A Comparison of Two Shallow Water Models with Non-Conforming Adaptive Grids: classical tests

In an effort to study the applicability of adaptive mesh refinement (AMR) techniques to atmospheric models an interpolation-based spectral element shallow water model on a cubed-sphere grid is compared to a block-structured finite volume method in latitude-longitude geometry. Both models utilize a non-conforming adaptation approach which doubles the resolution at fine-coarse mesh interfaces. The underlying AMR libraries are quad-tree based and ensure that neighboring regions can only differ by one refinement level. The models are compared via selected test cases from a standard test suite for the shallow water equations. They include the advection of a cosine bell, a steady-state geostrophic flow, a flow over an idealized mountain and a Rossby-Haurwitz wave. Both static and dynamics adaptations are evaluated which reveal the strengths and weaknesses of the AMR techniques. Overall, the AMR simulations show that both models successfully place static and dynamic adaptations in local regions without requiring a fine grid in the global domain. The adaptive grids reliably track features of interests without visible distortions or noise at mesh interfaces. Simple threshold adaptation criteria for the geopotential height and the relative vorticity are assessed.Comment: 25 pages, 11 figures, preprin

Incremental and Transitive Discrete Rotations

A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from \ZZ[i] to \ZZ[i], whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotate d copies of an image for angles in-between 0 and a destination angle. The di scretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm whic h computes incrementally a discretized rotation. The suggested method uses o nly integer arithmetic and does not compute any sine nor any cosine. More pr ecisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be th e key ingredient that will make the resulting procedure optimally fast and e xact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of defin itions for discrete geometry

Directional edge and texture representations for image processing

An efficient representation for natural images is of fundamental importance in image processing and analysis. The commonly used separable transforms such as wavelets axe not best suited for images due to their inability to exploit directional regularities such as edges and oriented textural patterns; while most of the recently proposed directional schemes cannot represent these two types of features in a unified transform. This thesis focuses on the development of directional representations for images which can capture both edges and textures in a multiresolution manner. The thesis first considers the problem of extracting linear features with the multiresolution Fourier transform (MFT). Based on a previous MFT-based linear feature model, the work extends the extraction method into the situation when the image is corrupted by noise. The problem is tackled by the combination of a "Signal+Noise" frequency model, a refinement stage and a robust classification scheme. As a result, the MFT is able to perform linear feature analysis on noisy images on which previous methods failed. A new set of transforms called the multiscale polar cosine transforms (MPCT) are also proposed in order to represent textures. The MPCT can be regarded as real-valued MFT with similar basis functions of oriented sinusoids. It is shown that the transform can represent textural patches more efficiently than the conventional Fourier basis. With a directional best cosine basis, the MPCT packet (MPCPT) is shown to be an efficient representation for edges and textures, despite its high computational burden. The problem of representing edges and textures in a fixed transform with less complexity is then considered. This is achieved by applying a Gaussian frequency filter, which matches the disperson of the magnitude spectrum, on the local MFT coefficients. This is particularly effective in denoising natural images, due to its ability to preserve both types of feature. Further improvements can be made by employing the information given by the linear feature extraction process in the filter's configuration. The denoising results compare favourably against other state-of-the-art directional representations

A fast algorithm for the computation of 2-D forward and inverse MDCT

International audienceA fast algorithm for computing the two-dimensional (2-D) forward and inverse modified discrete cosine transform (MDCT and IMDCT) is proposed. The algorithm converts the 2-D MDCT and IMDCT with block size M N into four 2-D discrete cosine transforms (DCTs) with block size ðM=4Þ ðN=4Þ. It is based on an algorithm recently presented by Cho et al. [An optimized algorithm for computing the modified discrete cosine transform and its inverse transform, in: Proceedings of the IEEE TENCON, vol. A, 21–24 November 2004, pp. 626–628] for the efficient calculation of onedimensional MDCT and IMDCT. Comparison of the computational complexity with the traditional row–column method shows that the proposed algorithm reduces significantly the number of arithmetic operations

FASTLens (FAst STatistics for weak Lensing) : Fast method for Weak Lensing Statistics and map making

With increasingly large data sets, weak lensing measurements are able to measure cosmological parameters with ever greater precision. However this increased accuracy also places greater demands on the statistical tools used to extract the available information. To date, the majority of lensing analyses use the two point-statistics of the cosmic shear field. These can either be studied directly using the two-point correlation function, or in Fourier space, using the power spectrum. But analyzing weak lensing data inevitably involves the masking out of regions or example to remove bright stars from the field. Masking out the stars is common practice but the gaps in the data need proper handling. In this paper, we show how an inpainting technique allows us to properly fill in these gaps with only $N \log N$ operations, leading to a new image from which we can compute straight forwardly and with a very good accuracy both the pow er spectrum and the bispectrum. We propose then a new method to compute the bispectrum with a polar FFT algorithm, which has the main advantage of avoiding any interpolation in the Fourier domain. Finally we propose a new method for dark matter mass map reconstruction from shear observations which integrates this new inpainting concept. A range of examples based on 3D N-body simulations illustrates the results.Comment: Final version accepted by MNRAS. The FASTLens software is available from the following link : http://irfu.cea.fr/Ast/fastlens.software.ph