Unknown Radial Distortion Centers in Multiple View Geometry Problems

The radial undistortion model proposed by Fitzgibbon and the radial fundamental matrix were early steps to extend classical epipolar geometry to distorted cameras. Later minimal solvers have been proposed to find relative pose and radial distortion, given point correspondences between images. However, a big drawback of all these approaches is that they require the distortion center to be exactly known. In this paper we show how the distortion center can be absorbed into a new radial fundamental matrix. This new formulation is much more practical in reality as it allows also digital zoom, cropped images and camera-lens systems where the distortion center does not exactly coincide with the image center. In particular we start from the setting where only one of the two images contains radial distortion, analyze the structure of the particular radial fundamental matrix and show that the technique also generalizes to other linear multi-view relationships like trifocal tensor and homography. For the new radial fundamental matrix we propose different estimation algorithms from 9,10 and 11 points. We show how to extract the epipoles and prove the practical applicability on several epipolar geometry image pairs with strong distortion that - to the best of our knowledge - no other existing algorithm can handle properly.

Effects and determinants of household debt in Portugal

As the amount of debt has gradually increased, particularly in recent years, Portugal is currently one of the European countries exhibiting one of the highest levels of overall indebtedness, including in both sovereign and private sectors. Indeed, this condition is the outcome of increasing levels of debt assumed not only by the government, but also by companies and families, being the later mostly due to mortgage loans and due charges. This paper focuses on the study of borrowing by Portuguese households. The research has been made in respect to the notion of debt, the consequences of recent developments in debt, among other factors. In order to analyse the factors that are most associated with debt, a study was developed using two multiple regression models, one using a longer time series and another shorter, evaluating the effect of several variables, such as consumption, savings, unemployment, inflation and interest rates, in order to check whether they could be associated with a higher level of debt.