start-ver=1.4 cd-journal=joma no-vol=46 cd-vols= no-issue= article-no= start-page=21 end-page=33 dt-received= dt-revised= dt-accepted= dt-pub-year=2012 dt-pub=201201 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Optimal Computation of 3-D Similarity: Gauss-Newton vs.Gauss-Helmert en-subtitle= kn-subtitle= en-abstract= kn-abstract=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. en-copyright= kn-copyright= en-aut-name=KanataniKenichi en-aut-sei=Kanatani en-aut-mei=Kenichi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=NiitsumaHirotaka en-aut-sei=Niitsuma en-aut-mei=Hirotaka kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= affil-num=1 en-affil= kn-affil=Department of Computer Science, Okayama University affil-num=2 en-affil= kn-affil=Department of Computer Science, Okayama University END start-ver=1.4 cd-journal=joma no-vol=46 cd-vols= no-issue= article-no= start-page=1 end-page=9 dt-received= dt-revised= dt-accepted= dt-pub-year=2012 dt-pub=201201 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Optimal Computation of 3-D Similarity from Space Data with Inhomogeneous Noise Distributions en-subtitle= kn-subtitle= en-abstract= kn-abstract=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. en-copyright= kn-copyright= en-aut-name=KanataniKenichi en-aut-sei=Kanatani en-aut-mei=Kenichi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=NiitsumaHirotaka en-aut-sei=Niitsuma en-aut-mei=Hirotaka kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= affil-num=1 en-affil= kn-affil=Department of Computer Science, Okayama University affil-num=2 en-affil= kn-affil=Department of Computer Science, Okayama University END start-ver=1.4 cd-journal=joma no-vol=45 cd-vols= no-issue= article-no= start-page=36 end-page=45 dt-received= dt-revised= dt-accepted= dt-pub-year=2011 dt-pub=201101 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Optimal Computation of 3-D Rotation under Inhomogeneous Anisotropic Noise en-subtitle= kn-subtitle= en-abstract= kn-abstract=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 grenormalizationh. 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. en-copyright= kn-copyright= en-aut-name=KanataniKenichi en-aut-sei=Kanatani en-aut-mei=Kenichi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=NiitsumaHirotaka en-aut-sei=Niitsuma en-aut-mei=Hirotaka kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= affil-num=1 en-affil= kn-affil=Department of Computer Science, Okayama University affil-num=2 en-affil= kn-affil=Department of Computer Science, Okayama University END start-ver=1.4 cd-journal=joma no-vol=45 cd-vols= no-issue= article-no= start-page=15 end-page=26 dt-received= dt-revised= dt-accepted= dt-pub-year=2011 dt-pub=201101 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Hyper Least Squares and Its Applications en-subtitle= kn-subtitle= en-abstract= kn-abstract=We present a new least squares (LS) estimator, called gHyperLSh, 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. en-copyright= kn-copyright= en-aut-name=KanataniKenichi en-aut-sei=Kanatani en-aut-mei=Kenichi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=RangrajanPrasanna en-aut-sei=Rangrajan en-aut-mei=Prasanna kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= en-aut-name=SugayaYasuyuki en-aut-sei=Sugaya en-aut-mei=Yasuyuki kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=3 ORCID= en-aut-name=NiitsumaHirotaka en-aut-sei=Niitsuma en-aut-mei=Hirotaka kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=4 ORCID= affil-num=1 en-affil= kn-affil=Department of Computer Science, Okayama University affil-num=2 en-affil= kn-affil=Department of Electrical Engineering, Southern Methodist University affil-num=3 en-affil= kn-affil=Department of Computer Science and Engineering, Toyohashi University of Technology affil-num=4 en-affil= kn-affil=Department of Computer Science, Okayama University END start-ver=1.4 cd-journal=joma no-vol=44 cd-vols= no-issue= article-no= start-page=50 end-page=59 dt-received= dt-revised= dt-accepted= dt-pub-year=2010 dt-pub=201001 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=High Accuracy Homography Computation without Iterations en-subtitle= kn-subtitle= en-abstract= kn-abstract=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 ghyperaccurateh estimator which is unbiased up to second order noise terms. Then, we introduce a computational simplification, which we call gTaubin approximationh, 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. en-copyright= kn-copyright= en-aut-name=KanataniKenichi en-aut-sei=Kanatani en-aut-mei=Kenichi kn-aut-name=‹à’JŒ’ˆê kn-aut-sei=‹à’J kn-aut-mei=Œ’ˆê aut-affil-num=1 ORCID= en-aut-name= en-aut-sei= en-aut-mei= kn-aut-name=NiitsumaHirotaka kn-aut-sei=Niitsuma kn-aut-mei=Hirotaka aut-affil-num=2 ORCID= en-aut-name= en-aut-sei= en-aut-mei= kn-aut-name=RangrajanPrasanna kn-aut-sei=Rangrajan kn-aut-mei=Prasanna aut-affil-num=3 ORCID= affil-num=1 en-affil= kn-affil=Department of Computer Science Okayama University affil-num=2 en-affil= kn-affil=Department of Computer Science Okayama University affil-num=3 en-affil= kn-affil=Department of Electrical Engineering Southern Methodist University END start-ver=1.4 cd-journal=joma no-vol=44 cd-vols= no-issue= article-no= start-page=32 end-page=41 dt-received= dt-revised= dt-accepted= dt-pub-year=2010 dt-pub=201001 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Optimization without Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation en-subtitle= kn-subtitle= en-abstract= kn-abstract=We present an alternative approach to what we call the gstandard optimizationh, which minimizes a cost function by searching a parameter space. Instead, the input is gorthogonally projectedh in the joint input space onto the manifold defined by the gconsistency constrainth, 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. en-copyright= kn-copyright= en-aut-name=KanataniKenichi en-aut-sei=Kanatani en-aut-mei=Kenichi kn-aut-name=‹à’JŒ’ˆê kn-aut-sei=‹à’J kn-aut-mei=Œ’ˆê aut-affil-num=1 ORCID= en-aut-name= en-aut-sei= en-aut-mei= kn-aut-name=NiitsumaHirotaka kn-aut-sei=Niitsuma kn-aut-mei=Hirotaka aut-affil-num=2 ORCID= en-aut-name= en-aut-sei= en-aut-mei= kn-aut-name=SugayaYasuyuki kn-aut-sei=Sugaya kn-aut-mei=Yasuyuki aut-affil-num=3 ORCID= affil-num=1 en-affil= kn-affil=Department of Computer Science Okayama University affil-num=2 en-affil= kn-affil=Department of Computer Science Okayama University affil-num=3 en-affil= kn-affil=Department of Information and Computer Sciences Toyohashi University of Technology END