Image compression with anisotropic diffusion

Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widely-used JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec

Shape-Driven Interpolation With Discontinuous Kernels: Error Analysis, Edge Extraction, and Applications in Magnetic Particle Imaging

Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications, such as medical imaging. In this paper, we study a radial basis function type of method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The results of the conducted numerical experiments are in line with the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising in order to obtain edge-preserving image reconstructions in which ringing artifacts are reduced

Towards optical intensity interferometry for high angular resolution stellar astrophysics

Most neighboring stars are still detected as point sources and are beyond the angular resolution reach of current observatories. Methods to improve our understanding of stars at high angular resolution are investigated. Air Cherenkov telescopes (ACTs), primarily used for Gamma-ray astronomy, enable us to increase our understanding of the circumstellar environment of a particular system. When used as optical intensity interferometers, future ACT arrays will allow us to detect stars as extended objects and image their surfaces at high angular resolution. Optical stellar intensity interferometry (SII) with ACT arrays, composed of nearly 100 telescopes, will provide means to measure fundamental stellar parameters and also open the possibility of model-independent imaging. A data analysis algorithm is developed and permits the reconstruction of high angular resolution images from simulated SII data. The capabilities and limitations of future ACT arrays used for high angular resolution imaging are investigated via Monte-Carlo simulations. Simple stellar objects as well as stellar surfaces with localized hot or cool regions can be accurately imaged. Finally, experimental efforts to measure intensity correlations are expounded. The functionality of analog and digital correlators is demonstrated. Intensity correlations have been measured for a simulated star emitting pseudo-thermal light, resulting in angular diameter measurements. The StarBase observatory, consisting of a pair of 3 m telescopes separated by 23 m, is described.Comment: PhD dissertatio

Full-field optical measurement of curvatures in ultra-thin-film–substrate systems in the range of geometrically nonlinear deformations

This article describes coherent gradient sensing (CGS) as an optical, full-field, real-time, nonintrusive, and noncontact technique for the measurement of curvatures and nonuniform curvature changes in film-substrate systems. The technique is applied to the study of curvature fields in thin Al films (6 mum) deposited on thin circular silicon wafers (105 mum) of "large" in-plane dimensions (50.8 mm in diameter) subjected to thermal loading histories. The loading and geometry is such that the system experiences deformations that are clearly within the nonlinear range. The discussion is focused on investigating the limits of the range of the linear relationship between the thermally induced mismatch strain and the substrate curvature, on the degree to which the substrate curvature becomes spatially nonuniform in the range of geometrically nonlinear deformation, and finally, on the bifurcation of deformation mode from axial symmetry to asymmetry with increasing mismatch strain. Results obtained on the basis of both simple models and more-detailed finite-element simulations are compared with the full-field CGS measurements with the purpose of validating the analytical and numerical models

Analytical method to measure three-dimensional strain patterns in the left ventricle from single slice displacement data

Background: Displacement encoded Cardiovascular MR (CMR) can provide high spatial resolution measurements of three-dimensional (3D) Lagrangian displacement. Spatial gradients of the Lagrangian displacement field are used to measure regional myocardial strain. In general, adjacent parallel slices are needed in order to calculate the spatial gradient in the through-slice direction. This necessitates the acquisition of additional data and prolongs the scan time. The goal of this study is to define an analytic solution that supports the reconstruction of the out-of-plane components of the Lagrangian strain tensor in addition to the in-plane components from a single-slice displacement CMR dataset with high spatio-temporal resolution. The technique assumes incompressibility of the myocardium as a physical constraint. Results: The feasibility of the method is demonstrated in a healthy human subject and the results are compared to those of other studies. The proposed method was validated with simulated data and strain estimates from experimentally measured DENSE data, which were compared to the strain calculation from a conventional two-slice acquisition. Conclusion: This analytical method reduces the need to acquire data from adjacent slices when calculating regional Lagrangian strains and can effectively reduce the long scan time by a factor of two

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Jumping with variably scaled discontinuous kernels (VSDKs)

open3In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images.openStefano De Marchi, Francesco Marchetti, Emma Perracchione,DE MARCHI, Stefano; Marchetti, Francesco; Perracchione, Emm

Jumping with variably scaled discontinuous kernels (VSDKs)

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Spatio-temporal correlations of jets using high-speed particle image velocimetry

The major source of aircraft noise at take-off is jet noise. If jet noise is not adequately addressed environmental impact concerns will constrain the planned growth of the air transport system. A considerable amount of research worldwide has therefore been aimed at identifying ways to reduce jet noise including development of a predictive tool that can estimate the noise generated by new nozzle designs. Current noise prediction techniques, however, still require the input of empirically calibrated noise source models and their performance is still inadequate. In addition, development of detailed noise source identification measurements and the associated understanding of how to control (and reduce) the noise at the source has been limited. The fundamental turbulence property which acts as the source of propagating noise in shear layers is the two-point space-time velocity correlation (Rijkl). Very few measurements exist for this property to guide model development. It is therefore the aim of the work reported in this thesis to provide new experimental data that helps identify the turbulence sources located within the shear layer of jets. The technique of Partical Imaging Velocimetry (PIV) is used to capture directly the flowfield and all relevant turbulent statistics... cont'd