From fns to heiv: a link between two vision parameter estimation methods

Abstract

Abstract-Problems requiring accurate determination of parameters from image-based quantities arise often in computer vision. Two recent, independently developed frameworks for estimating such parameters are the FNS and HEIV schemes. Here, it is shown that FNS and a core version of HEIV are essentially equivalent, solving a common underlying equation via different means. The analysis is driven by the search for a nondegenerate form of a certain generalized eigenvalue problem, and effectively leads to a new derivation of the relevant case of the HEIV algorithm. This work may be seen as an extension of previous efforts to rationalize and interrelate a spectrum of estimators, including the renormalization method of Kanatani and the normalized eight-point method of Hartley.

Enforcing consistency constraints in uncalibrated multiple homography estimation using latent variables

“…The approximation leading from the EML cost function to the AML cost function involves, among other things, the elimination of the nuisance parameters (cf. [8,10,11,26,31,35]). …”

“…,w I ] for any iterative method operating on η is obtained by modifying the specific solution given above. The modification reflects the fact that X admits only an approximate representation as in (10). The steps of the initialisation procedure are detailed in Algorithm 1.…”

“…[8,10,11,26,31,35]). Importantly, when m, the common length of the f(z n , β)'s, surpasses the codimension r of the submanifolds of the form {z ∈ R k | f(z, β) = 0} with β representing parameters under which the data might have been generated, the inverses Σ(z n , β) −1 in the above expression for J AML must be replaced by, say, the r-truncated pseudo-inverses Σ(z n , β) + r [17,26].…”

Optimal Estimation of Homogeneous Vectors

“…A further approach is the heteroscedastic errors-in-variables (HEIV) model which was introduced by Leedan [11] and refined by Matei and Meer [7]. A recent paper [12] illustrated the link between FNS and HEIV. Consequently, HEIV shows similar properties and problems (for instance, Nestares et al report convergence problems in [13]).…”

Further Improving Geometric Fitting

“…Extending this view, Chojnacki et al [4] asserted that renormalization is an approximate method for ML and proposed a new method called FNS (fundamental numerical scheme) for directly computing ML [5]. They also pointed out that in this respect the HEIV (heteroscedastic errors-in-variables) of Leedan and Meer [18] falls in the same category [6], too. From these observations, Chojnacki et al [4] asserted superiority of the FNS and the HEIV over renormalization.…”