Singular analysis of stereoscopic serif

The article is devoted to the development of a methodological apparatus for evaluating the influence of geometric survey conditions on the ratio of errors in the planned and altitude components of determining the terrain points’ spatial coordinates. For this purpose, an approach based on the singular analyzing the matrix of the system of equations for intersecting lines in the basis plane is used. It is shown that singular numbers characterize the ellipse of coordinate definition errors in plane and height. They are calculated for the most typical cases of stereo surveying. It is noted that the geometric conditions of stereo photography are mainly determined by the serif angle. By interpolating the results of calculating singular numbers, the formulas characterizing the direct and inverse dependence of the ratio of the stereo-cut in plan and height on the serif angle errors’ are obtained. The author considers the practical issues of using the developed methodological apparatus to justify the solutions related to the assessment of the ratio between the errors in determining the plane coordinates and the heights of terrain points from stereo images.

НАСЛЕДОВАНИЕ СИНГУЛЯРНЫХ ВЕКТОРОВ ПРИ ПОПОЛНЕНИИ МАТРИЦЫ СТОЛБЦОМ

Let the matrix A1 be obtained from the matrix A by adding a column to it on the right. The possibility of inheritance of singular numbers and the corresponding singular vectors when passing from matrix A to matrix A1 is investigated. The singular value decompositions of the matrix A are based on the scalar and vector properties of the square symmetric matrices ATA and AAT. The article deals with the singular value decomposition of the matrix A, which has more rows than columns, and the decomposition is based on the analysis of the ATA matrix.

Estimates for the Singular Values of −Δ‎ + q when q is Complex

This chapter considers the Schrödinger operator −Δ‎ + q with q complex. In this case the operator is not self-adjoint and so the analysis of Chapter XI does not apply. It is the distribution of the singular values that is now considered, the technique used being again the localization to cubes forming a covering of Ω‎, together with the Max–Min Principle. Some results are obtained concerning the sequence class lp to which the singular numbers and eigenvalues belong.

General Principles of Integrity Checking of Digital Images and Application for Steganalysis

The new common approach for integrity checking of digital images is developed. The new features of formal parameters defining image are revealed, theoretically grounded and practically tested. The characteristics of the mutual arrangement of left and right singular vectors corresponding to the largest singular value of the image’s matrix (block of matrix) and the vector composed of singular numbers is obtained. Formal parameters are obtained using normal singular decomposition of matrix (block of matrix) which is uniquely determined. It is shown that for most blocks of original image (no matter lossy or lossless) the angle between the left (right) mentioned singular vector and vector composed of singular numbers is defined by the angle between the n-optimal vector and the vector of standard basis of the range corresponding dimension. It is shown that the determined feature brakes for the mentioned formal parameters in a non-original image. This shows the integrity violation of the image, i.e. the existence of the additional information embedded using steganography algorithms. So this can be used as a basis for development of new universal steganography methods and algorithms, and one example of the realization is proposed. The efficiency of the proposed algorithm won’t depend on the details of steganography method used for embedding. All the obtained results can be easily adapted for the digital video and audio analysis.