The divergence of the BFGS and Gauss Newton methods

Mascarenhas, Walter F.

doi:10.1007/s10107-013-0720-6

Cited by 26 publications

(17 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Moore-Penrose Quasi-Newton methods attempt to build an approximation of the Hessian matrix (or its inverse) that incorporates second order information by incorporating first order information as the optimisation proceeds. The Broyden-Fletcher-Goldfarb-Shanno Algorithm (BFGS) [49,126,136,144] is one of the most famous quasi-Newton algorithms for unconstrained optimization. Moving away from deterministic algorithms, intelligence-oriented algorithms (Genetic algorithms, Swarm algorithms) [97,173,179,180] with their simplicity are another way to search the solution of extreme problems with many local minima.…”

Section: Least Squares Methods (Lsm)mentioning

confidence: 99%

“…It automatically selects the most efficient algorithm for the computed mathematical problem. Matlab uses several algorithms depending on the type of problem to be solved: interior reflective Newton [17][18] [ 49,126,136,144] method, trust-region-dogleg, trust-region-reflective, Levenberg-Marquardt, simplex, BFGS, MiniMax, and so on.…”

Section: Least Squares Methods (Lsm)mentioning

confidence: 99%

See 1 more Smart Citation

Modelling and traceability for computationally-intensive precision engineering and metrology

et al. 2018

View full text Add to dashboard Cite

In contrast to measurements of the dimensions of machined parts realized by machine tools and characterized by CMMs, software results are not fully traceable and certified. Indeed, a computer is not a perfect machine and binary encoding of real numbers leads to rounding of successive intermediate calculations that may lead to globally false results. This is the case for poor implementations and poorly conditioned algorithms. Therefore, accurate geometric modelling and implementations will be detailed. Based on the works of National Metrology Institutes, the problem of software traceability will also be discussed. Some prospects for this complex task will finally be suggested.

show abstract

Section: Least Squares Methods (Lsm)mentioning

confidence: 99%

Section: Least Squares Methods (Lsm)mentioning

confidence: 99%

Modelling and traceability for computationally-intensive precision engineering and metrology

et al. 2018

View full text Add to dashboard Cite

show abstract

“…When coupled to our PnL algorithm, these two optimizations are referred as VPnL GN. Because of the non-linear nature of the functions f and g, the Gauss-Newton algorithm does not guarantee convergence to the global minimum with a random initialization [20]. However, convergence to optimal parameters can be achieved with an initialization close to the global minimum.…”

Section: Gauss-newton Optimization For the Rotation And The Translmentioning

confidence: 99%

Camera Pose Estimation Based on PnL With a Known Vertical Direction

Lecrosnier

Boutteau

Vasseur

et al. 2019

IEEE Robot. Autom. Lett.

View full text Add to dashboard Cite

In this paper, we address the problem of camera pose estimation using 2D and 3D line features, also known as PnL (Perspective-n-Line) with a known vertical direction. The minimal number of line correspondences required to estimate the complete camera pose is 3 (P3L) in the general case, yielding to a minimum of 8 possible solutions. Prior knowledge of the vertical direction, such as provided by common sensors (e.g. Inertial Measurement Unit, or IMU), reduces the problem to a 4 Degree of Freedom (DoF) problem and outputs a single solution. We benefit this fact to decouple the remaining rotation estimation and the translation estimation and we present a twofold contribution: (1) we present a linear formulation of the PnL problem in Plücker lines coordinates with a known vertical direction, including a Gauss-Newton-based orientation and location refinement to compensate IMU sensor noise. (2) we propose a new efficient RANdom SAmple Consensus (RANSAC) scheme for both feature pairing and outliers rejection based solely on rotation estimation from 2 line pairs. This greatly diminishes the computational cost compared to a RANSAC3 or RANSAC4 scheme. We evaluate our algorithms on synthetic data and on our own real dataset. Experimental results show state of the art results in term of accuracy and runtime, when facing 2D noise, 3D noise and vertical direction sensor noise. Index Terms-Computer Vision for Other Robotic Applications, Sensor Fusion, Localization I. INTRODUCTION Camera pose estimation consists in determining the position and the orientation of a camera with respect to a reference frame. This process requires known correspondences between real world features and their projection onto the image plane. When these features are points, we refer to the well-studied Perspective-n-Point (PnP) problem [1] [2] [3] [4] [5]. In the case of line features, we are facing the challenging, more recent and less studied Perspective-n-Line (PnL) problem. A recent review of the latter methods is presented in [6]. Once the 2D and 3D lines extraction process is completed, feature pairing is not a simple task: lines lack effective descriptors, and descriptor-based pose estimation methods are computationally more expensive.

show abstract

“…Then the optimal solution x a * , which naturally coincides with the actual location x a , can be achieved by using the iterative Gauss-Netwon method [26] with an arbitrary initial point…”

Section: Problem 1 (Tx Location Optimzation)mentioning

confidence: 99%

“…where (N r ≥ 4) to detect the TV-SV system clock difference, according to the solution of (23) in [26].…”

Section: Problem 1 (Tx Location Optimzation)mentioning

confidence: 99%

Multi-Point Vehicular Positioning via Millimeter-Wave Transmissions

Zhang

Wang

et al. 2019

2019 34th International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC)

View full text Add to dashboard Cite

Multi-point detection of the full-scale environment is an important issue in autonomous driving. The state-of-theart positioning technologies (such as RADAR and LIDAR) are incapable of real-time detection without line-of-sight. To address this issue, this paper presents a novel multi-point vehicular positioning technology via millimeter-wave (mmWave) transmission that exploits multi-path reflection from a target vehicle (TV) to a sensing vehicle (SV), which enables the SV to fast capture both the shape and location information of the TV in non-line-of-sight (NLoS) under the assistance of multi-path reflections. A phase-difference-of-arrival (PDoA) based hyperbolic positioning algorithm is designed to achieve the synchronization between the TV and SV. The stepped-frequency-continuous-wave (SFCW) is utilized as signals for multi-point detection of the TVs. Transceiver separation enables our approach to work in NLoS conditions and achieve much lower latency compared with conventional positioning techniques.

show abstract

The divergence of the BFGS and Gauss Newton methods

Cited by 26 publications

References 19 publications

Modelling and traceability for computationally-intensive precision engineering and metrology

Modelling and traceability for computationally-intensive precision engineering and metrology

Camera Pose Estimation Based on PnL With a Known Vertical Direction

Multi-Point Vehicular Positioning via Millimeter-Wave Transmissions

Contact Info

Product

Resources

About