### Abstract

By defining the structural entropy the von Neumann entropy of a charge distribution on a finite grid is divided into two parts. The first one, the extension entropy, is simply the logarithm of the occupation number (i.e., the number of the average grid sites occupied by the charge distribution), while the second part is the structural entropy itself. On a structural entropy versus participation ratio map the different types of localizations follow different, well characterized curves, and every distribution is represented by a vector on the map. By a structural entropy-filling factor map of any charge distributions on a finite grid (e.g., finite representation of an electron density, or a grayscale atomic force microscope (AFM) image) superstructures of different scale topologies with different decay types can be traced as well. However it is rather hard to distinguish elements of an additive superstructure, especially if the numerical parameters of the different scale patterns are necessary. In the AFM practice the background patterns are sometimes hard to compensate, and by simple structural entropy based calculations it is almost impossible to separate the superstructure of the atomic scale and the image-scale pattern. The reason is the following. Superstructures manifest on the structural entropy map as sum of the sub-structures vector, thus since none of the structures are known, only the sum of their vectors, the sub-vectors are not unique. Multiresolution or wavelet analysis (MRA) uses a system of basis functions with various time and frequency (space and spatial frequency) parameters for expanding functions. This system makes the time-frequency localization possible. Using some selected MRA resolution levels of the AFM image and carrying out the structural entropy based localization study on each of these levels will determine the decay type of the image at the length scales corresponding to the selected frequencies. This approach is promising for determining the large-scale patterns on AFM pictures.

Original language | English |
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Pages (from-to) | 1284-1288 |

Number of pages | 5 |

Journal | Progress in Electromagnetics Research Symposium |

Publication status | Published - Dec 26 2011 |

Event | Progress in Electromagnetics Research Symposium, PIERS 2011 Marrakesh - Marrakesh, Morocco Duration: Mar 20 2011 → Mar 23 2011 |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Electronic, Optical and Magnetic Materials

### Cite this

*Progress in Electromagnetics Research Symposium*, 1284-1288.