Least-squares fitting of a three-dimensional ellipsoid to noisy data

Malyugina, Alexandra; Igudesman, Konstantin; Chickrin, D.E.

doi:10.12988/ams.2014.49733

Cited by 11 publications

(7 citation statements)

References 5 publications

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“…Methods using algebraic errors achieve efficient computation but lack geometric understanding [46]. Nevertheless, in [40], the algebraic method of [47] was shown to outperform other methods. Hence, we fit ellipsoids for 16 cloud points of word sequences using the algebraic method [47] subject to constraint

A + B + C = 3

.…”

Section: Methodsmentioning

confidence: 99%

Selection of features and hidden Markov model parameters for English word recognition from Leap Motion air‐writing trajectories

2023

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Air‐writing recognition is relevant in areas such as natural human–computer interaction, augmented reality, and virtual reality. A trajectory is the most natural way to represent air writing. We analyze the recognition accuracy of words written in air considering five features, namely, writing direction, curvature, trajectory, orthocenter, and ellipsoid, as well as different parameters of a hidden Markov model classifier. Experiments were performed on two representative datasets, whose sample trajectories were collected using a Leap Motion Controller from a fingertip performing air writing. Dataset contains 840 English words from 21 classes, and dataset contains 1600 English words from 40 classes. A genetic algorithm was combined with a hidden Markov model classifier to obtain the best subset of features. Combination trajectory, orthocenter, writing direction, curvature provided the best feature set, achieving recognition accuracies on datasets and of 98.81% and 83.58%, respectively.

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A + B + C = 3

.…”

Section: Methodsmentioning

confidence: 99%

Selection of features and hidden Markov model parameters for English word recognition from Leap Motion air‐writing trajectories

2023

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“…I-B. Incorporating the measurement of a magnetometer with advanced calibration techniques such as [24] and [25] is interesting, deserving further investigation in the future. Noting that it takes negligible time to complete an FTM procedure (approximately 30 ms according to [5]), it is reasonable to assume that all measurements concerning the n-th walking step are synchronized.…”

Section: A Measurementsmentioning

confidence: 99%

Exploiting User Mobility for WiFi RTT Positioning: A Geometric Approach

Han

Kim

et al. 2020

Preprint

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Due to the massive deployment of WiFi APs and its accessibility to various positioning elements, WiFi positioning is a key enabler to provide seamless and ubiquitous location services to users. There are various kinds of WiFi positioning technologies, depending on the concerned positioning element. Among them, round-trip time (RTT) measured by a fine-timing measurement protocol has received great attention recently. It provides an acceptable ranging accuracy near the service requirements in favorable environments when a line-of-sight (LOS) path exists. Otherwise, a signal is detoured along with non-LOS paths, making the resultant ranging results different from the groundtruth. The difference between the two is called an RTT bias, which is the main reason for poor positioning performance. To address it, we aim at leveraging the history of user mobility detected by a smartphone's inertial measurement units, called pedestrian dead reckoning (PDR). Specifically, PDR provides the geographic relation among adjacent locations, guiding the resultant positioning estimates' sequence not to deviate from the user trajectory. To this end, we describe their relations as multiple geometric equations, enabling us to render a novel positioning algorithm with acceptable accuracy. The algorithm is designed into two phases. First, an RTT bias of each AP can be compensated by leveraging the geometric relation mentioned above. It provides a user's relative trajectory defined on the local coordinate system of the concerned AP. Second, the user's absolute trajectory can be found by rotating every relative trajectory to be aligned, called trajectory alignment. The proposed algorithm gives a unique position when the number of detected steps and APs is at least 4 and 3 for linear mobility and 5 and 2 for arbitrary mobility. Various field experiments extensively verify the proposed algorithm's effectiveness that the average positioning error is approximately 0.369 (m) and 1.705 (m) in LOS and NLOS environments, respectively.

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“…This problem is also investigated in a variety of 2D cases in which fitting a quadratic surface to a set of data is desired [1][2][3][4][5]. Quadratic surfaces almost fit every set of 3D data [6] which have a wide range of application such as 3D reconstruction [7], pose estimation [8], restricted stereo correspondence problem [9] and object recognition analysis and sensor calibration [10][11][12][13]. One of the most important applications of ellipsoid fitting in sensor calibration is three-axis magnetic sensor calibration [10].…”

Section: Introductionmentioning

confidence: 99%

“…One of the most important applications of ellipsoid fitting in sensor calibration is three-axis magnetic sensor calibration [10]. These sensors are widely used in navigation systems to measure the intensity of the earth magnetic field [12]. Moreover, many electronic devices such as smartphones and smartwatches [14,15] use the 3D ellipsoid fitting method to calibrate their sensors.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Robust ellipsoid fitting method based on optimization of a novel nonlinear cost function in navigation systems

Mirzaei

Hosseini

2019

J Braz. Soc. Mech. Sci. Eng.

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Low-cost sensors based on micro-electro-mechanical systems (MEMS) are typically used for attitude determination in navigation systems especially magnetometer which is used for heading determination. The measured value of the MEMS magnetometer is subjected to different kinds of error such as random noise, constant bias, non-orthogonality, scale factor deviation and more importantly hard iron and soft iron effects. Therefore, in order to reach more accurate measurement, highprecision calibration is needed. One of the most common methods for calibrating MEMS magnetic sensors is least squares ellipsoid fitting. But, the common least squares ellipsoid fitting method can be inefficient for real-time applications in the presence of colored noise and outliers. In this paper, a modified ellipsoid fitting method is proposed in which a nonlinear optimization is developed to minimize a novel cost function. In the cost function of the proposed robust method, the effect of outliers and noise is considered and the standard deviation of the data is kept minimum. Finally, the efficiency of the new algorithms is demonstrated through the experimental results.

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Least-squares fitting of a three-dimensional ellipsoid to noisy data

Cited by 11 publications

References 5 publications

Selection of features and hidden Markov model parameters for English word recognition from Leap Motion air‐writing trajectories

Selection of features and hidden Markov model parameters for English word recognition from Leap Motion air‐writing trajectories

Exploiting User Mobility for WiFi RTT Positioning: A Geometric Approach

Robust ellipsoid fitting method based on optimization of a novel nonlinear cost function in navigation systems

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