What is the generalform for the non-linear minimization problem?

This is a generalform that encompasses both the point–to–point metric (Ci = wiI2×2, wi being a weight), and point-to-line metric (Ci = winin T i , where ni ∈ R 2 is the versor normal to the line).

What is the ICP variant for converged robotics?

To the author’s knowledge, PLICP is the only ICP variant used in robotics localization for which such properties have been shown.

How can one obtain a solution to the nonlinear minimization problem?

Expand (18) to obtain ∑i(Mix − πi) T Ci (Mix − πi) =∑i(xTMTi CiMix + π T i Ciπi − 2π T i CiMix)Ignoring the constant terms, the new function to minimize isxT( ∑iMTi CiMi)︸︷︷︸Mx +( ∑i−2πTi CiMi)︸︷︷︸gx (19)By defining the matrix W = [ 02×2 02×2 02×2 I2×2 ] , the constraint x23 + x 2 4 = 1 can be written as xTWx = 1, and the problem becomes:min xxTMx + gTx (20)subject to xTWx = 1 (21)The solution will be derived using Lagrange’s multipliers.

What is the simplest form for the nonlinear minimization problem?

Given the vector pi = (pi0, pi1), define the following matrix Mi:Mi = [ 1 0 pi0 −pi1 0 1 pi1 pi0 ](17)Then (16) can be written in matrix form asmin x∑i(Mix − πi) T Ci (Mix − πi) (18)The following step simplifies the expression to obtain a simple quadratic form.

What is the time to perform a realignment?

One possible solution investigated is to use a quick fast global algorithm like GPM for a first realignment, and then finish the job with PLICP.

Open AccessProceedings Article10.1109/ROBOT.2008.4543181

An ICP variant using a point-to-line metric

Andrea Censi

- 19 May 2008

- pp 19-25

709

TL;DR: PLICP as discussed by the authors is an iterative closest/corresponding point (ICP) variant that uses a point-to-line metric, and an exact closed-form for minimizing such metric.

Abstract: This paper describes PLICP, an ICP (iterative closest/corresponding point) variant that uses a point-to-line metric, and an exact closed-form for minimizing such metric. The resulting algorithm has some interesting properties: it converges quadratically, and in a finite number of steps. The method is validated against vanilla ICP, IDC (iterative dual correspondences), and MBICP (Metric-Based ICP) by reproducing the experiments performed in Minguez et al. (2006). The experiments suggest that PLICP is more precise, and requires less iterations. However, it is less robust to very large initial displacement errors. The last part of the paper is devoted to purely algorithmic optimization of the correspondence search; this allows for a significant speed-up of the computation. The source code is available for download.

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Most frequently asked questions

1. What have the authors contributed in "An icp variant using a point-to-line metric" ?

This paper describes PLICP, an ICP ( Iterative Closest/Corresponding Point ) variant that uses a point-to-line metric, and an exact closed-form for minimizing such metric.. The last part of the paper is devoted to purely algorithmic optimization of the correspondence search ; this allows for a significant speed-up of the computation.. The experiments suggest that PLICP is more precise, and requires less iterations.

2. What is the way to solve the realignment problem of PLICP?

To overcome the realignment problems of PLICP, one possibility is to use a global algorithm for a quick first coarse realignment, and then use PLICP for the final convergence.

3. What are the parameters used for the naive algorithm?

For the naive algorithm, the parameters used are max |t| = 0.5m e max |θ| = 25◦, which are the actual maximum translation and rotation found in the log (they are the ‘optimal’ parameters).

4. how to find the solution in a three-dimensional space?

The three-dimensional solution (tx, ty, θ) will be found in the four-dimensional space x = [x1, x2, x3, x4] , [tx, ty, cos θ, sin θ] by imposing the constraint x 2 3 +x 2 4 = 1.

Fig. 1. Near convergence, the point-to-line metric approximates the distance to the surface better than the point-to-point metric used in vanilla ICP.

Fig. 4. Fig. 4(a) shows the naive algorithm that serially visits the points in yt−1; a bound for max |∆ϕ| can be found by equation (33). The smart algorithm does not visit the points serially. To find ji 1 , it starts at point ji−1

Fig. 5. Pseudo-C source code for the smart algorithm. This is basically a prettied-up version of the code in file icp/icp_corr_tricks.c.

Fig. 2. This figure illustrates the difference in convergence behavior of PLICP and ICP. Establishing point-to-segments correspondences induces a partition of the state space in discrete regions Qi = { q | M(q) = Ci }. The points in each Qi produce the same correspondences Ci: since the next PLICP iteration depends only on the matching information, the successor is unique for each region (bold arrows). PLICP can converge either to a stable point (q ∞ ) or be captured in a loop (from qℓ in the figure). Vanilla ICP has a very different behavior. The next step in ICP depends on the particular points matched inside the segments, so ICP allows different trajectories in every Ci area. Moreover, because the computation is stopped when the motion is smaller than a threshold, in general the “stable” solution q ∞ is never reached.

Fig. 3. The results for MBICP, IDC, ICP, are taken from [6].

Citations

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A method for registration of 3-D shapes

Paul J. Besl, +1 more

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TL;DR: In this paper, the authors describe a general-purpose representation-independent method for the accurate and computationally efficient registration of 3D shapes including free-form curves and surfaces, based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point.

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Efficient variants of the ICP algorithm

Szymon Rusinkiewicz, +1 more

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TL;DR: An implementation is demonstrated that is able to align two range images in a few tens of milliseconds, assuming a good initial guess, and has potential application to real-time 3D model acquisition and model-based tracking.

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Proceedings Article•10.1109/ROBOT.1991.132043

Object modeling by registration of multiple range images

YangQuan Chen, +1 more

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TL;DR: The authors propose an approach that works on range data directly and registers successive views with enough overlapping area to get an accurate transformation between views and performs a functional that does not require point-to-point matches.

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The Player/Stage Project: Tools for Multi-Robot and Distributed Sensor Systems

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Proceedings Article•10.1109/ICPR.2002.1047997

The Trimmed Iterative Closest Point algorithm

Dmitry Chetverikov, +3 more

- 11 Aug 2002

TL;DR: A new natural and simple, robustified extension of the popular Iterative Closest Point (ICP) algorithm is presented, called the Trimmed ICP (TrICP), based on the consistent use of the least trimmed squares (LTS) approach in all phases of the operation.

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An ICP variant using a point-to-line metric

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AI Agents for this Paper

Most frequently asked questions

1. What have the authors contributed in "An icp variant using a point-to-line metric" ?

2. What is the way to solve the realignment problem of PLICP?

3. What are the parameters used for the naive algorithm?

4. how to find the solution in a three-dimensional space?

5. What is the generalform for the non-linear minimization problem?

6. What is the ICP variant for converged robotics?

7. How can one obtain a solution to the nonlinear minimization problem?

8. What is the simplest form for the nonlinear minimization problem?

9. What is the time to perform a realignment?

Figures

Citations

Poses as Queries: Image-to-LiDAR Map Localization with Transformers

Unleash the Potential of 3D Point Cloud Modeling with A Calibrated Local Geometry-driven Distance Metric

Calibration of Laser Range Finders for mobile robot localization in ITER

Construction of level of details in garment image skeleton

IMU preintegration for 2D SLAM problems using Lie Groups

References

A method for registration of 3-D shapes

Efficient variants of the ICP algorithm

Object modeling by registration of multiple range images

The Player/Stage Project: Tools for Multi-Robot and Distributed Sensor Systems

The Trimmed Iterative Closest Point algorithm

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