On the Numerical Evaluation of Loop Integrals With Mellin-Barnes Representations

An improved method is presented for the numerical evaluation of multi-loop integrals in dimensional regularization. The technique is based on Mellin-Barnes representations, which have been used earlier to develop algorithms for the extraction of ultraviolet and infrared divergencies. The coefficients of these singularities and the non-singular part can be integrated numerically. However, the numerical integration often does not converge for diagrams with massive propagators and physical branch cuts. In this work, several steps are proposed which substantially improve the behavior of the numerical integrals. The efficacy of the method is demonstrated by calculating several two-loop examples, some of which have not been known before.Comment: 13 pp. LaTe

Input Information in the Approximate Calculation of Two-Dimensional Integral from Highly Oscillating Functions (Irregular Case)

Nowadays, methods for digital signal and image processing are widely used in scientific and technical areas. Current stage of research in astronomy, radiology, computed tomography, holography, and radar is characterized by broad use of digital technologies, algorithms, and methods. Correspondingly, an issue of development of new or improvement of known mathematical models arose, especially for new types of input information. There are the cases when input information about function is given on the set of traces of the function on planes, the set of traces of the function on lines, and the set of values of the function in the points. The paper is dedicated to the improvement of mathematical models of digital signal processing and imaging by the example of constructing formulas of approximate calculation of integrals of highly oscillating functions of two variables (irregular case). The feature of the proposed methods is using the input information about function as a set of traces of function on lines. The estimation of proposed method has been done for the Lipschitz class and class of differentiable functions. The proposed formula is based on the algorithm, which is also effective for a class of discontinuous functions

Camera Calibration Based on the Common Self-polar Triangle of Sphere Images

Sphere has been used for camera calibration in recent years. In this paper, a new linear calibration method is proposed by using the common self-polar triangle of sphere images. It is shown that any two of sphere images have a common self-polar triangle. Accordingly, a simple method for locating the vertices of such triangles is presented. An algorithm for recovering the vanishing line of the support plane using these vertices is developed. This allows to find out the imaged circular points, which are used to calibrate the camera. The proposed method starts from an existing theory in projective geometry and recovers five intrinsic parameters without calculating the projected circle center, which is more intuitive and simpler than the previous linear ones. Experiments with simulated data, as well as real images, show that our technique is robust and accurate