Planar projection

Abstract: Omnidirectional vision systems can provide panoramic alertness in surveillance, improve navigational capabilities, and produce panoramic images for multimedia. Catadioptric realizations of omnidirectional vision combine reflective surfaces and lenses. A particular class of them, the central panoramic systems, preserve the uniqueness of the projection viewpoint. In fact, every central projection system including the well known perspective projection on a plane falls into this category. In this paper, we provide a unifying theory for all central catadioptric systems. We show that all of them are isomorphic to projective mappings from the sphere to a plane with a projection center on the perpendicular to the plane. Subcases are the stereographic projection equivalent to parabolic projection and the central planar projection equivalent to every conventional camera. We define a duality among projections of points and lines as well as among different mappings. This unification is novel and has a significant impact on the 3D interpretation of images. We present new invariances inherent in parabolic projections and a unifying calibration scheme from one view. We describe the implied advantages of catadioptric systems and explain why images arising in central catadioptric systems contain more information than images from conventional cameras. One example is that intrinsic calibration from a single view is possible for parabolic catadioptric systems given only three lines. Another example is metric rectification using only affine information about the scene.

Abstract: We propose a method for automatically guiding patch-based image completion using mid-level structural cues. Our method first estimates planar projection parameters, softly segments the known region into planes, and discovers translational regularity within these planes. This information is then converted into soft constraints for the low-level completion algorithm by defining prior probabilities for patch offsets and transformations. Our method handles multiple planes, and in the absence of any detected planes falls back to a baseline fronto-parallel image completion algorithm. We validate our technique through extensive comparisons with state-of-the-art algorithms on a variety of scenes.

Abstract: The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analyses of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique least square projections (LSP). From an initial projection of the control points, LSP defines the positioning of their neighboring points through a numerical solution that aims at preserving a similarity relationship between the points given by a metric in mD. In order to perform the projection, a small number of distance calculations are necessary, and no repositioning of the points is required to obtain a final solution with satisfactory precision. The results show the capability of the technique to form groups of points by degree of similarity in 2D. We illustrate that capability through its application to mapping collections of textual documents from varied sources, a strategic yet difficult application. LSP is faster and more accurate than other existing high-quality methods, particularly where it was mostly tested, that is, for mapping text sets.