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

Scale Invariant Interest Points with Shearlets

Shearlets are a relatively new directional multi-scale framework for signal analysis, which have been shown effective to enhance signal discontinuities such as edges and corners at multiple scales. In this work we address the problem of detecting and describing blob-like features in the shearlets framework. We derive a measure which is very effective for blob detection and closely related to the Laplacian of Gaussian. We demonstrate the measure satisfies the perfect scale invariance property in the continuous case. In the discrete setting, we derive algorithms for blob detection and keypoint description. Finally, we provide qualitative justifications of our findings as well as a quantitative evaluation on benchmark data. We also report an experimental evidence that our method is very suitable to deal with compressed and noisy images, thanks to the sparsity property of shearlets

Discrete spherical means of directional derivatives and Veronese maps

We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using the Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation

High-order discretization of backward anisotropic diffusion and application to image processing

Anisotropic diffusion is a well recognized tool in digital image processing, including edge detection and focusing. We present here a particular nonlinear time-dependent operator together with an appropriate high-order discretization for the space variable. In just a single step, the procedure emphasizes the contour lines encircling the objects, paving the way to accurate reconstructions at a very low cost. One of the main features of such an approach is the possibility of relying on a rather large set of invariant discontinuous images, whose edges can be determined without introducing any approximation

Esquemas de diferencias finitas en el procesamiento de imágenes

Digital Image processing has been a research area of interest in the last decades, standing out for its applications in the analysis of diagnostic images and astronomical images. In this paper, we perform an overview of edge detection methods through finite-difference to present edge detection as a problem-based learning strategy for numerical differentiation, in order to improve the students’ skills in modeling and algorithmic thinking in numerical analysis courses. In addition, we present image restoration through finite-difference as a problem involving partial differential equations and software tools.El procesamiento de imágenes digitales ha sido un área de investigación de interés en las últimas décadas, destacándose por sus aplicaciones en el análisis de imágenes diagnósticas e imágenes astronómicas. En este artículo, realizamos una descripción general de los métodos de detección de bordes a través de diferencias finitas, con el fin de presentar la detección de bordes como una estrategia de enseñanza de los esquemas de diferencias finitas mediante aprendizaje basado en problemas, buscando desarrollar competencias de modelamiento matemático y pensamiento algorítmico en estudiantes de análisis numérico. Además, presentamos la restauración de imágenes mediante diferencias finitas como un problema que involucra ecuaciones diferenciales parciales y herramientas de software

Discrete Signal Processing on Graphs: Frequency Analysis

Signals and datasets that arise in physical and engineering applications, as well as social, genetics, biomolecular, and many other domains, are becoming increasingly larger and more complex. In contrast to traditional time and image signals, data in these domains are supported by arbitrary graphs. Signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs. This paper studies the concepts of low and high frequencies on graphs, and low-, high-, and band-pass graph filters. In traditional signal processing, there concepts are easily defined because of a natural frequency ordering that has a physical interpretation. For signals residing on graphs, in general, there is no obvious frequency ordering. We propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low-, high-, and band-pass graph signals and filters. We study the design of graph filters with specified frequency response, and illustrate our approach with applications to sensor malfunction detection and data classification

Cellular neural networks, Navier-Stokes equation and microarray image reconstruction

Copyright @ 2011 IEEE.Although the last decade has witnessed a great deal of improvements achieved for the microarray technology, many major developments in all the main stages of this technology, including image processing, are still needed. Some hardware implementations of microarray image processing have been proposed in the literature and proved to be promising alternatives to the currently available software systems. However, the main drawback of those proposed approaches is the unsuitable addressing of the quantification of the gene spot in a realistic way without any assumption about the image surface. Our aim in this paper is to present a new image-reconstruction algorithm using the cellular neural network that solves the Navier–Stokes equation. This algorithm offers a robust method for estimating the background signal within the gene-spot region. The MATCNN toolbox for Matlab is used to test the proposed method. Quantitative comparisons are carried out, i.e., in terms of objective criteria, between our approach and some other available methods. It is shown that the proposed algorithm gives highly accurate and realistic measurements in a fully automated manner within a remarkably efficient time

Low computational complexity variable block size (VBS) partitioning for motion estimation using the Walsh Hadamard transform (WHT)

Variable Block Size (VBS) based motion estimation has been adapted in state of the art video coding, such as H.264/AVC, VC-1. However, a low complexity H.264/AVC encoder cannot take advantage of VBS due to its power consumption requirements. In this paper, we present a VBS partition algorithm based on a binary motion edge map without either initial motion estimation or Rate-Distortion (R-D) optimization for selecting modes. The proposed algorithm uses the Walsh Hadamard Transform (WHT) to create a binary edge map, which provides a computational complexity cost effectiveness compared to other light segmentation methods typically used to detect the required region