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Estimating homographies without normalization
Abstract
Existing linear methods for estimating homographies, rely on coordinate normalization, to reduce the error in the estimated homography. Unfortunately, the estimated homography depends on the choice of the normalization. The proposed extension to the (linear) Taubin estimator is the perfect substitute for such methods, as it does not rely on coordinate normalization, and produces homographies whose error is consistent with existing methods. Also, unlike existing linear methods, the proposed Taubin estimator is theoretically unbiased, and unaffected by similarity transformations of the correspondences in the two views. In addition, it can be adapted to estimate other quantities such as trifocal tensors.
... For both methods the normalization of the measurements is a key step to improve the quality of the estimated homography [18]. However, the normalization has some disadvantages [20]: First, the normalization matrices are calculated from noisy measurements and are sensitive to outliers, and second, for a given measurement the noise affecting each point is independent of the others. However, in normalized measurements this independence is removed [2]. ...
... However, in normalized measurements this independence is removed [2]. A method is proposed in [20] to overcome this problem by avoiding the normalization and using a Taubin estimator instead, obtaining similar results as the normalized one. ...
In this paper, we investigate the influence of the spatial configuration of a number of $n \geq 4$ control points on the accuracy and robustness of space resection methods, e.g. used by a fiducial marker for pose estimation. We find robust configurations of control points by minimizing the first order perturbed solution of the DLT algorithm which is equivalent to minimizing the condition number of the data matrix. An empirical statistical evaluation is presented verifying that these optimized control point configurations not only increase the performance of the DLT homography estimation but also improve the performance of planar pose estimation methods like IPPE and EPnP, including the iterative minimization of the reprojection error which is the most accurate algorithm. We provide the characteristics of stable control point configurations for real-world noisy camera data that are practically independent on the camera pose and form certain symmetric patterns dependent on the number of points. Finally, we present a comparison of optimized configuration versus the number of control points.
... Following correspondence matching, one proceeds to obtain a refined estimate of the homography using the Taubin estimator proposed in [81]. ...
... The pixel coordinates of the corresponding points in the two images may be used to obtain a refined estimate of the warp . A repurposed version of the Taubin estimator of [81] is used to this end. ...
The image captured by an imaging system is subject to constraints imposed by the wave nature of light and the geometry of image formation. The former limits the resolving power of the imager while the latter results in a loss of size and range information. The body of work presented in this dissertation strives to overcome the aforementioned limits. The suite of techniques and apparatus ideas disclosed in the work afford imagers the unique ability to capture spatial detail lost to optical blur, while also recovering range information.
A recurring theme in the work is the notion of imaging under patterned illumination. The Moiré fringes arising from the heterodyning of the object detail and the patterned illumination, are used to improve the resolving power of the imager. The deformations in the phase of the detected illumination pattern, aid in the recovery of range information.
The work furnishes a comprehensive mathematical model for imaging under patterned illumination that accommodates blur due to the imaging/illumination optics, and the perspective foreshortening observed at macroscopic scales. The model discloses the existence of a family of active stereo arrangements that jointly support super resolution (improvement of resolving power) and scene recovery (recovery of range information).
The work also presents a new description of the theoretical basis for super resolution. The description confirms that an improvement in resolving power results from the computational engineering of the imager impulse response. The above notion is explored further, in developing a strategy for engineering the impulse response of an imager, using patterned illumination. It is also established that optical aberrations are not an impediment to super resolution.
Furthermore, the work advances the state-of-the-art in scene recovery by establishing that a broader class of sinusoidal patterns may be used to recover range information, while circumventing the extensive calibration process employed by current approaches.
The work concludes by examining an extreme example of super resolution using patterned illumination. In particular, a strategy that overcomes the severe anisotropy in the resolving power of a single-lens imager is examined. Spatial frequency analysis of the reconstructed image confirms the effectiveness of lattice illumination in engineering a computational imager with near isotropic resolving power.
... However, the accuracy of LS is very much limited. Recently, a new approach for increasing the accuracy of LS has been proposed in several †1 Okayama University †2 Southern Methodist University †3 Toyohashi University of Technology applications 1),12),22)- 24) . In this paper, we call it HyperLS and present a unified formulation and clarify various theoretical issues that have not been fully studied so far. ...
... Hence, the Taubin method 26) , which is defined for a single constraint equation, cannot be applied. However, the use of the above N T as N plays the same role of the Taubin method 26) for ellipse fitting and fundamental matrix computation, as first pointed out by Rangarajan and Papamichalis 24) . As before, we can let σ = 1 in the matrix N T in actual computation. ...
We present a general framework of a special type of least squares (LS) es-timator, which we call "HyperLS," for parametiper estimation that frequently arises in computer vision applications. It minimizes the algebraic distance un-der a special scale normalization, which is derived by a detailed error analysis in such a way that statistical bias is removed up to second order noise terms. We discuss in detail many theoretical issues involved in its derivation. By nu-merical experiments, we show that HyperLS is far superior to the standard LS and comparable in accuracy to maximum likelihood (ML), which is known to produce highly accurate results but may fail to converge if poorly initialized. We conclude that HyperLS is a perfect candidate for ML initialization.
... Since the DLT method gives a biased solution, higher accurate direct methods [16,29] have been proposed by analyzing algebraic error to remove the statistical bias. These methods are based on the minimization of independent and isotropic Gaussian noise, therefore, they assume that the point correspondences are not contaminated by outliers. ...
... The homography is computed using established techniques in computer vision [53] that rely on matching corresponding features in the camera image of a reference projector pattern, such as a grid of squares. ...
Macroscopic imagers are subject to constraints imposed by the wave nature of light and the geometry of image formation. The former limits the resolving power while the latter results in a loss of absolute size and shape information. The suite of methods outlined in this work enables macroscopic imagers the unique ability to capture unresolved spatial detail while recovering topographic information. The common thread connecting these methods is the notion of imaging under patterned illumination. The notion is advanced further to develop computational imagers with resolving power that is decoupled from the constraints imposed by the collection optics and the image sensor. These imagers additionally feature support for multiscale reconstruction.
... However, the Taubin estimator is defined only for a single constraint , such as the circle/ellipse equation and the epipolar equation, while a homography is described by multiple equations. It was only recently that Rangarajan and Papamichalis [14] revealed the existence of a " Taubin-like " estimator for homographies, but they failed to rigorously analyze the accuracy of their estimator. On the other hand, Al-Sharadqah and Chernov [1], Rangarajan and Kanatani [13], and Kanatani and Rangarajan [8] recently proposed a very accurate LS estimator for circle and ellipse fitting based on the perturbation theory of Kanatani [6]; it eliminates the bias of the fitted circle/ellipse up to second order noise terms. ...
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.
If we take two images of a planar surface from two different places, the two images are related by a mapping called a homography. Computing a homography from point correspondences over two images is one of the most fundamental processes of computer vision. This is because, among other things, the 3D positions of the planar surface we are viewing and the two cameras that took the images can be computed from the computed homography. Such applications are discussed in Chaps. 7 and 8. This chapter describes the principles and typical computational procedures for accurately computing the homography by considering the statistical properties of the noise involved in correspondence detection. As in ellipse fitting and fundamental matrix computation, the methods are classified into algebraic (least squares, iterative reweight, the Taubin method, renormalization, HyperLS, and hyper-renormalization) and geometric (FNS, geometric distance minimization, and hyperaccurate correction). We also describe the RANSAC procedure for removing wrong correspondences (outliers).
Because circular objects are projected to ellipses in images, ellipse fitting is a first step for 3-D analysis of circular objects in computer vision applications. For this reason, the study of ellipse fitting began as soon as computers came into use for image analysis in the 1970s, but it is only recently that optimal computation techniques based on the statistical properties of noise were established. These include renormalization (1993), which was then improved as FNS (2000) and HEIV (2000). Later, further improvements, called hyperaccurate correction (2006), HyperLS (2009), and hyper-renormalization (2012), were presented. Today, these are regarded as the most accurate fitting methods among all known techniques. This book describes these algorithms as well implementation details and applications to 3-D scene analysis.
We also present general mathematical theories of statistical optimization underlying all ellipse fitting algorithms, including rigorous covariance and bias analyses and the theoretical accuracy limit. The results can be directly applied to other computer vision tasks including computing fundamental matrices and homographies between images.
This book can serve not simply as a reference of ellipse fitting algorithms for researchers, but also as learning material for beginners who want to start computer vision research. The sample program codes are downloadable from the website: https://sites.google.com/a/morganclaypool.com/ellipse-fitting-for-computer-vision-implementation-and-applications.
Table of Contents: Preface / Introduction / Algebraic Fitting / Geometric Fitting / Robust Fitting / Ellipse-based 3-D Computation / Experiments and Examples / Extension and Generalization / Accuracy of Algebraic Fitting / Maximum Likelihood and Geometric Fitting / Theoretical Accuracy Limit / Answers / Bibliography / Authors' Biographies / Index
A new method for the non-iterative computation of a homography matrix is described. Rearrangement of the equations leads to a block partitioned sparse matrix, facilitating a residualization based on orthogonal matrix projections. This improves the handling of the error structure of the linear system of equations. The vanishing line is treated as the principal component in the estimation process. This estimate is more robust, since the position of the vanishing line depends only on the relative position and orientation of the camera to the observed plane, and is invariant to the structure of the points observed on the plane. A flop-count indicates that the new method is 11 times faster for four point correspondences, and converges to a factor of 5 for a large number of points. Furthermore, a new non-iterative method of treating error in both images is derived. Combining the forward H and reverse G projections in a suitable manner eliminates the systematic bias of the estimation, and the first order error: a strict bound on the error reduction is derived. This can be achieved faster than a classical DLT due to the improved numerical efficiency. Results of Monte-Carlo simulations are presented to verify the performance.
The author addresses the problem of parametric representation and estimation of complex planar curves in 2-D surfaces in 3-D, and nonplanar space curves in 3-D. Curves and surfaces can be defined either parametrically or implicitly, with the latter representation used here. A planar curve is the set of zeros of a smooth function of two variables x - y , a surface is the set of zeros of a smooth function of three variables x - y - z , and a space curve is the intersection of two surfaces, which are the set of zeros of two linearly independent smooth functions of three variables x - y - z For example, the surface of a complex object in 3-D can be represented as a subset of a single implicit surface, with similar results for planar and space curves. It is shown how this unified representation can be used for object recognition, object position estimation, and segmentation of objects into meaningful subobjects, that is, the detection of `interest regions' that are more complex than high curvature regions and, hence, more useful as features for object recognition
We describe a theoretically optimal algorithm for computing the homography between two images in relation to image mosaicing applications. First, we derive a theoretical accuracy bound based on a mathematical model of image noise and do simulation to confirm that our renormalization technique effectively attains that bound; our algorithm is optimal in that sense. Then, we apply our technique to mosaicing of images with small overlaps. By using real images, we show how our algorithm reduces the instability of the image mapping. 1. Introduction A homography is a mapping that typically occurs between two perspective images of a planar surface in the scene. Computing homography plays an essential role in aerial image registration and analysis of runway and road images viewed from airplanes and vehicles. Since far-away scenes can effectively be regarded as planar surfaces, we can integrate multiple images into one continuous image by computing homographies; this process is known as image m...
Let and be the projections of the same 3D object point in two different images (written in homogeneous coordinates). If all 3D points are restricted to lie on a plane, then the equation holds for all point correspondences , i=1,…,N (≃ denoting projective equivalence) with a 3×3-matrix A describing a mapping between two images of plane. The problem of estimating this mapping is known as homography estimation and constitutes a common problem in two-view motion analysis. In this paper, we will derive a new fast algorithm for homography estimation that takes image error models into account in order to improve estimation quality. In comparison to the well-known Least Squares (LS) estimation, the application of the Total Least Squares (TLS) method and a prior equilibration (which essentially consists in adjusting the error metric) leads to a considerable improvement in estimation quality. Starting out from the LS method, our approach is developed in several steps and results of each step are given, demonstrating the improvement achieved at each step. At the end, the outlier sensitivity is examined with an example for images with model violations (3D points not lying on the plane).
A rigorous accuracy analysis is given to var- ious techniques for estimating parameters of geometric models from noisy data. First, it is pointed out that parameter estimation for vision applications is very dif- ferent in nature from traditional statistical analysis and hence a different mathematical framework is necessary. After a general framework is formulated, typical numer- ical techniques are selected, and their accuracy is evalu- ated up to high order terms. As a byproduct, our analy- sis leads to a "hyperaccurate" method that outperforms existing methods.
This paper shows how to analytically calculate the statistical properties of the errors in estimated parameters. The basic tools to achieve this aim include first order approximation/perturbation techniques, such as matrix perturbation theory and Taylor Series. This analysis applies for a general class of parameter estimation problems that can be abstracted as a linear (or linearized) homogeneous equation.
Of course there may be many reasons why one might which to have such estimates. Here, we concentrate on the situation where one might use the estimated parameters to carry out some further statistical fitting or (optimal) refinement. In order to make the problem concrete, we take homography estimation as a specific problem. In particular, we show how the derived statistical errors in the homography coefficients, allow improved approaches to refining these coefficients through subspace constrained homography estimation (Chen and Suter in Int. J. Comput. Vis. 2008).
Indeed, having derived the statistical properties of the errors in the homography coefficients, before subspace constrained refinement, we do two things: we verify the correctness through statistical simulations but we also show how to use the knowledge of the errors to improve the subspace based refinement stage. Comparison with the straightforward subspace refinement approach (without taking into account the statistical properties of the homography coefficients) shows that our statistical characterization of these errors is both correct and useful.
The fundamental matrix is a basic tool in the analysis of scenes
taken with two uncalibrated cameras, and the eight-point algorithm is a
frequently cited method for computing the fundamental matrix from a set
of eight or more point matches. It has the advantage of simplicity of
implementation. The prevailing view is, however, that it is extremely
susceptible to noise and hence virtually useless for most purposes. This
paper challenges that view, by showing that by preceding the algorithm
with a very simple normalization (translation and scaling) of the
coordinates of the matched points, results are obtained comparable with
the best iterative algorithms. This improved performance is justified by
theory and verified by extensive experiments on real images