JaLCDOI 10.18926/19955
FullText URL Mem_Fac_Eng_OU_44_13.pdf
Author Kanatani, Kenichi| Sugaya Yasuyuki|
Abstract A new numerical scheme is presented for computing strict maximum likelihood (ML) of geometric fitting problems having an implicit constraint. Our approach is orthogonal projection of observations onto a parameterized surface defined by the constraint. Assuming a linearly separable nonlinear constraint, we show that a theoretically global solution can be obtained by iterative Sampson error minimization. Our approach is illustrated by ellipse fitting and fundamental matrix computation. Our method also encompasses optimal correction, computing, e.g., perpendiculars to an ellipse and triangulating stereo images. A detailed discussion is given to technical and practical issues about our approach.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2010-01
Volume volume44
Start Page 13
End Page 23
ISSN 1349-6115
language 英語
File Version publisher
NAID 120002309170
Author Kanatani, Kenichi|
Published Date 2004-10
Publication Title Pattern Analysis and Machine Intelligence
Content Type Journal Article
JaLCDOI 10.18926/46952
FullText URL mfe_38_1-2_039_059.pdf
Author Kanatani, Kenichi|
Abstract We investigate the meaning of "statistical methods" for geometric inference based on image feature points. Tracing back the origin of feature uncertainty to image processing operations, we discuss the implications of asymptotic analysis in reference to "geometric fitting" and "geometric model selection", We point out that a correspondence exists between the standard statistical analysis and the geometric inference problem. We also compare the capability of the "geometric AIC" and the "geometric MDL' in detecting degeneracy. Next, we review recent progress in geometric fitting techniques for linear constraints, describing the "FNS method", the "HEIV method", the "renormalization method", and other related techniques. Finally, we discuss the "Neyman-Scott problem" and "semiparametric models" in relation to geometric inference. We conclude that applications of statistical methods requires careful considerations about the nature of the problem in question.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2004-03
Volume volume38
Issue issue1-2
Start Page 39
End Page 59
ISSN 0475-0071
language 英語
File Version publisher
NAID 80016889442
JaLCDOI 10.18926/14123
FullText URL Mem_Fac_Eng_OU_40_1_53.pdf
Author Kanatani, Kenichi| Sugaya, Yasuyuki| Hanno Ackermann|
Abstract In order to reconstruct 3-D Euclidean shape by the Tomasi-Kanade factorization, one needs to specify an affine camera model such as orthographic, weak perspective, and paraperspective. We present a new method that does not require any such specific models. We show that a minimal requirement for an affine camera to mimic perspective projection leads to a unique camera model, which we call a symmetric affine camera, which has two free functions. We determine their values from input images by linear computation and demonstrate by experiments that an appropriate camera model is automatically selected.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2006-01
Volume volume40
Issue issue1
Start Page 53
End Page 63
ISSN 0475-0071
language 英語
File Version publisher
NAID 120002308664
JaLCDOI 10.18926/14087
FullText URL Mem_Fac_Eng_OU_41_1_73.pdf
Author Kanatani, Kenichi|
Abstract A rigorous accuracy analysis is given to various techniques for estimating parameters of geometric models from noisy data for computer vision applications. First, it is pointed out that parameter estimation for vision applications is very different in nature from traditional statistical analysis and hence a different mathematical framework is necessary in such a domain. After general theories on estimation and accuracy are given, typical existing techniques are selected, and their accuracy is evaluated up to higher order terms. This leads to a “hyperaccurate” method that outperforms existing methods.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2007-01
Volume volume41
Issue issue1
Start Page 73
End Page 92
ISSN 0475-0071
language 英語
File Version publisher
NAID 120002308410
JaLCDOI 10.18926/46970
FullText URL mfe_37_1_025_032.pdf
Author Kanazawa, Yasushi| Kanatani, Kenichi|
Abstract We present a new method for detecting point matches between two images without using any combinatorial search. Our strategy is to impose various local and non-local constraints as "soft" constraints by introducing their "confidence" measures via "mean-field approximations". The computation is a cascade of evaluating the confidence values and sorting according to them. In the end, we impose the "hard" epipolar constraint by RANSAC. We also introduce a model selection procedure to test if the image mapping can be regarded as a homography. We demonstrate the effectiveness of our method by real image examples.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2002-11
Volume volume37
Issue issue1
Start Page 25
End Page 32
ISSN 0475-0071
language 英語
File Version publisher
NAID 80015664456
JaLCDOI 10.18926/49320
FullText URL mfe_047_001_018.pdf
Author Kanatani, Kenichi|
Abstract We summarize techniques for optimal geometric estimation from noisy observations for computer vision applications. We first discuss the interpretation of optimality and point out that geometric estimation is different from the standard statistical estimation. We also describe our noise modeling and a theoretical accuracy limit called the KCR lower bound. Then, we formulate estimation techniques based on minimization of a given cost function: least squares (LS), maximum likelihood (ML), which includes reprojection error minimization as a special case, and Sampson error minimization. We describe bundle adjustment and the FNS scheme for numerically solving them and the hyperaccurate correction that improves the accuracy of ML. Next, we formulate estimation techniques not based on minimization of any cost function: iterative reweight, renormalization, and hyper-renormalization. Finally, we show numerical examples to demonstrate that hyper-renormalization has higher accuracy than ML, which has widely been regarded as the most accurate method of all. We conclude that hyper-renormalization is robust to noise and currently is the best method.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2013-01
Volume volume47
Start Page 1
End Page 18
ISSN 1349-6115
language 英語
Copyright Holders Copyright © by the authors
File Version publisher
NAID 120005232372
JaLCDOI 10.18926/14124
FullText URL Mem_Fac_Eng_OU_40_1_64.pdf
Author Kanatani, Kenichi|
Abstract This article summarizes recent advancements of the theories and techniques for 3-D reconstruction from multiple images. We start with the description of the camera imaging geometry as perspective projection in terms of homogeneous coordinates and the definition of the intrinsic and extrinsic parameters of the camera. Next, we described the epipolar geometry for two, three, and four cameras, introducing such concepts as the fundamental matrix, epipolars, epipoles, the trifocal tensor, and the quadrifocal tensor. Then, we present the self-calibration technique based on the stratified reconstruction approach, using the absolute dual quadric constraint. Finally, we give the definition of the affine camera model and a procedure for 3-D reconstruction based on it.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2006-01
Volume volume40
Issue issue1
Start Page 64
End Page 77
ISSN 0475-0071
language 英語
File Version publisher
NAID 120002308332
JaLCDOI 10.18926/19957
FullText URL Mem_Fac_Eng_OU_44_32.pdf
Author Kanatani, Kenichi| Niitsuma Hirotaka| Sugaya Yasuyuki|
Abstract 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.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2010-01
Volume volume44
Start Page 32
End Page 41
ISSN 1349-6115
language 英語
File Version publisher
NAID 120002309124
JaLCDOI 10.18926/14155
FullText URL Mem_Fac_Eng_39_1_63.pdf
Author Kanatani, Kenichi|
Abstract Geometric fitting is one of the most fundamental problems of computer vision. In [8], the author derived a theoretical accuracy bound (KCR lower bound) for geometric fitting in general and proved that maximum likelihood (ML) estimation is statistically optimal. Recently, Chernov and Lesort [3] proved a similar result, using a weaker assumption. In this paper, we compare their formulation with the author’s and describe the background of the problem. We also review recent topics including semiparametric models and discuss remaining issues.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2005-01
Volume volume39
Issue issue1
Start Page 63
End Page 70
ISSN 0475-0071
language 英語
File Version publisher
NAID 120002308366
JaLCDOI 10.18926/48127
FullText URL mfe_046_021_033.pdf
Author Kanatani, Kenichi| Niitsuma, Hirotaka|
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.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2012-01
Volume volume46
Start Page 21
End Page 33
ISSN 1349-6115
language 英語
Copyright Holders Copyright © by the authors
File Version publisher
NAID 80022451622
JaLCDOI 10.18926/48125
FullText URL mfe_046_001_009.pdf
Author Kanatani, Kenichi| Niitsuma, Hirotaka|
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.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2012-01
Volume volume46
Start Page 1
End Page 9
ISSN 1349-6115
language 英語
Copyright Holders Copyright © by the authors
File Version publisher
NAID 80022451620
JaLCDOI 10.18926/44498
FullText URL mfe_045_036_045.pdf
Author Kanatani, Kenichi| Niitsuma, Hirotaka|
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 “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.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2011-01
Volume volume45
Start Page 36
End Page 45
ISSN 1349-6115
language 英語
Copyright Holders Copyright © by the authors
File Version publisher
NAID 80021759250
Author Kanatani, Kenichi|
Published Date 2001-7
Publication Title Computer Vision
Content Type Journal Article
JaLCDOI 10.18926/47001
FullText URL mfe_36_1_059_077.pdf
Author Kanatani, Kenichi|
Abstract Contrasting "geometric fitting", for which the noise level is taken as the asymptotic variable, with "statistical inference", for which the number of observations is taken as the asymptotic variable, we give a new definition of the "geometric AIC" and the "geometric MDL" as the counterparts of Akaike's AIC and Rissanen's MDL. We discuss various theoretical and practical problems that emerge from our analysis. Finally, we show, doing experiments using synthetic and real images, that the geometric MDL does not necessarily outperform the geometric AIC and that the two criteria have very different characteristics.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2001-12
Volume volume36
Issue issue1
Start Page 59
End Page 77
ISSN 0475-0071
language 英語
File Version publisher
NAID 80012855281
JaLCDOI 10.18926/19956
FullText URL Mem_Fac_Eng_OU_44_24.pdf
Author Kanatani, Kenichi| Sugaya Yasuyuki|
Abstract We present an improved version of the MSL method of Sugaya and Kanatani for multibody motion segmentation. We replace their initial segmentation based on heuristic clustering by an analytical computation based on GPCA, fitting two 2-D affine spaces in 3-D by the Taubin method. This initial segmentation alone can segment most of the motions in natural scenes fairly correctly, and the result is successively optimized by the EM algorithm in 3-D, 5-D, and 7-D. Using simulated and real videos, we demonstrate that our method outperforms the previous MSL and other existing methods. We also illustrate its mechanism by our visualization technique.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2010-01
Volume volume44
Start Page 24
End Page 31
ISSN 1349-6115
language 英語
File Version publisher
NAID 120002309159
JaLCDOI 10.18926/19958
FullText URL Mem_Fac_Eng_OU_44_42.pdf
Author Kanatani, Kenichi| Rangrajan Prasanna|
Abstract This paper presents a new method for fitting an ellipse to a point sequence extracted from images. It is widely known that the best fit is obtained by maximum likelihood. However, it requires iterations, which may not converge in the presence of large noise. Our approach is algebraic distance minimization; no iterations are required. Exploiting the fact that the solution depends on the way the scale is normalized, we analyze the accuracy to high order error terms with the scale normalization weight unspecified and determine it so that the bias is zero up to the second order. We demonstrate by experiments that our method is superior to the Taubin method, also algebraic and known to be highly accurate.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2010-01
Volume volume44
Start Page 42
End Page 49
ISSN 1349-6115
language 英語
File Version publisher
NAID 120002309054
JaLCDOI 10.18926/44496
FullText URL mfe_045_015_026.pdf
Author Kanatani, Kenichi| Rangrajan, Prasanna| Sugaya, Yasuyuki| Niitsuma, Hirotaka|
Abstract 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.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2011-01
Volume volume45
Start Page 15
End Page 26
ISSN 1349-6115
language 英語
Copyright Holders Copyright © by the authors
File Version publisher
NAID 120002905952
JaLCDOI 10.18926/19959
FullText URL Mem_Fac_Eng_OU_44_50.pdf
Author Kanatani, Kenichi| Niitsuma Hirotaka| Rangrajan Prasanna|
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 “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.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2010-01
Volume volume44
Start Page 50
End Page 59
ISSN 1349-6115
language 英語
File Version publisher
NAID 120002308986
JaLCDOI 10.18926/14055
FullText URL Mem_Fac_Eng_OU_42_10.pdf
Author Kanatani, Kenichi|
Abstract The author introduced the "geometric AIC" and the "geometric MDL" as model selection criteria for geometric fitting problems. These correspond to Akaike’s "AIC" and Rissanen's "BIC", respectively, well known in the statistical estimation framework. Another criterion well known is Schwarz’ "BIC", but its counterpart for geometric fitting has been unknown. This paper introduces the corresponding criterion, which we call the "geometric BIC", and shows that it is of the same form as the geometric MDL. We present the underlying logical reasoning of Bayesian estimation.
Publication Title Memoirs of the Faculty of Engineering, Okayama University
Published Date 2008-01
Volume volume42
Issue issue1
Start Page 10
End Page 17
ISSN 0475-0071
language 英語
File Version publisher
NAID 120002308447