6DOF point cloud alignment using geometric algebra-based adaptive filtering

Al-Nuaimi, Anas; Steinbach, Eckehard; Lopes, Wilder B.; Lopes, Cássio G.

doi:10.1109/wacv.2016.7477642

Cited by 18 publications

(20 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this work, we setting G as an identity matrix. In order to obtained GA-NLMS adaptive rule, we need to calculate a multivector derivative from the equation (21). we know that the differential operator ∂ w has the algebra properties of the multivector in GA space [40].…”

Section: The Proposed Algorithm a Normalized Least Mean Square mentioning

confidence: 99%

“…In GA space, geometric algebra provides an efficient computing framework without using coordinate information, it simplifies the complexity of computation [15]- [19]. Recently, the LMS filter algorithm based on GA theory is proposed, GA-LMS algorithm has been applied for 3-D point-clouds registration problem and the 6 degrees of freedom(6-DOF) recovery transformation [20], [21]. GA-LMS algorithm has been widely used in signal processing, however, the convergence rate is slow.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Normalized Adaptive Filtering Algorithm Based on Geometric Algebra

Wang

Liang

et al. 2020

IEEE Access

View full text Add to dashboard Cite

In this paper, we extend the original Normalized Least Mean Fourth (NLMF) and Normalized Least Mean Square (NLMS) adaptive filtering algorithms into Geometric Algebra (GA) space to enable them to process multidimensional signals. We redefine the cost functions and propose the GA based NLMF and NLMS algorithms (GA-NLMF & GA-NLMS). We take full advantage of the ability of GA to represent multidimensional signals in GA space. GA-NLMS minimizes the cost function of the normalized mean square of the error signal, and remain stable as the input signal of the filter increases. GA-NLMS has fast convergence rate but higher steady-state error. The GA-NLMF algorithm minimizes the cost function of the normalized mean fourth of the error signal. Simulation results show that our proposed GA-NLMS adaptive filtering algorithm outperforms original NLMS algorithm in terms of convergence rate and steady-state error, and GA-NLMF outperforms both NLMF and GA-NLMS algorithms. GA-NLMF has faster convergence rate and lower steady state error, which is proved in the experiments.

show abstract

Section: The Proposed Algorithm a Normalized Least Mean Square mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Normalized Adaptive Filtering Algorithm Based on Geometric Algebra

Wang

Liang

et al. 2020

IEEE Access

View full text Add to dashboard Cite

show abstract

“…where D, A k , X, B k are general multivectors. The term of the cost function minimized in [11], [12] (subject to r r = rr = 1) to estimate the relative rotation between 3D PCDs. In this paper it is studied the case in which X = U k , A k = 1, B k = W k (general multivectors), so that…”

Section: B General Cost Function In Gamentioning

confidence: 99%

Geometric-Algebra Adaptive Filters

Lopes¹,

Lopes

2019

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language wellsuited for the description of geometric transformations. Also, differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows for applying the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex numbers, quaternions, etc. Relying on those characteristics (among others), a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing very good agreement for different levels of measurement noise.

show abstract

“…Based on the achieved results, the use of GAAFs in computer vision is quite promising [44,46] since the alignment of PCDs is a subtask present in many applications. However, the mean-square analysis of the GAAFs for pose estimation is rather challenging and only incipient results were achieved so far.…”

Section: Bunny Registrationmentioning

confidence: 96%

“…Given a proper selection of the step-size value, after a number of iterations, the vectors y k and r k−1 x k r k−1 are (almost) aligned. Any misalignment is due to the existence of outliers which arise during the feature matching stage (refer to [46] for techniques to minimize the influence of outliers). Once the AF has converged, the final estimate of r is applied to all the points in X (Source PCD), aligning it with Y (Target PCD).…”

Section: Ga Least-mean Squares (Ga-lms)mentioning

confidence: 99%

Geometric-algebra adaptive filters.

Lopes

View full text Add to dashboard Cite

This document introduces a new class of adaptive filters, namely GeometricAlgebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics, among others, a general leastsquares cost function is posed, from which two types of GAAFs are designed. The first one, called standard, provides a generalization of regular adaptive filters for any subalgebra of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing almost perfect agreement for different levels of measurement noise. The second type, called pose estimation, is designed to estimate rigid transformations -rotation and translation -in n-dimensional spaces. The GA-LMS performance is assessed in a 3-dimensional registration problem, in which it is able to estimate the rigid transformation that aligns two point clouds that share common parts.

show abstract

6DOF point cloud alignment using geometric algebra-based adaptive filtering

Cited by 18 publications

References 24 publications

A Normalized Adaptive Filtering Algorithm Based on Geometric Algebra

A Normalized Adaptive Filtering Algorithm Based on Geometric Algebra

Geometric-Algebra Adaptive Filters

Geometric-algebra adaptive filters.

Contact Info

Product

Resources

About