Vision and Lie's approach to invariance

Gool, Luc Van; Moons, Theo; Pauwels, Eric J.; Oosterlinck, André

doi:10.1016/0262-8856(95)99715-d

Cited by 73 publications

(52 citation statements)

References 3 publications

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“…It seems that not much work has been done on matching points directly using projective transforms. Perhaps the difficulty of the problem is best illustrated by the nonexistence result for finite projective invariants shown in [5]. However, in [6], a set of infinite projective invariants has been proposed, and some of its implication have recently been investigated in the work of [7,3] on registering binary images.…”

Section: Related Workmentioning

confidence: 99%

A Direct Method for Estimating Planar Projective Transform

Chi

Yang

2011

Computer Vision – ACCV 2010

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Abstract. Estimating planar projective transform (homography) from a pair of images is a classical problem in computer vision. In this paper, we propose a novel algorithm for direct registering two point sets in R 2 using projective transform without using intensity values. In this very general context, there is no easily established correspondences that can be used to estimate the projective transform, and most of the existing techniques become either inadequate or inappropriate. While the planar projective transforms form an eight-dimensional Lie group, we show that for registering 2D point sets, the search space for the homographies can be effectively reduced to a three-dimensional space. To further improve on the running time without significantly reducing the accuracy of the registration, we propose a matching cost function constructed using local polynomial moments of the point sets and a coarse to fine approach. The resulting registration algorithm has linear time complexity with respect to the number of input points. We have validated the algorithm using points sets collected from real images. Preliminary experimental results are encouraging and they show that the proposed method is both efficient and accurate.

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Section: Related Workmentioning

confidence: 99%

A Direct Method for Estimating Planar Projective Transform

Chi

Yang

2011

Computer Vision – ACCV 2010

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“…More specifically, it is assumed that a "nuisance factor" (represented by a Lie group [14]) acts on the features according to a parameter vector p. In other words, this nuisance factor changes the feature…”

Section: Invariant Operators and Bayes Ratementioning

confidence: 99%

“…Thus, the invariant operator g(x) maps all features belonging to an orbit of the group actions into the same point. Based on this definition, [14] shows that the all invariant operators must satisfy the following partial differential equation:…”

Section: Invariant Operators and Bayes Ratementioning

confidence: 99%

“…A main concern is obviously finding operators that are only invariant to the intended parameters: everyone is familiar with the paradox of the trivial invariant, which maps any feature to a constant values, and therefore is in- * This work was supported by NSF under contract IIS-0222943. variant to any possible transformation! A systematic approach to the design of invariants was presented in [14]. By describing the effect of nuisance parameters as orbits of Lie group actions, [14] determines the number of independent invariants (which is equal to the number of independent parameters needed to fix the position of a point in measurement space, minus the orbit dimension) and shows that designing invariants is equivalent to finding the solutions of a system of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…A systematic approach to the design of invariants was presented in [14]. By describing the effect of nuisance parameters as orbits of Lie group actions, [14] determines the number of independent invariants (which is equal to the number of independent parameters needed to fix the position of a point in measurement space, minus the orbit dimension) and shows that designing invariants is equivalent to finding the solutions of a system of partial differential equations. This theory is very general and can be applied to a wide variety of geometric and photometric features.…”

Section: Introductionmentioning

confidence: 99%

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Invariant operators, small samples, and the bias-variance dilemma

Shi

Manduchi

Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004.

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Fast hierarchical noniterative registration algorithm

Alexander

1999

Int. J. Imaging Syst. Technol.

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This note describes a fast algorithm for registering pairs of images from a time sequence of images. The algorithm solves a linear regression problem based on a linearization of the image matching equation, to obtain the registration coefficients. The problem of ill‐posedness caused by differentiation of a noisy image sampled on a finite lattice is solved by means of a patch algorithm. The algorithm uses an integrated form of the linearized displacement equation. Registration is simultaneously carried out on a set of computationally fast prefilters providing three downsampled bandpass images for each input image. The filters are multilevel and permit an efficient and versatile hierarchical registration procedure. Downsampling the images before registration significantly reduces computation time. As a final step, registration is repeated using full‐size images. Results for two‐dimensional (2D) images using a six‐parameter affine registration transformation and a 12‐parameter second‐order polynomial transformation indicate the method is 2.5 times faster (12.7 s/256 × 256 image pair) than a previous iterative method (28 s/pair) and, like the previous method, is robust to noise. The method may easily be generalized to 3D image registration and more general transformations, and is well‐suited to parallel processing. © 1999 John Wiley & Sons, Inc. Int J Imaging Syst Technol 10, 242–257, 1999

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Vision and Lie's approach to invariance

Cited by 73 publications

References 3 publications

A Direct Method for Estimating Planar Projective Transform

A Direct Method for Estimating Planar Projective Transform

Invariant operators, small samples, and the bias-variance dilemma

Fast hierarchical noniterative registration algorithm

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