arXiv subjects

Muhammad Safdar

Publications and source records attributed to Muhammad Safdar.

4 recordsLinked to original sources

arXiv2014

Linearization from complex Lie point transformations

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimension $d$, with $d\leq 4$. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue in $\R^{3}$ of the linearizability criteria in $\R^2$.

math.CA↗

arXiv2014

Comparing CSI and PCA in Amalgamation with JPEG for Spectral Image Compression

Continuing our previous research on color image compression, we move towards spectral image compression. This enormous amount of data needs more space to store and more time to transmit. To manage this sheer amount of data, researchers have investigated different techniques so that image quality can be conserved and compressibility can be improved. The principle component analysis (PCA) can be employed to reduce the dimensions of spectral images to achieve high compressibility and performance. Due to processing complexity of PCA, a simple interpolation technique called cubic spline interpolation (CSI) was considered to reduce the dimensionality of spectral domain of spectral images. The CSI and PCA were employed one by one in the spectral domain and were amalgamated with the JPEG, which was employed in spatial domain. Three measures including compression rate (CR), processing time (Tp) and color difference CIEDE2000 were used for performance analysis. Test results showed that for a fixed value of compression rate, CSI based algorithm performed poor in terms of dE00, in comparison with PCA, but is still reliable because of small color difference. On the other hand it has lower complexity and is computationally much better as compared to PCA based algorithm, especially for spectral images with large size.

cs.MM↗

arXiv2011

On the symmetry solutions of two-dimensional systems not solvable by standard symmetry analysis

A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are linearizable, complex-linearizable and solvable systems. We also present the underlying concept diagrammatically that provides an analogue in $\Re^{3}$ of the geometric linearizability criteria in $\Re^2$.

math.CA↗

arXiv2011

Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An "optimal (or simplest) canonical form" of linear systems had been established to obtain the symmetry structure, namely with 5, 6, 7, 8 and 15 dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, a "reduced optimal canonical form" is obtained. This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6, 7 and 15-dimensional algebras for these systems and illustrate our results with examples.

math.CA↗