Faculty of Engineering, Okayama University
Acta Medica Okayama
1349-6115
46
2012
Optimal Computation of 3-D Similarity: Gauss-Newton vs.Gauss-Helmert
21
33
EN
Kenichi
Kanatani
Hirotaka
Niitsuma
10.18926/48127
Because 3-D data are acquired using 3-D sensing such as stereo vision and laser range finders, they have inhomogeneous and anisotropic noise. This paper studies optimal computation of the similarity (rotation, translation, and scale change) of such 3-D data. We first point out that the Gauss-Newton and the Gauss-Helmert methods, regarded as different techniques, have similar structures. We then combine them to define what we call the modified Gauss-Helmert method and do stereo vision simulation to show that it is superior to either of the two in convergence performance. Finally, we show an application to real GPS geodetic data and point out that the widely used homogeneous and isotropic noise model is insufficient and that GPS geodetic data are prone to numerical problems.
No potential conflict of interest relevant to this article was reported.
Faculty of Engineering, Okayama University
Acta Medica Okayama
1349-6115
46
2012
Optimal Computation of 3-D Similarity from Space Data with Inhomogeneous Noise Distributions
1
9
EN
Kenichi
Kanatani
Hirotaka
Niitsuma
10.18926/48125
We optimally estimate the similarity (rotation, translation, and scale change) between two sets of 3-D data in the presence of inhomogeneous and anisotropic noise. Adopting the Lie algebra representation of the 3-D rotational change, we derive the Levenberg-Marquardt procedure for simultaneously optimizing the rotation, the translation, and the scale change. We test the performance of our method using simulated stereo data and real GPS geodetic sensing data. We conclude that the conventional method assuming homogeneous and isotropic noise is insufficient and that our simultaneous optimization scheme can produce an accurate solution.
No potential conflict of interest relevant to this article was reported.
Faculty of Engineering, Okayama University
Acta Medica Okayama
1349-6115
45
2011
Optimal Computation of 3-D Rotation under Inhomogeneous Anisotropic Noise
36
45
EN
Kenichi
Kanatani
Hirotaka
Niitsuma
10.18926/44498
We present a new method for optimally computing the 3-D rotation from two sets of 3-D data.
Unlike 2-D data, the noise in 3-D data is inherently inhomogeneous and anisotropic, reflecting the characteristics of the 3-D sensing used. To cope with this, Ohta and Kanatani introduced a technique called “renormalization”. Following them, we represent a 3-D rotation in terms of a quaternion and compute an exact maximum likelihood solution using the FNS of Chojnacki et
al. As an example, we consider 3-D data obtained by stereo vision and optimally compute the 3-D rotation by analyzing the noise characteristics of stereo reconstruction. We show that the widely used method is not suitable for 3-D data. We confirm that the renormalization of Ohta and
Kanatani indeed computes almost an optimal solution and that, although the difference is small, the proposed method can compute an even better solution.
No potential conflict of interest relevant to this article was reported.
Faculty of Engineering, Okayama University
Acta Medica Okayama
1349-6115
45
2011
Hyper Least Squares and Its Applications
15
26
EN
Kenichi
Kanatani
Prasanna
Rangrajan
Yasuyuki
Sugaya
Hirotaka
Niitsuma
10.18926/44496
We present a new least squares (LS) estimator, called “HyperLS”, specifically designed for parameter estimation in computer vision applications. It minimizes the algebraic distance under a special scale normalization, which is derived by rigorous error analysis in such a way that statistical bias is removed up to second order noise terms. Numerical experiments suggest that our HyperLS is far superior to the standard LS and comparable in accuracy to maximum likelihood (ML), which is known to produce highly accurate results in image applications but may fail to converge if poorly initialized. Our HyperLS is a perfect candidate for ML initialization. In addition, we discuss how
image-based inference problems have different characteristics form conventional statistical applications, with a view to serving as a bridge between mathematicians and computer engineers.
No potential conflict of interest relevant to this article was reported.
Faculty of Engineering, Okayama University
Acta Medica Okayama
1349-6115
44
2010
High Accuracy Homography Computation without Iterations
50
59
EN
Kenichi
Kanatani
10.18926/19959
We present highly accurate least-squares (LS) alternatives to the theoretically optimal maximum likelihood (ML) estimator for homographies between two images. Unlike ML, our estimators are non-iterative and yield solutions even in the presence of large noise. By rigorous error analysis, we derive a “hyperaccurate” estimator which is unbiased up to second order noise terms. Then, we introduce a computational simplification, which we call “Taubin approximation”, without incurring a loss in accuracy. We experimentally demonstrate that our estimators have accuracy surpassing the traditional LS estimator and comparable to the ML estimator.
No potential conflict of interest relevant to this article was reported.
Faculty of Engineering, Okayama University
Acta Medica Okayama
1349-6115
44
2010
Optimization without Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation
32
41
EN
Kenichi
Kanatani
10.18926/19957
We present an alternative approach to what we call the “standard optimization”, which minimizes a cost function by searching a parameter space. Instead, the input is “orthogonally projected” in the joint input space onto the manifold defined by the “consistency constraint”, which demands that any minimal subset of observations produce the same result. This approach avoids many difficulties encountered in the standard optimization. As typical examples, we apply it to line
fitting and multiview triangulation. The latter produces a new algorithm far more efficient than existing methods. We also discuss optimality of our approach.
No potential conflict of interest relevant to this article was reported.