Error Analysis in Homography Estimation by First Order Approximation Tools: A General Technique

Chen, Pei; Suter, David

doi:10.1007/s10851-008-0113-2

Cited by 13 publications

(18 citation statements)

References 30 publications

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“…Under this condition, the covariance matrix of an estimate x is such that Λ x = Λ ± x −1 x and, moreover, Λ x = P ⊥ x Λ 0 x P ⊥ x , where P ⊥ x is the kl × kl symmetric projection matrix given by P ⊥ x = I kl − x −2 xx and Λ 0 x is a kl × kl symmetric matrix that we shall refer to as a pre-covariance matrix. An argument leading to the above assertion, together with explicit expressions for Λ 0 x in two specific cases, can be found in Appendices A and B; see also [3,28]. As P ⊥ x x = 0 and x P ⊥ x = 0 , the matrix Λ x satisfies Λ x x = 0 and x Λ x = 0 , and, in particular, is singular.…”

Section: Approximate Maximum Likelihood Cost Function and Scale Invarmentioning

confidence: 95%

“…It is readily verified that H can be expressed in terms of H as H = r(H) = (vec(H)) (3) Here, given an n × m matrix S and a positive integer r that divides m, S (r) denotes the r-wise vector transposition of S, that is, the n × (m/r) matrix obtained by performing a block transposition on S, with blocks comprising length-r column vectors [19]. 1 In MATLAB parlance,…”

Section: Rank-four Constraint Enforcementmentioning

confidence: 99%

“…3 For each i i 0 , determine two closest eigenvalues µ (1) ii0 and µ (2) ii0 of X −1 i X i0 , and set µ ii0 = (µ (1) ii0 + µ (2) ii0 )/2. 4 Take forb a left singular vector of the 3 × 6(I − 1) juxtaposition (horizontal concatenation) of the matrices µ (1) ii0 X i − X i0 and µ (2) ii0 X i − X i0 , i i i0 , corresponding to the biggest singular value.…”

Section: Initialisation Proceduresmentioning

confidence: 99%

“…To investigate the performance of the proposed method on real data, we utilised the Model House sequence from the Oxford dataset, 3 the Lady Symon and Old Classics Wing scenes from the AdelaideRMF dataset 4 [45], and additionally two traffic scenes from the Traffic Signs dataset 5 [30].…”

Section: Real Datamentioning

confidence: 99%

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Enforcing consistency constraints in uncalibrated multiple homography estimation using latent variables

Chojnacki

Szpak

Brooks

et al. 2015

Machine Vision and Applications

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Springer is a green publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights. Abstract An approach is presented for estimating a set of interdependent homography matrices linked together by latent variables. The approach allows enforcement of all underlying consistency constraints while accounting for the arbitrariness of the scale of each individual matrix. The input data is assumed to be in the form of a set of homography matrices individually estimated from image data with no regard to the consistency constraints, appended by a set of error covariances, each characterising the uncertainty of a corresponding homography matrix. A statistically motivated cost function is introduced for upgrading, via optimisation, the input data to a set of homography matrices satisfying the constraints. The function is invariant to a change of any of the individual scales of the input matrices. The proposed approach is applied to the particular problem of estimating a set of homography matrices induced by multiple planes in the 3D scene between two views. An optimisation algorithm for this problem is developed that operates on natural underlying latent variables, with the use of those variables ensuring that all consistency constraints are satisfied. Experimental results indicate that the algorithm outperforms previous schemes proposed for the same task and is fully comparable in accuracy with the 'gold standard' bundle adjustment technique, rendering the whole approach both of practical and theoretical interest. With a view to practical application, it is shown that the proposed algorithm can be incorporated into the familiar random sampling and consensus technique, so that the resulting modified scheme is capable of robust fitting of fully consistent homographies to data with outliers. Publishing in a subscription-based journal

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Section: Approximate Maximum Likelihood Cost Function and Scale Invarmentioning

confidence: 95%

Section: Rank-four Constraint Enforcementmentioning

confidence: 99%

Section: Initialisation Proceduresmentioning

confidence: 99%

Section: Real Datamentioning

confidence: 99%

See 2 more Smart Citations

Enforcing consistency constraints in uncalibrated multiple homography estimation using latent variables

Chojnacki

Szpak

Brooks

et al. 2015

Machine Vision and Applications

View full text Add to dashboard Cite

show abstract

“…Any such Λ 0 xi is meant to carry the bulk of information about the relative importance of the individual entries of x i (see [6], [7] for the raw covariance matrices of homography estimates and [8], [9] and Section VI for the raw covariance matrices of fundamental matrix estimates). Upon upgrading every Λ 0 xi to a corresponding corrected covariance matrix…”

Section: B Aml Cost Functionmentioning

confidence: 99%

Multi-projective Parameter Estimation for Sets of Homogeneous Matrices

Chojnacki

Hill

Hengel

et al. 2009

2009 Digital Image Computing: Techniques and Applications

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Abstract-A number of problems in computer vision require the estimation of a set of matrices, each of which is defined only up to an individual scale factor and represents the parameters of a separate model, under the assumption that the models are intrinsically interconnected. One example of such a set is a family of fundamental matrices sharing an infinite homography. Here an approach is presented to estimating a general set of interdependent matrices defined to within separate scales. The input data is assumed to consist of individually estimated matrices for particular models, which when considered collectively may fail to satisfy the constraints representing the inter-model relationships. Two cost functions are proposed for upgrading, via optimisation, the data of this sort to a collection of matrices satisfying the intermodel constraints. One of these functions incorporates error covariances. Each function is invariant to any change of scale for the input estimates. The proposed approach is applied to the particular problem of estimating a set of fundamental matrices of the form of the example set above. Experimental results are given which demonstrate the effectiveness of the approach.

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Rank Constraints for Homographies over Two Views: Revisiting the Rank Four Constraint

Chen

Suter

2008

Int J Comput Vis

Self Cite

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It is well known that one can collect the coefficients of five (or more) homographies between two views into a large, rank deficient matrix. In principle, this implies that one can refine the accuracy of the estimates of the homography coefficients by exploiting the rank constraint. However, the standard rank-projection approach is impractical for two different reasons: it requires many homographies to even score a modest gain; and, secondly, correlations between the errors in the coefficients will lead to poor estimates.In this paper we study these problems and provide solutions to each. Firstly, we show that the matrices of the homography coefficients can be recast into two parts, each consistent with ranks of only one. This immediately establishes the prospect of realistically (that is, with as few as only three or four homographies) exploiting the redundancies of the homographies over two views. We also tackle the remaining issue: correlated coefficients. We compare our approach with the "gold standard"; that is, non-linear bundle adjustment (initialized from the ground truth estimate-the ideal initialization). The results confirm our theory and show one can implement rank-constrained projection and come close to the gold standard in effectiveness. Indeed, our algorithm (by itself), or our algorithm further refined by a bundle adjustment stage; may be a practical algorithm: providing generally better results than the "standard" DLT (direct linear transformation) algorithm, and even better than the bundle adjustment result with the DLT result as the starting point. Our unoptimized version has roughly the same cost as bundle adjustment and yet can generally produce close to the "gold standard" estimate (as illustrated by comparison with bundle adjustment initialized from the ground truth).Independent of the merits or otherwise of our algorithm, we have illuminated why the naive approach of direct rankprojection is relatively doomed to failure. Moreover, in revealing that there are further rank constraints, not previously known; we have added to the understanding of these issues, and this may pave the way for further improvements.

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Error Analysis in Homography Estimation by First Order Approximation Tools: A General Technique

Cited by 13 publications

References 30 publications

Enforcing consistency constraints in uncalibrated multiple homography estimation using latent variables

Enforcing consistency constraints in uncalibrated multiple homography estimation using latent variables

Multi-projective Parameter Estimation for Sets of Homogeneous Matrices

Rank Constraints for Homographies over Two Views: Revisiting the Rank Four Constraint

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