On Geometric Alignment in Low Doubling Dimension

In real-world, many problems can be formulated as the alignment between two geometric patterns. Previously, a great amount of research focus on the alignment of 2D or 3D patterns, especially in the field of computer vision. Recently, the alignment of geometric patterns in high dimension finds several novel applications, and has attracted more and more attentions. However, the research is still rather limited in terms of algorithms. To the best of our knowledge, most existing approaches for high dimensional alignment are just simple extensions of their counterparts for 2D and 3D cases, and often suffer from the issues such as high complexities. In this paper, we propose an effective framework to compress the high dimensional geometric patterns and approximately preserve the alignment quality. As a consequence, existing alignment approach can be applied to the compressed geometric patterns and thus the time complexity is significantly reduced. Our idea is inspired by the observation that high dimensional data often has a low intrinsic dimension. We adopt the widely used notion "doubling dimension" to measure the extents of our compression and the resulting approximation. Finally, we test our method on both random and real datasets, the experimental results reveal that running the alignment algorithm on compressed patterns can achieve similar qualities, comparing with the results on the original patterns, but the running times (including the times cost for compression) are substantially lower

On Geometric Alignment in Low Doubling Dimension

In real-world, many problems can be formulated as the alignment between two geometric patterns. Previously, a great amount of research focus on the alignment of 2D or 3D patterns, especially in the field of computer vision. Recently, the alignment of geometric patterns in high dimension finds several novel applications, and has attracted more and more attentions. However, the research is still rather limited in terms of algorithms. To the best of our knowledge, most existing approaches for high dimensional alignment are just simple extensions of their counterparts for 2D and 3D cases, and often suffer from the issues such as high complexities. In this paper, we propose an effective framework to compress the high dimensional geometric patterns and approximately preserve the alignment quality. As a consequence, existing alignment approach can be applied to the compressed geometric patterns and thus the time complexity is significantly reduced. Our idea is inspired by the observation that high dimensional data often has a low intrinsic dimension. We adopt the widely used notion "doubling dimension" to measure the extents of our compression and the resulting approximation. Finally, we test our method on both random and real datasets, the experimental results reveal that running the alignment algorithm on compressed patterns can achieve similar qualities, comparing with the results on the original patterns, but the running times (including the times cost for compression) are substantially lower

HAADF-STEM block-scanning strategy for local measurement of strain at the nanoscale

Lattice strain measurement of nanoscale semiconductor devices is crucial for the semiconductor industry as strain substantially improves the electrical performance of transistors. High resolution scanning transmission electron microscopy (HR-STEM) imaging is an excellent tool that provides spatial resolution at the atomic scale and strain information by applying Geometric Phase Analysis or image fitting procedures. However, HR-STEM images regularly suffer from scanning distortions and sample drift during image acquisition. In this paper, we propose a new scanning strategy that drastically reduces artefacts due to drift and scanning distortion, along with extending the field of view. The method allows flexible tuning of the spatial resolution and decouples the choice of field of view from the need for local atomic resolution. It consists of the acquisition of a series of independent small subimages containing an atomic resolution image of the local lattice. All subimages are then analysed individually for strain by fitting a nonlinear model to the lattice images. The obtained experimental strain maps are quantitatively benchmarked against the Bessel diffraction technique. We demonstrate that the proposed scanning strategy approaches the performance of the diffraction technique while having the advantage that it does not require specialized diffraction cameras

The chiral and flavour projection of Dirac-Kahler fermions in the geometric discretization

It is shown that an exact chiral symmetry can be described for Dirac-Kahler fermions using the two complexes of the geometric discretization. This principle is extended to describe exact flavour projection and it is shown that this necessitates the introduction of a new operator and two new structures of complex. To describe simultaneous chiral and flavour projection, eight complexes are needed in all and it is shown that projection leaves a single flavour of chiral field on each.Comment: v2: 17 pages, Latex. 5 images eps. Added references, reformatted and clarification of some point

Shape Completion using 3D-Encoder-Predictor CNNs and Shape Synthesis

We introduce a data-driven approach to complete partial 3D shapes through a combination of volumetric deep neural networks and 3D shape synthesis. From a partially-scanned input shape, our method first infers a low-resolution -- but complete -- output. To this end, we introduce a 3D-Encoder-Predictor Network (3D-EPN) which is composed of 3D convolutional layers. The network is trained to predict and fill in missing data, and operates on an implicit surface representation that encodes both known and unknown space. This allows us to predict global structure in unknown areas at high accuracy. We then correlate these intermediary results with 3D geometry from a shape database at test time. In a final pass, we propose a patch-based 3D shape synthesis method that imposes the 3D geometry from these retrieved shapes as constraints on the coarsely-completed mesh. This synthesis process enables us to reconstruct fine-scale detail and generate high-resolution output while respecting the global mesh structure obtained by the 3D-EPN. Although our 3D-EPN outperforms state-of-the-art completion method, the main contribution in our work lies in the combination of a data-driven shape predictor and analytic 3D shape synthesis. In our results, we show extensive evaluations on a newly-introduced shape completion benchmark for both real-world and synthetic data

Perturbative, Non-Supersymmetric Completions of the Little Higgs

The little Higgs mechanism produces a light 100 GeV Higgs while raising the natural cutoff from 1 TeV to 10 TeV. We attempt an iterative little Higgs mechanism to produce multiple factors of 10 between the cutoff and the 100 GeV Higgs mass in a perturbative theory. In the renormalizable sector of the theory, all quantum corrections to the Higgs mass proportional to mass scales greater than 1 TeV are absent -- this includes quadratically divergent, log-divergent, and finite loops at all orders. However, even loops proportional to scales just a factor of 10 above the Higgs (or any other scalar) mass come with large numerical factors that reintroduce fine-tuning. Top loops, for example, produce an expansion parameter of not 1/(4 pi) but 1/5. The geometric increase in the number of fields at higher energies simply exacerbates this problem. We build a complete two-stage model up to 100 TeV, show that direct sensitivity of the electroweak scale to the cutoff is erased, and estimate the tuning due to large numerical factors. We then discuss the possibility, in a toy model with only scalar and gauge fields, of generating a tower of little Higgs theories and show that the theory quickly becomes a large-N gauge theory with ~ N fundamental scalars. We find evidence that at least this toy model could successfully generate light scalars with an exponentially large cutoff in the absence of supersymmetry or strong dynamics. The fine-tuning is not completely eliminated, but evidence suggests that this result is model dependent. We then speculate as to how one might marry a working tower of fields of this type at high scales to a realistic theory at the weak scale.Comment: 26 (+1) pages, 9 figure