Self-assembly of the discrete Sierpinski carpet and related fractals

It is well known that the discrete Sierpinski triangle can be defined as the nonzero residues modulo 2 of Pascal's triangle, and that from this definition one can easily construct a tileset with which the discrete Sierpinski triangle self-assembles in Winfree's tile assembly model. In this paper we introduce an infinite class of discrete self-similar fractals that are defined by the residues modulo a prime p of the entries in a two-dimensional matrix obtained from a simple recursive equation. We prove that every fractal in this class self-assembles using a uniformly constructed tileset. As a special case we show that the discrete Sierpinski carpet self-assembles using a set of 30 tiles

Fractals for the Classroom. Part 1: Introduction to Fractals and Chaos

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

Illustration of vascular structures for augmented reality in liver surgery

We present methods for intraoperative visualization of vascular structures in liver surgery. The underlying concept combines conventional augmented reality approaches with illustrative rendering techniques. Our methods reduce the visual complexity of vascular structures, and accentuate spatial relations. The proposed visualization techniques are embedded in a clinical prototype application that has already been used in the operating room for preliminary evaluations. To verify the expressiveness of our illustration methods, we performed a user study with controlled lab conditions. The study revealed a clear advantage in distance assessment for the proposed illustrative approach in comparison to conventional rendering techniques

Fast unsupervised hot-spot detection in 1H-MR spectroscopic imaging data using ICA

Independent Component Analysis (ICA) is a blind source separation technique that has previously been applied to various time-varying signals. It may in particular be utilized to study 1H-MR spectroscopic imaging (MRSI) data. The work presented firstly investigates preprocessing and parameterization for ICA on simulated data to assess different strategies. We then applied ICA processing to 2D/3D brain and prostate MRSI data obtained from two healthy volunteers and 17 patients. We conducted a correlation analysis of the mixing and separating matrices resulting from ICA processing with maps obtained from metabolite quantitations in order to elucidate the relationship between quantitative and ICA results. We found that the mixing matrices corresponding to the estimated independent components highly correlate with the metabolite maps for some cases, and for others differ. We provide explanations and speculations for that and propose a scheme to utilize the knowledge for hot-spot detection. From our experience, ICA is much faster than the calculation of metabolic maps. Additionally, water and lipid contaminations are on the way removed from the data; the user needs not manually exclude spectroscopic voxels from processing or analysis. ICA results show hot spots in the data, even where quantitation-based metabolic maps are difficult to assess due to noisy data or macromolecule distortions

Contrast sensitivity in mammographic softcopy reading

In mammographic softcopy reading, assessment of contrast resolution is mainly performed with phantoms, including detection tasks with a homogeneous image background. For tasks in visual perception a processing hierarchy is assumed, where detection tasks represent the base level. The results of investigations based on detection tasks might not allow predictions on the sensitivity for recognizing low-contrast patterns in a situation with complex images. We introduce the MCS method (Mammographic Contrast Sensitivity) for determining the contrast sensitivity function (CSF) in mammograms. Gabor patterns and digits are used as visual targets. The observers have to cope with an orientation discrimination task for the Gabor patterns and an identification task for the digits. The contrast thresholds are measured by a psychophysical staircase procedure at six spatial frequencies up to 16 cycles per degree. A study with eight observers was performed to show the applicability of the MCS method. The results of the observer study with several mammographic cases show that the approach is applicable independent of the chosen images. The results for Gabor pattern targets were different from those with digits, both in overall sensitivity and in the shape of the contrast sensitivity function. Sensitivity to pattern recognition is thus not reliably predicted from the Gabor CSF, and a more complex target like a digit or a character should be preferred. The measurement of a contrast sensitivity function does not take more than 4 minutes. The results can be used to appraise the effects of viewing conditions with an aim of drawing conclusions for mammographic softcopy reading