On the effect of image denoising on galaxy shape measurements

Weak gravitational lensing is a very sensitive way of measuring cosmological parameters, including dark energy, and of testing current theories of gravitation. In practice, this requires exquisite measurement of the shapes of billions of galaxies over large areas of the sky, as may be obtained with the EUCLID and WFIRST satellites. For a given survey depth, applying image denoising to the data both improves the accuracy of the shape measurements and increases the number density of galaxies with a measurable shape. We perform simple tests of three different denoising techniques, using synthetic data. We propose a new and simple denoising method, based on wavelet decomposition of the data and a Wiener filtering of the resulting wavelet coefficients. When applied to the GREAT08 challenge dataset, this technique allows us to improve the quality factor of the measurement (Q; GREAT08 definition), by up to a factor of two. We demonstrate that the typical pixel size of the EUCLID optical channel will allow us to use image denoising.Comment: Accepted for publication in A&A. 8 pages, 5 figure

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Does median filtering truly preserve edges better than linear filtering?

Image processing researchers commonly assert that "median filtering is better than linear filtering for removing noise in the presence of edges." Using a straightforward large-$n$ decision-theory framework, this folk-theorem is seen to be false in general. We show that median filtering and linear filtering have similar asymptotic worst-case mean-squared error (MSE) when the signal-to-noise ratio (SNR) is of order 1, which corresponds to the case of constant per-pixel noise level in a digital signal. To see dramatic benefits of median smoothing in an asymptotic setting, the per-pixel noise level should tend to zero (i.e., SNR should grow very large). We show that a two-stage median filtering using two very different window widths can dramatically outperform traditional linear and median filtering in settings where the underlying object has edges. In this two-stage procedure, the first pass, at a fine scale, aims at increasing the SNR. The second pass, at a coarser scale, correctly exploits the nonlinearity of the median. Image processing methods based on nonlinear partial differential equations (PDEs) are often said to improve on linear filtering in the presence of edges. Such methods seem difficult to analyze rigorously in a decision-theoretic framework. A popular example is mean curvature motion (MCM), which is formally a kind of iterated median filtering. Our results on iterated median filtering suggest that some PDE-based methods are candidates to rigorously outperform linear filtering in an asymptotic framework.Comment: Published in at http://dx.doi.org/10.1214/08-AOS604 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Automatic, computer aided geometric design of free-knot, regression splines

A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality

Outlier robust corner-preserving methods for reconstructing noisy images

The ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother. Unfortunately, they usually seem to be at odds with each other; one can only improve one property at the cost of the other. By combining M-smoothing and least-squares-trimming, the TM-smoother is introduced as a means to unify corner-preserving properties and outlier robustness. To identify edge- and corner-preserving properties, a new theory based on differential geometry is developed. Further, robustness concepts are transferred to image processing. In two examples, the TM-smoother outperforms other corner-preserving smoothers. A software package containing both the TM- and the M-smoother can be downloaded from the Internet.Comment: Published at http://dx.doi.org/10.1214/009053606000001109 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A multiresolution framework for local similarity based image denoising

In this paper, we present a generic framework for denoising of images corrupted with additive white Gaussian noise based on the idea of regional similarity. The proposed framework employs a similarity function using the distance between pixels in a multidimensional feature space, whereby multiple feature maps describing various local regional characteristics can be utilized, giving higher weight to pixels having similar regional characteristics. An extension of the proposed framework into a multiresolution setting using wavelets and scale space is presented. It is shown that the resulting multiresolution multilateral (MRM) filtering algorithm not only eliminates the coarse-grain noise but can also faithfully reconstruct anisotropic features, particularly in the presence of high levels of noise

Differential invariant signatures and flows in computer vision : a symmetry group approach

Includes bibliographical references (p. 40-44).Supported by the National Science Foundation. DMS-9204192 DMS-8811084 ECS-9122106 Supported by the Air Force Office of Scientific Research. AFOSR-90-0024 Supported by the Rothschild Foundation-Yad Hanadiv and by Image Evolutions, Ltd.Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum