Positive definite symmetric matrices (so-called tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular manifold of constant curvature without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have

Wireless sensor networks have attracted attention from a diverse set of researchers, due to the unique combination of distributed, resource and data processing constraints. However, until now, the lack of real sensor network deployments have resulted in ad-hoc assumptions on a wide range of issues including topology characteristics and data distribution. As deployments of sensor networks become more widespread [1, 2], many of these assumptions need to be revisited. This paper

Abstract—Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuoustime signal from a sequence of corrupted discrete-time samples. In this paper, a general formulation of this problem is developed that treats the interpolation problem from ideal, noisy samples, and the deconvolution problem in which the signal is filtered prior to sampling, in a unified way. The signal reconstruction is performed in a shift-invariant subspace spanned by the integer shifts of a generating function, where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter. Several alternative approaches to designing the correction filter are suggested, which differ in their assumptions on the signal and noise. The classical deconvolution solutions (least-squares, Tikhonov, and Wiener) are adapted to our particular situation, and new methods that are optimal in a minimax sense are also proposed. The solutions often have a similar structure and can be computed simply and efficiently by digital filtering. Some concrete examples of reconstruction filters are presented, as well as simple guidelines for selecting the free parameters (e.g., regularization) of the various algorithms. Index Terms—Deconvolution, interpolation, minimax reconstruction, sampling. I.

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We present a unified framework for interpolation and regularisation of scalar- and tensor-valued images. This framework is based on elliptic partial differential equations (PDEs) and allows rotationally invariant models. Since it does not require a regular grid, it can also be used for tensor-valued scattered data interpolation and for tensor field inpainting. By choosing suitable differential operators, interpolation methods using radial basis functions are covered. Our experiments show that a novel interpolation technique based on anisotropic diffusion with a diffusion tensor should be favoured: It outperforms interpolants with radial basis functions, it allows discontinuity-preserving interpolation with no additional oscillations, and it respects positive semidefiniteness of the input tensor data.

While methods based on partial differential equations (PDEs) and variational techniques are powerful tools for denoising and inpainting digital images, their use for image compression was mainly focussing on pre- or postprocessing so far. In our paper we investigate their potential within the decoding step. We start with the observation that edge-enhancing diffusion (EED), an anisotropic nonlinear diffusion filter with a diffusion tensor, is well-suited for scattered data interpolation: Even when the interpolation data are very sparse, good results are obtained that respect discontinuities and satisfy a maximum– minimum principle. This property is exploited in our studies on PDE-based image compression. We use an adaptive triangulation method based on B-tree coding for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the EED process. They can be coded in a compact and elegant way that reflects the B-tree structure. Our experiments illustrate that for high compression rates and non-textured images, this PDE-based approach gives visually better results than the widely-used JPEG coding.

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

ABSTRACT We present a semiautomated software solution to the problem of extending the lateral field of view of a classical microscope. The initial requirements are a set of overlapping images, along with their user-provided coarse mosaic. Our solution then refines this initial mosaic in a fully automatic fashion. We rely on a highly accurate registration engine to perform the pairwise registration of the individual images, and on an efficient strategy to minimize the amount of computations while maintaining the highest possible global quality. We describe these ingredients, which we make available as a free multiplatform user-friendly software package. We also highlight why and how the specific aspects of the present microscopy application differ from those encountered while creating more common mosaics such as panoramas. Finally, we present experimental results that illustrate and validate our method on a real biological sample. We conclude by showing that we are able to reach subpixel accuracy. Microsc. Res. Tech. 70:135–146, 2007. VC 2006 Wiley-Liss, Inc.

When involved in the visual design of graphical user interfaces, graphic designers can do more than providing static graphics for programmers to incorporate into applications. We describe a technique that allows them to provide examples of graphical objects at various key sizes using their usual drawing tool, then let the system interpolate their resizing behavior. We relate this technique to current practices of graphic designers, provide examples of its use and describe the underlying inference algorithm. We show how the mathematical properties of the algorithm allows the system to be predictable and explain how it can be combined with more traditional layout mechanisms. ACM Classification: H5.2 [Information Interfaces and

The main objective of the proposed research is to evaluate the invariance of the cooccurrence texture model with respect to patient size in Computed Tomography (CT) data. Since patients ’ scans can have high variation in pixel spacing, in order to standardize all patients’ texture descriptors, we investigate the effect of four interpolation techniques in reducing the pixel spacing variance: nearest neighbor, bilinear, cubic and B-spline methods. After applying interpolation, the co-occurrence texture model and nine Haralick texture descriptors are calculated to quantify the texture appearance of the soft tissues. The differences in the texture are evaluated before and after interpolation using Analysis of Variance (ANOVA) and the General Linear Model (GLM). Our preliminary results on liver images from five different patients show that the co-occurrence texture model is not affected by the difference in pixel spacing; therefore, computer-aided segmentation, retrieval and diagnosis tools that are based on co-occurrence texture models are expected to work on multiple patients of different sizes. 1