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.

Partidos políticos e direito de oposição a Pirro

Art. 114º da Constituição da República Portuguesa: “1. Os partidos políticos participam nos órgãos baseados no sufrágio universal e directo, de acordo com a sua representatividade eleitoral. 2. É reconhecido às minorias o direito de oposição democrática, nos termos da Constituição e da lei. 3. Os partidos políticos representados na Assembleia da República e que não façam parte do Governo gozam, designadamente, do direito de serem informados regular e directamente pelo Governo sobre o andamento dos principais assuntos de interesse público, de igual direito gozando os partidos políticos representados nas Assembleias Legislativas das regiões autónomas e em quaisquer outras assembleias designadas por eleição directa relativamente aos correspondentes executivos de que não façam parte.”. § Article 114 of the Constitution of the Portuguese Republic. "1. Political parties shall participate in organs based on direct and universal suffrage, according to their electoral representation. 2. It is recognized minorities the right to democratic opposition, under the Constitution and the law. 3. Political parties represented in Parliament and not forming part of the Government shall, in particular, the right to be informed regularly and directly by the Government on the progress of the main matters of public interest, equal rights enjoying the political parties represented in Legislative Assemblies of the autonomous regions and any other assemblies formed through direct elections with respect to the corresponding executives who are not part. ".