We consider mathematical aspects of the logarithmic spiral and its utility in turbulence modeling. We consider mathematically the set of point crossings resulting from a linear intersection through the center of a logarithmic spiral. We derive analytically the fractal dimension as a function of scale for this set of crossings. We also derive analytically the power spectrum of the thresholded fu...

We study spectral properties of the flow ẋ = 1/F (x, y), ẏ = 1/λF(x, y) on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs (F, λ) the spectrum of the flow on the torus is singular continuous with all spectral measures support...

The aim of this paper is to illustrate how we can use the ideas of the fractal geometry to describe some structural properties of myosin. The fractal dimension of the chain B backbone of myosin has been determined. It is 1.12 ± 0.012 for m<100. The spectral analysis was also used to determine the value for the spectral coefficient, –0.268 ± 0.04, and it reflects fractal aspects of the chain B o...

Two measures are commonly used to describe scale-invariant complexity in images: fractal dimension (D) and power spectrum decay rate (β). Although a relationship between these measures has been derived mathematically, empirical validation across measurements is lacking. Here, we determine the relationship between D and β for 1and 2-dimensional fractals. We find that for 1-dimensional fractals, ...

A hyperspectral image can be considered as an image cube where the third dimension is the spectral domain represented by hundreds of spectral wavelengths. A hyperspectral image pixel is actually a column vector with dimension equal to the number of spectral bands and contains valuable spectral information that can be used to detect and identify a variety of nature and man-made material. Some sp...

This work proposes the development and study of a novel technique for the generation of fractal descriptors used in texture analysis. The novel descriptors are obtained from a multiscale transform applied to the Fourier technique of fractal dimension calculus. The power spectrum of the Fourier transform of the image is plotted against the frequency in a loglog scale and a multiscale transform i...

The so-called spectral dimension is a scale-dependent number associated with both geometries and field theories that has recently attracted much attention, driven largely, though not exclusively, by investigations of causal dynamical triangulations (CDT) and Hořava gravity as possible candidates for quantum gravity. We advocate the use of the spectral dimension as a probe for the kinematics of ...

We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis–Hastings algorithm that adds an accept– reject mechanism to a Markov ch...