This paper presents a unified image processing and analysis framework for cortical thickness in characterizing a clinical population. The emphasis is placed on the development of data smoothing and analysis framework. The human brain cortex is a highly convoluted surface. Due to the convoluted non-Euclidean surface geometry, data smoothing and analysis on the cortex are inherently difficult. Wh...

The elastodynamic image forces on edge and screw dislocations in the presence of a planar-free surface are derived. The explicit form of the elastodynamic fields of an injected, quiescent screw dislocation are also derived. The resulting image forces are affected by retardation effects: the dislocations experience no image force for a period of time defined by the arrival and reflection at the ...

We present a fast algorithm with small memory storage to compute the spatiotemporal boundary integral equation method (ST-BIEM) particularly for elastodynamic problem. The time complexity of convolution and consumption store kernel convolved variables are originally $\mathcal O(N^2M)$ in ST-BIEM given number discretized fault elements $N$ steps $M$. Such huge costs reduced be O(N \log N)$ by ou...

In the present paper, we show that (under some minor technical assumption) Complex Gaussian multiplicative chaos defined as complex exponential of a log-correlated field can be obtained by taking limit convoluted with smoothing kernel. We consider two types chaos: e?X for log correlated X and ?=?+i?, ?,??R e?X+i?Y Y independent fields ?,??R. Our result is valid in range Psub:={?2+?2<d}?{|?|?( d...

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed relies on interpolating function around singularity terms solutions underlying homogeneous PDE, so as recast singular nearly inte...

Gaussian kernel smoothing has been widely used in 3D whole brain imaging analysis as a way to increase signal-to-noise ratio. Gaussian kernel is isotropic in Euclidian space. However, data obtained on the convoluted brain cortex fails to be isotropic in the Euclidean sense. On the curved surface, a straight line between two points is not the shortest distance so one may incorrectly assign less ...

We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R , whose solution is interpreted in the mild sense. The noise Ẇ is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f). When H > 1/2, the equation generalizes the Itô-sense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the ...

We present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold. Gaussian kernel smoothing, which weights neighboring observations according to their 3D Euclidean distance, has been widely used in 3D brain images to increase the signal-to-noise ratio. When the observations lie on a convoluted brain surface, however, it is more natural ...