Islamic geometric patterns | EPFL Graph Search

Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create a representation of an important Islamic figure according to many holy scriptures. The geometric designs in Islamic art are often built on combinations of repeated squares and circles, which may be overlapped and interlaced, as can arabesques (with which they are often combined), to form intricate and complex patterns, including a wide variety of tessellations. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, or may retreat into the background around other motifs. The complexity and variety of patterns used evolved from simple stars and lozenges in the ninth century, through a variety of 6- to 13-point patterns by the 13th century, and finally to include also 14- and 16-point stars in the sixteenth century. Geometric patterns occur in a variety of forms in Islamic art and

Analytical Methods, Correlative Microscopy and Software Tools for Quantitative Single Molecule Localization Microscopy

Arun Shivanandan

Single Molecule Localization Microscopy (SMLM) techniques, such as PhotoActivated Localization Microscopy (PALM) and STochastic Optical Reconstruction Microscopy(STORM), can get around the diffraction limited resolution of conventional fluorescent microscopy (FM). However, there are a number of possible sources of imaging errors in SMLM which can significantly impact such studies. They include labeling artifacts, a limited detection efficiency of label molecules(of about 40-60% in PALM, e.g.,) and an uncertainty in localization in the range of 20-50nm. This thesis describes a rigorous review of such sources of error. Also, the SMLM readout is different from FM, the tools used in FM may not be directly applicable to SMLM. Accurate and precise quantitative imaging with SMLM requires analytical, experimental and software tools that address such issues. We describe analytical methods that accounts for two major sources of errors in analysis of membrane protein organization with SMLM. We model limited detection efficiency as independent subsampling of the set of label molecules. We then use a theoretical property of commonly used second order properties in quantitative SMLM, such as Ripleyâs K-function, L(r) - r function and the Pair Correlation Function (PCF), to show that they are invariant to such subsampling. We derive expressions for their stochastic estimators due to subsampling, and characterize the errors. The results can be extended to co-localization analysis as well. We then describe a method that estimate the true locations of points given the observed ones in clusters. We characterize the relative Mean Squared Error of the combined approach, and find that it can significantly reduce the errors in quantification. We apply these methods on data on clustering due to photoblinking of individual fluorophores, and data with redundant labeling. We then study the theoretical properties of a function that has been proposed as an estimator of cluster size. We also describe a method to identify the cluster model from data. SMLM provides single molecule resolution images of specific molecular species. Atomic Force Microscopy ( AFM), on the other hand, provides nonspecific, high resolution spatial profile information. Correlative AFM-SMLM can provide not only validation of SMLM, but also the complementary information so obtained can be used to design innovative experiments. We describe in vitro imaging of actin filaments with an AFM-SMLM correlative tool, that could provide information about sources of imaging inhomogeneity in SMLM. The tool were also used to image live mammalian cells, and can be used to obtain nanoscale information about mechanical properties of cells and also as a tool for nanomanipulation. Co-localization is a common spatial interaction quantification method used in biological imaging. It is possible to use tools from spatial statistics to obtain better statistical power on tests of spatial interaction, compared to conventional co-localization measures. Such tools can also handle SMLM data better, since they work on point patterns rather than images. We describe an ImageJ/Fiji plugin that implements a spatial statistics framework that extends the concept of co-localization, by means of a model based on Gibbs function of interaction potentials. We describe the application of this software on both confocal microscopy data of virus-endosomes, and SMLM data on GPCR protein-clathrin.

EPFL2015

Principal component component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids

Katharine Felicity Turner

Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the persistent homology rank functions. For a point pattern X we construct a filtration of spaces by taking the union of balls of radius a centred on points in X, X-a = U-x is an element of x B(x, a). The rank function beta(k)(X) : {(a, b) is an element of R-2 : a R is then defined by beta(k)(X)(a, b) = rank (l* : H-k(X-a) -> H-k(X-b)) where l* is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square. (C) 2016 Elsevier B.V. All rights reserved.

Elsevier2016

Data coding tools for color-coded vector nanolithography

Charles Baur, Raphaël Foschia, Sébastien Grange, Andrzej Kulik, Saveen Kumar

We propose and demonstrate the ability and efficiency of using a universal file format for a nanolithography pattern. A problem faced by the physicists working in the field of nanolithography is a lack of a flexible pattern design software (possibly open–source) that could be applied in combination with a broad range of commercial scanning probe microscope (SPM) systems. The current nanolithography software packages are device–specific and not portable. Therefore, it is impossible to make a lithography pattern and share it with fellow physicists working on a networked sub-system. In this paper we describe the software designed to read and interpret a nanolithography pattern stored in a Windows Metafile (WMF) standard graphic format and next to draw it on a substrate using an SPM tip. The nanolithography parameters like height, velocity, feedback force, etc. are coded in the color of the WMF onto the RGB channels of the image establishing a distinct relation between a graphical feature (color) and the used nanolithography scheme (voltage, height, etc.). The concept enables preparation of complex patterns using any standard graphic software and aids an intuitive recognition of the mode and parameters set for a pattern. The advantages of using a WMF over other approaches and the universal scope of the software are discussed.

2004

Analytical Methods, Correlative Microscopy and Software Tools for Quantitative Single Molecule Localization Microscopy

Arun Shivanandan

EPFL2015