In this paper, we take up the long-standing problem of how to recover 3-D shapes represented by a 2-D image, such as the image on the retina of the eye, or in a video camera. Our approach is biologically grounded in a theory of how the human visual system solves this problem, focusing on shapes that are mirror symmetrical in 3-D. A 3-D mirror-symmetrical shape can be recovered from a single 2-D orthographic or perspective image by applying several a priori constraints: 3-D mirror symmetry, 3-D compactness, and planarity of contours. From the computational point of view, the application of a 3-D symmetry constraint is challenging because it requires establishing 3-D symmetry correspondence among features of a 2-D image, which itself is asymmetrical for almost all viewing directions relative to the 3-D symmetrical shape. We describe new invariants of a 3-D to 2-D projection for the case of a pair of mirror-symmetrical planar contours, and we formally state and prove the necessary and sufficient conditions for detection of this type of symmetry in a single orthographic and perspective image.
Rigorous nonlinear analysis of the physical model of Costas loop is very difficult task, so for analysis, simplified mathematical models and numerical simulation are widely used. In the work it is shown that the use of simplified mathematical models, and the application of non rigorous methods of analysis may lead to wrong conclusions concerning the operability of Costas loop.