Characterizing the Uncertainty of the Fundamental Matrix

This paper deals with the analysis of the uncertainty of the fundamental matrix. The basic idea is to compute the fundamental matrix and its uncertainty in the same time. We shall show two different methods. The first one is a statistical approach. As in all statistical methods the precision of the results depends on the number of analyzed samples. This means that we can always improve our results if we increase the number of samples but this process is very time consuming. We propose a much simpler method which gives results which are close to the results of the statistical methods. At the end of paper we shall show some experimental results obtained with synthetic and real data

Optimal estimation of joint parameters in phase space

We address the joint estimation of the two defining parameters of a displacement operation in phase space. In a measurement scheme based on a Gaussian probe field and two homodyne detectors, it is shown that both conjugated parameters can be measured below the standard quantum limit when the probe field is entangled. We derive the most informative Cram\'er-Rao bound, providing the theoretical benchmark on the estimation and observe that our scheme is nearly optimal for a wide parameter range characterizing the probe field. We discuss the role of the entanglement as well as the relation between our measurement strategy and the generalized uncertainty relations.Comment: 8 pages, 3 figures; v2: references added and sections added to the supplemental material; v3: minor changes (published version

On the fitting of surfaces to data with covariances

Copyright © 2000 IEEEWe consider the problem of estimating parameters of a model described by an equation of special form. Specific models arise in the analysis of a wide class of computer vision problems, including conic fitting and estimation of the fundamental matrix. We assume that noisy data are accompanied by (known) covariance matrices characterizing the uncertainty of the measurements. A cost function is first obtained by considering a maximum-likelihood formulation and applying certain necessary approximations that render the problem tractable. A Newton-like iterative scheme is then generated for determining a minimizer of the cost function. Unlike alternative approaches such as Sampson's method or the renormalization technique, the new scheme has as its theoretical limit the minimizer of the cost function. Furthermore, the scheme is simply expressed, efficient, and unsurpassed as a general technique in our testing. An important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimation approaches.Wojciech Chojnacki, Michael J. Brooks, Anton van den Hengel and Darren Gawle

Incompatibility of Observables as State-Independent Bound of Uncertainty Relations

For a pair of observables, they are called "incompatible", if and only if the commutator between them does not vanish, which represents one of the key features in quantum mechanics. The question is, how can we characterize the incompatibility among three or more observables? Here we explore one possible route towards this goal through Heisenberg's uncertainty relations, which impose fundamental constraints on the measurement precisions for incompatible observables. Specifically, we quantify the incompatibility by the optimal state-independent bounds of additive variance-based uncertainty relations. In this way, the degree of incompatibility becomes an intrinsic property among the operators, but not on the quantum state. To justify our case, we focus on the incompatibility of spin systems. For an arbitrary setting of two or three linearly-independent Pauli-spin operators, the incompatibility is analytically solved, the spins are maximally incompatible if and only if they are orthogonal to each other. On the other hand, the measure of incompatibility represents a versatile tool for applications such as testing entanglement of bipartite states, and EPR-steering criteria.Comment: Comments are welcom