Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115472013Overviews of Optimization Techniques for Geometric Estimation118ENKenichiKanatani10.18926/49320We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115462012Optimal Computation of 3-D Similarity: Gauss-Newton vs.Gauss-Helmert2133ENKenichiKanataniHirotakaNiitsuma10.18926/48127Because 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 UniversityActa Medica Okayama1349-6115462012Calibration of Ultra-Wide Fisheye Lens Cameras by Eigenvalue Minimization1020ENKenichiKanatani10.18926/48126We present a new technique for calibrating ultra-wide fisheye lens cameras by imposing the constraint that collinear points be rectified to be collinear, parallel lines to be parallel, and orthogonal lines to be orthogonal. Exploiting the fact that line fitting reduces to an eigenvalue problem, we do a rigorous perturbation analysis to obtain a Levenberg-Marquardt procedure for the optimization. Doing experiments, we point out that spurious solutions exist if collinearity and parallelism alone are imposed. Our technique has many desirable properties. For example, no metric information is required about the reference pattern or the camera position, and separate stripe patterns can be displayed on a video screen to generate a virtual grid, eliminating the grid point extraction processing.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115462012Optimal Computation of 3-D Similarity from Space Data with Inhomogeneous Noise Distributions19ENKenichiKanataniHirotakaNiitsuma10.18926/48125We 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 UniversityActa Medica Okayama0475-00713612001Automatic Detection of Circular Objects by Ellipse Growing107116ENKenichiKanataniNaoyaOhta10.18926/47004We present a new method for automatically detecting circular objects in images: we detect an osculating circle to an elliptic arc using a Hough transform, iteratively deforming it into an ellipse, removing outlier pixels, and searching for a separate edge. The voting space is restricted to one and two dimensions for efficiency, and special weighting schemes are
introduced to enhance the accuracy. We demonstrate the effectiveness of our method using real images. Finally, we apply our method to the calibration of a turntable for 3-D object shape reconstruction.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713612001Evaluation and Selection of Models for Motion Segmentation91106ENKenichiKanataniNaoyaOhta10.18926/47003We present a theoretically optimal linear algorithm for 3-D reconstruction from point correspondences over two views. We also present a similarly constructed optimal linear algorithm for 3-D reconstruction from optical flow. We then compare the performance of the two algorithms by simulation and real-image experiments using the same data. This is the first impartial comparison ever done in the sense that the two algorithms are both optimal, extracting the information contained in the data to a maximum possible degree. We observe that the finite motion solution is always superior to the optical flow solution and conclude
that the finite motion algorithm should be used for 3-D reconstruction.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713612001Evaluation and Selection of Models for Motion Segmentation7990ENKenichiKanatani10.18926/47002We first present an improvement of Kanatani's subspace separation [8] for motion segmentation by newly introducing the affine space constraint. We point out that this improvement does not always fare well due to the effective noise it introduces. In order to judge which solution to adopt if different segmentations are obtained, we present two criteria: one is the standard F test; the other is model selection using the geometric AIC of Kanatani [7] and the geometric MDL of Matsunaga and Kanatani [13]. We test these criteria doing real image experiments.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713612001Model Selection for Geometric Fitting: Geometric Ale and Geometric MDL5977ENKenichiKanatani10.18926/47001Contrasting "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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713712002Automatic Camera Model Selection for Multibody Motion Segmentation4149ENYasuyukiSugayaKenichiKanatani10.18926/46971We study the problem of segmenting independently moving objects in a video sequence. Several algorithms exist for classifying the trajectories of the feature points into independent motions, but the performance depends on the validity of the underlying camera imaging model. In this paper, we present a scheme for automatically selecting the best model using the geometric AIC before the segmentation stage, Using real video sequences,
we confirm that the segmentation accuracy indeed improves if the segmentation is based on the selected model. We also show that the trajectory data can be compressed into low-dimensional vectors using the selected model. This is very effective in reducing the computation time for a long video sequence.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713712002Robust Image Matching under a Large Disparity2532ENYasushiKanazawaKenichiKanatani10.18926/46970We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713712002For Geometric Inference from Images, What Kind of Statistical Model Is Necessary?1523ENKenichiKanatani10.18926/46969In order to facilitate smooth communications with researchers in other fields including statistics, this paper investigates the meaning of "statistical methods" for geometric inference based on image feature points, We point out that statistical analysis does not make sense unless the underlying "statistical ensemble" is clearly defined. We trace back the origin of feature uncertainty to image processing operations for computer vision in general and discuss the implications of asymptotic analysis for performance evaluation in reference to "geometric fitting", "geometric model selection", the "geometric AIC", and the "geometric MDL". Referring to such statistical concepts as "nuisance parameters", the "Neyman-Scott problem", and "semiparametric models", we point out that simulation experiments for performance evaluation will lose meaning without carefully considering the assumptions involved and intended applications.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-0071381-22004Factorization without Factorization: Complete Recipe6171ENKenichiKanataniYasuyukiSugaya10.18926/46953The Tomasi-Kanade factorization for reconstructing the 3-D shape of the feature points tracked through a video stream is widely regarded as based on factorization of a matrix by SVD (singular value decomposition). This paper points out that the core principle is the affine camera approximation to the imaging geometry and that SVD is merely one means of numerical computation. We first describe the geometric structure of the problem and then give a complete programming scheme for 3-D reconstruction.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-0071381-22004Uncertainty Modeling and Geometric Inference3959ENKenichiKanatani10.18926/46952We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115452011Optimal Computation of 3-D Rotation under Inhomogeneous Anisotropic Noise3645ENKenichiKanataniHirotakaNiitsuma10.18926/44498We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115452011Bundle Adjustment for 3-D Reconstruction: Implementation and Evaluation2735ENKenichiKanataniYasuyukiSugaya10.18926/44497We describe in detail the algorithm of bundle adjustment for 3-D reconstruction from multiple
images based on our latest research results. The main focus of this paper is on the handling of camera rotations and the efficiency of computation and memory usage when the number of variables is very large; an appropriate consideration of this is the core of the implementation of bundle adjustment. Computing the fundamental matrix from two views and reconstructing the 3-D structure from multiple views, we evaluate the performance of our algorithm and discuses technical issues of bundle adjustment implementation.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115452011Hyper Least Squares and Its Applications1526ENKenichiKanataniPrasannaRangrajanYasuyukiSugayaHirotakaNiitsuma10.18926/44496We 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.No potential conflict of interest relevant to this article was reported.Acta Medica Okayama2001Motion segmentation by subspace separation and model selection586591ENKenichiKanatani<p>Reformulating the Costeira-Kanade algorithm as a pure mathematical theorem independent of the Tomasi-Kanade factorization, we present a robust segmentation algorithm by incorporating such techniques as dimension correction, model selection using the geometric AIC, and least-median fitting. Doing numerical simulations, we demonstrate that oar algorithm dramatically outperforms existing methods. It does not involve any parameters which need to be adjusted empirically </p>
No potential conflict of interest relevant to this article was reported.IEEE Computer SocietyActa Medica Okayama1550-61852005Further improving geometric fitting213ENKenichiKanatani<p>We give a formal definition of geometric fitting in a way that suits computer vision applications. We point out that the performance of geometric fitting should be evaluated in the limit of small noise rather than in the limit of a large number of data as recommended in the statistical literature. Taking the KCR lower bound as an optimality requirement and focusing on the linearized constraint case, we compare the accuracy of Kanatani's renormalization with maximum likelihood (ML) approaches including the FNS of Chojnacki et al. and the HEIV of Leedan and Meer. Our analysis reveals the existence of a method superior to all these. </p>No potential conflict of interest relevant to this article was reported.Acta Medica Okayama2004Uncertainty modeling and model selection for geometric inference13071319ENKenichiKanatani<p>We first 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" and point out that a correspondence exists between the standard statistical analysis and the geometric inference problem. Then, we derive the "geometric AIC" and the "geometric MDL" as counterparts of Akaike's AIC and Rissanen's MDL. We show by experiments that the two criteria have contrasting characteristics in detecting degeneracy. </p>
No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115442010High Accuracy Homography Computation without Iterations5059ENKenichiKanatani10.18926/19959We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115442010Hyperaccurate Ellipse Fitting without Iterations4249ENKenichiKanatani10.18926/19958This 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115442010Optimization without Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation3241ENKenichiKanatani10.18926/19957We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115442010Improved Multistage Learning for Multibody Motion Segmentation2431ENKenichiKanatani10.18926/19956We 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama1349-6115442010Unified Computation of Strict Maximum Likelihood for Geometric Fitting1323ENKenichiKanatani10.18926/19955A 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713912005Optimality of Maximum Likelihood Estimation for GeometricFitting and the KCR Lower Bound6370ENKenichiKanatani10.18926/14155Geometric 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 authorfs and describe the background of the problem. We also review recent topics including semiparametric models and discuss remaining issues.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00713912005Extracting Moving Objects from a Moving Camera VideoSequence5662ENYasuyukiSugayaKenichiKanatani10.18926/14153We present a new method for extracting objects moving independently of the background from a video sequence taken by a moving camera. We first extract and track feature points through the sequence and select the trajectories of background points by exploiting geometric constraints
based on the affine camera model. Then, we generate a panoramic image of the background and compare it with the individual frames. We describe our image processing and thresholding techniques.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714012006Overview of 3-D Reconstruction from Images6477ENKenichiKanatani10.18926/14124This 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714012006Uncalibrated Factorization Using a Variable Symmetric Affine Camera5363ENKenichiKanataniYasuyukiSugaya10.18926/14123In 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.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714012006Generating Dense Point Matches Using Epipolar Geometry4452ENYasuyukiSugayaKenichiKanataniYasushiKanazawa10.18926/14122Dense point matches are generated over two images by rectifying the two images to align epipolar lines horizontally, and horizontally sliding a template. To overcome inherent limitations of 2-D search, we incorporate the gnaturalness of the 3-D shapeh implied by the resulting matches.
After stating our rectification procedure, we introduce our multi-scale template matching scheme and our outlier removal technique using tentatively reconstructed 3-D shapes. Doing real image experiments, we discuss the performance of our method and remaining issues.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714112007Statistical Optimization for Geometric Fitting: TheoreticalAccuracy Bound and High Order Error Analysis7392ENKenichiKanatani10.18926/14087A 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 ghyperaccurateh method that outperforms existing methods.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714112007Experimental Evaluation of Geometric Fitting Algorithms6372ENKenichiKanataniYasuyukiSugaya10.18926/14086The convergence performance of typical numerical schemes for geometric fitting for computer vision applications is compared. First, the problem and the associated KCR lower bound are stated. Then, three well known fitting algorithms are described: FNS, HEIV, and renormalization.
To these, we add a special variant of Gauss-Newton iterations. For initialization of iterations, random choice, least squares, and Taubinfs method are tested. Numerical simulations and real image experiments and conducted for fundamental matrix computation and ellipse
fitting, which reveals different characteristics of each method.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714212008Fundamental Matrix Computation: Theory and Practice1835ENKenichiKanatani10.18926/14056We classify and review existing algorithms for computing the fundamental matrix from point correspondences and propose new effective schemes: 7-parameter Levenberg-Marquardt (LM) search, EFNS, and EFNS-based bundle adjustment. Doing experimental comparison, we show that EFNS and the 7-parameter LM search exhibit the best performance and that additional bundle adjustment does not increase the accuracy to any noticeable degree.No potential conflict of interest relevant to this article was reported.Faculty of Engineering, Okayama UniversityActa Medica Okayama0475-00714212008Geometric BIC1017ENKenichiKanatani10.18926/14055The author introduced the "geometric AIC" and the "geometric MDL" as model selection criteria for geometric fitting problems. These correspond to Akaikefs "AIC" and Rissanen's "BIC", respectively, well known in the statistical estimation framework. Another criterion well
known is Schwarzf "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.No potential conflict of interest relevant to this article was reported.