General matrix representations for B-splines

Qin, Kaihuai

doi:10.1007/s003710050206

Cited by 98 publications

(68 citation statements)

References 14 publications

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“…are the cumulative basis functions for the B-splines, derived from the matrix representation of the De Boor-Cox formula [25].B j is the j-th entry (0 based) of the cubic polynomial vector.…”

Section: A Cumulative B-splinesmentioning

confidence: 99%

Continuous-Time Trajectory Estimation for Event-based Vision Sensors

Mueggler¹,

Gallego²,

Scaramuzza³

2015

Robotics: Science and Systems XI

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Abstract-Event-based vision sensors, such as the Dynamic Vision Sensor (DVS), do not output a sequence of video frames like standard cameras, but a stream of asynchronous events. An event is triggered when a pixel detects a change of brightness in the scene. An event contains the location, sign, and precise timestamp of the change. The high dynamic range and temporal resolution of the DVS, which is in the order of micro-seconds, make this a very promising sensor for high-speed applications, such as robotics and wearable computing. However, due to the fundamentally different structure of the sensor's output, new algorithms that exploit the high temporal resolution and the asynchronous nature of the sensor are required. In this paper, we address ego-motion estimation for an event-based vision sensor using a continuous-time framework to directly integrate the information conveyed by the sensor. The DVS pose trajectory is approximated by a smooth curve in the space of rigid-body motions using cubic splines and it is optimized according to the observed events. We evaluate our method using datasets acquired from sensor-in-the-loop simulations and onboard a quadrotor performing flips. The results are compared to the ground truth, showing the good performance of the proposed technique.

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Section: A Cumulative B-splinesmentioning

confidence: 99%

Continuous-Time Trajectory Estimation for Event-based Vision Sensors

Mueggler¹,

Gallego²,

Scaramuzza³

2015

Robotics: Science and Systems XI

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“…Given a time s i ≤ s(t) < s i+1 we define u(t) = s(t) − s i . Using this time formulation and based on the matrix representation for the De Boor -Cox formula [26], we can write the matrix representation of a cumulative basisB(u) and it's time derivativesḂ(u),B(u) as:…”

Section: Cumulative Cubic B-splinesmentioning

confidence: 99%

Spline Fusion: A continuous-time representation for visual-inertial fusion with application to rolling shutter cameras

Lovegrove

Patron-Perez

Sibley

2013

Procedings of the British Machine Vision Conference 2013

140

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This paper describes a general continuous-time framework for visual-inertial simultaneous localization and mapping and calibration. We show how to use a spline parameterization that closely matches the torque-minimal motion of the sensor. Compared to traditional discrete-time solutions, the continuous-time formulation is particularly useful for solving problems with high-frame rate sensors and multiple unsynchronized devices. We demonstrate the applicability of the method for multi-sensor visual-inertial SLAM and calibration by accurately establishing the relative pose and internal parameters of multiple unsynchronized devices. We also show the advantages of the approach through evaluation and uniform treatment of both global and rolling shutter cameras within visual and visual-inertial SLAM systems.

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“…The cubic B-spline is a popular choice because it has C 2 continuity, which is important when evaluating the state for acceleration. Furthermore, it is simple to include B-splines in our formulation using the matrix representations developed by [16].…”

Section: A Hierarchical Waveletsmentioning

confidence: 99%

A hierarchical wavelet decomposition for continuous-time SLAM

Anderson

Dellaert

Barfoot

2014

2014 IEEE International Conference on Robotics and Automation (ICRA)

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Abstract-This paper proposes using hierarchical wavelets as a basis in parametric continuous-time batch estimation. The need for a continuous-time robot pose in the simultaneous localization and mapping (SLAM) problem has arisen as stateof-the-art batch SLAM algorithms attempt to handle more challenging hardware; specifically, the continuous-time framework is particularly beneficial when using high-rate sensors, multiple unsynchronized sensors, or scanning sensors, such as lidar and rolling-shutter cameras, during motion. Although the traditional discrete-time SLAM formulation can be adapted by using temporal pose interpolation, approaches using the continuous-time framework are able to generate smooth robot trajectories with less state variables. In this paper, we focus on the parametric approach using temporal basis functions to develop a finite-element representation of the continuous-time robot trajectory. While the majority of current implementations have utilized a uniformly spaced B-spline basis, we note that trajectory richness is often quite variable; in this paper, we show how a hierarchical system of wavelet basis functions can be used to increase the resolution of the solution only in the temporally local regions of the trajectory that require additional detail. We validate our approach by contrasting uniform Bsplines and wavelets in a six-dimensional pose-graph SLAM experiment, using both simulated and real data.

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General matrix representations for B-splines

Cited by 98 publications

References 14 publications

Continuous-Time Trajectory Estimation for Event-based Vision Sensors

Continuous-Time Trajectory Estimation for Event-based Vision Sensors

Spline Fusion: A continuous-time representation for visual-inertial fusion with application to rolling shutter cameras

A hierarchical wavelet decomposition for continuous-time SLAM

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