@article {15762,
	title = {Implicitly-weighted total least squares},
	journal = {Linear Algebra and its Applications},
	volume = {435},
	year = {2011},
	month = {2011/08/01/},
	pages = {560 - 577},
	abstract = {In a total least squares (TLS) problem, we estimate an optimal set of model parameters X , so that ( A - Δ A ) X = B - Δ B , where A is the model matrix, B is the observed data, and Δ A and Δ B are corresponding corrections. When B is a single vector, Rao (1997) and Paige and Strako{\v s} (2002) suggested formulating standard least squares problems, for which Δ A = 0 , and data least squares problems, for which Δ B = 0 , as weighted and scaled TLS problems. In this work we define an implicitly-weighted TLS formulation (ITLS) that reparameterizes these formulations to make computation easier. We derive asymptotic properties of the estimates as the number of rows in the problem approaches infinity, handling the rank-deficient case as well. We discuss the role of the ratio between the variances of errors in A and B in choosing an appropriate parameter in ITLS. We also propose methods for computing the family of solutions efficiently and for choosing the appropriate solution if the ratio of variances is unknown. We provide experimental results on the usefulness of the ITLS family of solutions.},
	keywords = {Data least squares, Errors in variables, least squares, Linear regression, total least squares, Variance estimation},
	isbn = {0024-3795},
	doi = {10.1016/j.laa.2010.06.020},
	url = {http://www.sciencedirect.com/science/article/pii/S0024379510003162},
	author = {Park,Sungwoo and O{\textquoteright}Leary,Dianne P.}
}
@conference {11965,
	title = {Statistical biases in optic flow},
	booktitle = {Computer Vision and Pattern Recognition, 1999. IEEE Computer Society Conference on.},
	volume = {1},
	year = {1999},
	month = {1999///},
	pages = {566 Vol. 1 - 566 Vol. 1},
	publisher = {IEEE},
	organization = {IEEE},
	abstract = {The computation of optical flow from image derivatives is biased in regions of non uniform gradient distributions. A least-squares or total least squares approach to computing optic flow from image derivatives even in regions of consistent flow can lead to a systematic bias dependent upon the direction of the optic flow, the distribution of the gradient directions, and the distribution of the image noise. The bias a consistent underestimation of length and a directional error. Similar results hold for various methods of computing optical flow in the spatiotemporal frequency domain. The predicted bias in the optical flow is consistent with psychophysical evidence of human judgment of the velocity of moving plaids, and provides an explanation of the Ouchi illusion. Correction of the bias requires accurate estimates of the noise distribution; the failure of the human visual system to make these corrections illustrates both the difficulty of the task and the feasibility of using this distorted optic flow or undistorted normal flow in tasks requiring higher lever processing},
	keywords = {Distributed computing, Frequency domain analysis, HUMANS, image derivatives, Image motion analysis, Image sequences, Least squares methods, Motion estimation, Optical computing, Optical distortion, optical flow, Optical noise, Ouchi illusion, perception of motion, Psychology, Spatiotemporal phenomena, statistical analysis, systematic bias, total least squares},
	isbn = {0-7695-0149-4},
	doi = {10.1109/CVPR.1999.786994},
	author = {Ferm{\"u}ller, Cornelia and Pless, R. and Aloimonos, J.}
}
@article {15823,
	title = {Tikhonov Regularization and Total Least Squares},
	journal = {SIAM Journal on Matrix Analysis and Applications},
	volume = {21},
	year = {1999},
	month = {1999///},
	pages = {185 - 194},
	abstract = {Discretizations of inverse problems lead to systems of linear equations with a highly ill-conditioned coefficient matrix, and in order to computestable solutions to these systems it is necessary to apply regularization methods. We show how Tikhonov{\textquoteright}s regularization method, which in its original formulation involves a least squares problem, can be recast in a total least squares formulation suited for problems in which both the coefficient matrix and the right-hand side are known only approximately. We analyze the regularizing properties of this method and demonstrate by a numerical example that, in certain cases with large perturbations, the new method is superior to standard regularization methods.},
	keywords = {bidiagonalization, discrete ill-posed problems, regularization, total least squares},
	doi = {10.1137/S0895479897326432},
	url = {http://link.aip.org/link/?SML/21/185/1},
	author = {Golub, Gene H. and Hansen,Per Christian and O{\textquoteright}Leary, Dianne P.}
}
@article {15796,
	title = {Regularization by Truncated Total Least Squares},
	journal = {SIAM Journal on Scientific Computing},
	volume = {18},
	year = {1997},
	month = {1997///},
	pages = {1223 - 1241},
	abstract = {The total least squares (TLS) method is a successful method for noise reduction in linear least squares problems in a number of applications. The TLS method is suited to problems in which both the coefficient matrix and the right-hand side are not precisely known. This paper focuses on the use of TLS for solving problems with very ill-conditioned coefficient matrices whose singular values decay gradually (so-called discrete ill-posed problems), where some regularization is necessary to stabilize the computed solution. We filter the solution by truncating the small singular values of the TLS matrix. We express our results in terms of the singular value decomposition (SVD) of the coefficient matrix rather than the augmented matrix. This leads to insight into the filtering properties of the truncated TLS method as compared to regularized least squares solutions. In addition, we propose and test an iterative algorithm based on Lanczos bidiagonalization for computing truncated TLS solutions.},
	keywords = {bidiagonalization, discrete ill-posed problems, regularization, total least squares},
	doi = {10.1137/S1064827594263837},
	url = {http://link.aip.org/link/?SCE/18/1223/1},
	author = {Fierro,R. D. and Golub, G. H and Hansen,P. C. and O{\textquoteright}Leary, Dianne P.}
}
