this case. After the synchronization, the average spatial deviation is reduced from 2.47 m to 0.42 m in the X-direction and 64.06 m to 2.34 m in the Y-direction, respectively. This study provides an economical and effective way to solve the problem of spatio-temporal synchronization of roadside sensors.

A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation

The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton–Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adopts a fracture energy-based arc-length method and an adaptive under-relaxation scheme. The arc-length method enables the simulation to overcome critical points (snap-back, snap-through instabilities) during the loading of a specimen. The use of an under-relaxation scheme stabilizes the solver by preventing the divergence due to an ill-behaving stiffness matrix. The efficiency of the proposed solver is further amplified with an adaptive mesh refinement scheme based on PHT-splines within the framework of isogeometric analysis. The numerical experiments presented in the manuscript demonstrate the efficacy of the solver. All the codes and data-sets accompanying this work will be made available on GitHub (https://github.com/rbharali/IGAFrac).

Data-Driven Priors for Robust PSSE via Gauss-Newton Unrolled Neural Networks

Renewable energy sources, elastic loads, and purposeful manipulation of meter readings challenge the monitoring and control of today’s power systems (PS). In this context, fast and robust state estimation (SE) is timely and of major importance to maintaining a comprehensive view of the system in real-time. Conventional PSSE solvers typically entail minimizing a nonlinear and nonconvex least-squares cost using e.g., the Gauss-Newton method. Those iterative solvers however, are sensitive to initialization and may converge to local minima. To overcome these hurdles, the present paper draws recent advances on image denoising to put forth a novel PSSE formulation with a data-driven regularization term capturing a deep neural network (DNN) prior. For the resultant regularized PSSE objective, a “Gauss-Newton-type” alternating minimization solver is developed first. To accommodate real-time monitoring, a novel end-to-end DNN is constructed subsequently by unrolling the proposed alternating minimization solver. The deep PSSE architecture can further account for the power network topology through a graph neural network (GNN) based prior. To further endow the physics-based DNN with robustness against bad data, an adversarial DNN training method is put forth. Numerical tests using real load data on the IEEE 118-bus benchmark system showcase the improved estimation and robustness performance of the proposed scheme compared with several state-of-the-art alternatives.

Micromagnetic Monte Carlo method with variable magnetization length based on the Landau–Lifshitz–Bloch equation for computation of large-scale thermodynamic equilibrium states

An efficient method for computing thermodynamic equilibrium states at the micromagnetic length scale is introduced using the Markov chain Monte Carlo method. Trial moves include not only rotations of vectors, but also a change in their magnetization length. The method is parameterized using the longitudinal susceptibility, reproduces the same Maxwell–Boltzmann distribution as the stochastic Landau–Lifshitz–Bloch equation, and is applicable both below and above the Curie temperature. The algorithm is fully parallel, can be executed on graphical processing units, and efficiently includes the long-range dipolar interaction. This method is generally useful for computing finite-temperature relaxation states for both uniform and non-uniform temperature profiles and can be considered as complementary to zero-temperature micromagnetic energy minimization solvers, with comparable computation time. Compared to the dynamic approach, it is shown that the micromagnetic Monte Carlo method is up to almost 20 times faster. Moreover, unlike quasizero temperature approaches that do not take into account the magnetization length distribution and stochasticity, the method is better suited for structures with unbroken symmetry around the applied field axis, granular films, and at higher temperatures and fields. In particular, applications to finite-temperature hysteresis loop modelling, chiral magnetic thin films, granular magnetic media, and artificial spin ices are discussed.

High-Dimensional Uncertainty Quantification via Tensor Regression With Rank Determination and Adaptive Sampling

Fabrication process variations can significantly influence the performance and yield of nanoscale electronic and photonic circuits. Stochastic spectral methods have achieved great success in quantifying the impact of process variations, but they suffer from the curse of dimensionality. Recently, low-rank tensor methods have been developed to mitigate this issue, but two fundamental challenges remain open: how to automatically determine the tensor rank and how to adaptively pick the informative simulation samples. This article proposes a novel tensor-regression method to address these two challenges. We use an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{q}/ \ell _{2}$ </tex-math></inline-formula> group-sparsity regularization to determine the tensor rank. The resulting optimization problem can be efficiently solved via an alternating minimization solver. We also propose a two-stage adaptive sampling method to reduce the simulation cost. Our method considers both exploration and exploitation via the estimated Voronoi cell volume and nonlinearity measurement, respectively. The proposed model is verified with synthetic and some realistic circuit benchmarks, on which our method can well capture the uncertainty caused by 19–100 random variables with only 100–600 simulation samples.

Fracture of thermo-elastic solids: Phase-field modeling and new results with an efficient monolithic solver

Thermally induced cracking occurs in many engineering problems such as drying shrinkage cracking of concrete, thermal shock induced fracture, micro cracking of two-phase composite materials etc. The computational simulation of such a fracture is complicated, but the use of phase-field models (PFMs) is promising as they can seamlessly model complex crack patterns like branching, merging, and fragmentation by treating the crack discontinuity as thin band of diffuse damage. Despite the success of phase-field models there are two major issues in previous PFMs of thermally induced fracture. Firstly, these models, which are mostly based on a PFM using a simple quadratic degradation function without any user-defined parameters, provide solutions that are sensitive to a length scale. Secondly, they are limited to brittle fracture only. As a solution, we extend the phase-field regularized cohesive zone model (PF-CZM) of Wu [JMPS, 103 (2017)], with a rational degradation function dependent on elasticity and fracture related material parameters, to thermoelastic solids. Furthermore, we present a monolithic BFGS algorithm to solve the three-field (displacements, phase-field and temperature) coupling equations altogether. Multiple thermally induced fracture problems, simulated within the framework of the finite element method, are presented with predictions in good agreement with previous findings and experiments. The proposed model is shown to give better performance in capturing the physics of thermally induced fracture: it provides length scale insensitive responses. And the monolithic solver is 4∼5 times faster than the conventional alternating minimization solver.

Minimal Solvers for Rectifying From Radially-Distorted Conjugate Translations.

This paper introduces minimal solvers that jointly solve for radial lens undistortion and affine-rectification using local features extracted from the image of coplanar translated and reflected scene texture, which is common in man-made environments. The proposed solvers accommodate different types of local features and sampling strategies, and three of the proposed variants require just one feature correspondence. State-of-the-art techniques from algebraic geometry are used to simplify the formulation of the solvers. The generated solvers are stable, small and fast. Synthetic and real-image experiments show that the proposed solvers have superior robustness to noise compared to the state of the art. The solvers are integrated with an automated system for rectifying imaged scene planes from coplanar repeated texture. Accurate rectifications on challenging imagery taken with narrow to wide field-of-view lenses demonstrate the applicability of the proposed solvers.

Efficient Relative Pose Estimation for Cameras and Generalized Cameras in Case of Known Relative Rotation Angle

We propose two minimal solvers to the problem of relative pose estimation for a camera with known relative rotation angle. In practice, such angle can be derived from the readings of a 3D gyroscope. Different from other relative pose formulations fusing a camera and a gyroscope, the use of relative rotation angle does not require extrinsic calibration between the two sensors. The first proposed solver is formulated for a calibrated regular camera and requires four-point correspondences from two views. The second solver extends the problem to a generalized camera and requires five-point correspondences. We represent the rotation part of the motion in terms of unit quaternions in order to construct polynomial equations encoding the epipolar constraints. The Grobner basis technique is then used to efficiently derive the polynomial solutions. Our first solver for regular cameras significantly improves the existing state-of-the-art solution. The second solver for generalized cameras is novel. The presented minimal solvers can be used in a hypothesize-and-test architecture such as RANSAC for reliable pose estimation. Experiments on synthetic and real datasets confirm that our algorithms are numerically stable, fast, and robust.

Minimal Solvers for Rectifying from Radially-Distorted Scales and Change of Scales

This paper introduces the first minimal solvers that jointly estimate lens distortion and affine rectification from the image of rigidly-transformed coplanar features. The solvers work on scenes without straight lines and, in general, relax strong assumptions about scene content made by the state of the art. The proposed solvers use the affine invariant that coplanar repeats have the same scale in rectified space. The solvers are separated into two groups that differ by how the equal scale invariant of rectified space is used to place constraints on the lens undistortion and rectification parameters. We demonstrate a principled approach for generating stable minimal solvers by the Gr\"obner basis method, which is accomplished by sampling feasible monomial bases to maximize numerical stability. Synthetic and real-image experiments confirm that the proposed solvers demonstrate superior robustness to noise compared to the state of the art. Accurate rectifications on imagery taken with narrow to fisheye field-of-view lenses demonstrate the wide applicability of the proposed method. The method is fully automatic.

Zeo19: A thermodynamic database for assessing zeolite stability during the corrosion of nuclear waste immobilization glasses

Stable, durable immobilization of radioactive wastes requires robust understanding of the sub-surface geochemical processes that occur in repository environments. For example, the accelerated dissolution (corrosion) of nuclear waste immobilization glasses (i.e., the so-called “Stage III” corrosion) following the precipitation of zeolitic phases is a significant issue that could result in radionuclide release. However, current uncertainties in establishing the tendency for the persistence of zeolites results in difficulties in estimating the chemical environments and state variables that favor zeolite precipitation. To assess the tendency for Stage III corrosion, we compiled a unified, internally-consistent thermodynamic database to estimate zeolite stability under conditions relevant to nuclear waste repositories (namely, p = 1 bar and T < 95 °C), i.e., for compounds including: analcime, clinoptilolite, mordenite, erionite, thomsonite, bikitaite, brewsterite, dachiardite, epistilbite, ferrierite, gonnardite, harmotome, leonhardite, paranatrolite, tetranatrolite, yugawaralite zeolite X, and zeolite P(Ca). The database, which features both existing and/or newly-derived thermodynamic properties, is integrated with a Gibbs (free) energy minimization (GEM) solver to estimate stable zeolite phase equilibria and their partitioning at equilibrium. The database offers favorable predictions of the solubility of the zeolite phases as a function of temperature. The validity of the database is ascertained by comparing newly-constructed equilibrium activity diagrams with experimental observations of zeolite formation during glass dissolution across conditions encompassing diverse solution chemistries, pH’s, and temperatures.

A phase-field damage model with micro inertia effect for the dynamic fracture of quasi-brittle solids

Being able to deal with complex crack patterns like branching, merging and fragmentation, phase-field models are promising in the computational modeling of dynamic fracture in solids. In order to account for the rate-dependent effect of quasi-brittle solids under high-rate loading, in this work we address a phase-field damage model for dynamic fracture by incorporate the micro inertia into the recently proposed unified phase-field damage theory. The macroscopic and microscopic balance equations are derived from Hamilton’s principle of least action and then solved by the alternate minimization solver. Several representative numerical examples are presented to demonstrate the capability of the proposed model to capture the failure process of brittle/quasi-brittle solids under dynamic loading scenarios. Ideal agreements are achieved against available experimental results and previous reports. In particular, the rate-dependence of brittle/quasi-brittle solids can be well captured.