We study the dynamics of polynomial automorphisms of$C$^{$k$}. To an algebraically stable automorphism we associate positive closed currents which are invariant under$f$, considering$f$as a rational map on$P$^{$k$}. These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.

Surface registration is widely used in machine vision and medical imaging, where 1-1 correspondences between surfaces are computed to study their variations. Surface maps are usually stored as the 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenience in data transmission and data storage, especially when a large set of surfaces are analyzed. To tackle this problem, we propose a novel representation of surface diffeomorphisms using Beltrami coefficients, which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of Beltrami coefficients on the source domain. Hence, every bijective surface map can be represented by a unique Beltrami coefficient. Conversely, given a Beltrami coefficient, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method introduced in this paper. Using this representation, 1/3 of the storage space is saved. We can further reduce the storage requirement by 90% by compressing the Beltrami coefficients using Fourier approximations. We test our algorithm on synthetic data, real human brain and hippocampal surfaces. Our results show high accuracy in the reconstructed data, while the amount of storage is greatly reduced. Our approach is compared with the Fourier compression of the coordinate functions using the same amount of data. The latter approach often shows jaggy results and cannot guarantee to preserve diffeomorphisms.

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There is an unmet medical need to identify neuroimaging biomarkers that is able to accurately diagnose and monitor Alzheimer's disease (AD) at very early stages and assess the response to AD-modifying therapies. To a certain extent, volumetric and functional magnetic resonance imaging (fMRI) studies can detect changes in structure, cerebral blood flow and blood oxygenation that are able to distinguish AD and mild cognitive impairment (MCI) subjects from normal controls. However, it has been challenging to use fully automated MRI analytic methods to identify potential AD neuroimaging biomarkers. We have thus proposed a method based on independent component analysis (ICA), for studying potential AD-related MR image features, coupled with the use of support vector machine (SVM) for classifying scans into categories of AD, MCI, and normal control (NC) subjects. The MRI data were selected from Open Access Series of Imaging Studies (OASIS) and the Alzheimer's Disease Neuroimaging Initiative (ADNI) databases. The experimental results showed that our ICA-based method can differentiate AD and MCI subjects from normal controls, although further methodological improvement in the analytic method and inclusion of additional variables may be required for optimal classification.

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