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Scale Space Multiresolution Analysis of Random Signals

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Scale Space Multiresolution Analysis of Random Signals Scale Space Multiresolution Analysis of Random Signals Lasse Holmstr¨om∗, Leena Pasanen Department of Mathematical Sciences, University of Oulu, Finland Reinhard Furrer Institute of Mathematics, University of Zurich,¨ Switzerland Stephan R. Sain National Center for Atmospheric Research, Boulder, Colorado, USA Abstract A method to capture the scale-dependent features in a random signal is pro- posed with the main focus on images and spatial fields defined on a regular grid. A technique based on scale space smoothing is used. However, where the usual scale space analysis approach is to suppress detail by increasing smoothing progressively, the proposed method instead considers differences of smooths at neighboring scales. A random signal can then be represented as a sum of such differences, a kind of a multiresolution analysis, each difference representing de- tails relevant at a particular scale or resolution. Bayesian analysis is used to infer which details are credible and which are just artifacts of random variation. The applicability of the method is demonstrated using noisy digital images as well as global temperature change fields produced by numerical climate prediction models. Keywords: Scale space smoothing, Bayesian methods, Image analysis, Climate research 1. Introduction In signal processing, smoothing is often used to suppress noise. An optimal level of smoothing is then of interest. In scale space analysis of noisy curves and images, instead of a single, in some sense “optimal” level of smoothing, a whole family of smooths is considered and each smooth is thought to provide information about the underlying object at a particular scale or resolution. The ∗Corresponding author ∗∗P.O.Box 3000, 90014 University of Oulu, Finland Tel. +358 50 563 7465 Fax +358 9 553 1730 Email address: [email protected] (Lasse Holmstr¨om ) URL: http://cc.oulu.fi/~llh/ (Lasse Holmstr¨om ) The electronic version of this paper includes Matlab software used in the computations Preprint submitted to Elsevier April 14, 2011 concept of scale space was introduced in the context of image analysis (see Lindeberg, 1994, and the references there) but during the past ten years it has emerged also as useful statistical data analysis procedure (cf. Holmstr¨om, 2010). The seminal idea was the SiZer technology introduced by Chaudhuri and Marron (1999, 2000) and the original concept has since then been extended to various directions, including Bayesian versions that can be used to analyze curves underlying noisy data (Er¨ast¨o and Holmstr¨om, 2005, Godtliebsen and Øig˚ard, 2005, Holmstr¨om, 2010). Recently, a Bayesian version for the analysis of images and spatial fields has also been proposed (Holmstr¨om and Pasanen, 2007, 2008, Pasanen and Holmstr¨om, 2008). The central idea in statistical scale space methods is to make inferences about the “statistically significant” features of the smooths of an object of which only a noisy observation is available. This is typically done by estimating a measure of local change of the smooths, such as the derivative or, in the Bayesian approach, exploring its posterior distribution. The analysis is carried out for a wide range of smoothing levels in the hope of discovering the salient scale- dependent features. Raising the smoothing level progressively suppresses details revealing increasingly large scale characteristics in the object underlying the data. By employing differences of smooths at neighboring scales, the method pro- posed in this paper attempts to separate the features into distinct scale cate- gories even more aggressively than the usual scale space procedure. The basic idea is illustrated with the simple example shown in Figure 1. The top panel in T the middle column shows a curve represented by a vector x = [x1,...,xn] of values on a discrete, equally spaced grid. The curve was constructed as the sum of the four underlying “detail” curves x1,..., x4 shown on the left. They can be thought to represent the features of the curve x in four different scales. In order to recover these scale-dependent detail curves, Nadaraya-Watson kernel smoothing (see Appendix B) was applied in the middle column to the curve x. The smoothing operator, an n × n matrix, is denoted by Sλ and λ denotes the smoothing level. Here λ1 = 0 (no smoothing) and Sλ1 x therefore is just x. The differences of these smooths are shown in the right column and they, together with the mean Sλ4 x, capture the original constituent detail curves reasonably well. The middle column can be thought to represent conventional scale space analysis where, in a sense, Sλi x contains the signal features for all scales λ ≥ λi. By considering differences of smooths, we attempt to remedy this by isolating, for two levels λi < λj , those features that are present at level λi but not at λj . One way to think about the contrast between usual scale space analysis and the difference of smooths approach is the distinction between low pass and band pass filtering. A related idea is the reroughing technique of Tukey (Tukey, 1977, Ch. 16) which, if the smoothing levels it uses are selected appropriately, can produce similar scale-dependent signal components. The purpose of this paper is to take the idea presented in this simple exam- ple further. We will develop a method that applies to more general situations, focusing on noisy images and random fields defined on regular grids. In contrast to the noiseless situation of the previous example, it then becomes essential to distinguish true features in the data from artifacts created by random fluctu- ation. Obviously, in order for the suggested idea to work well one must select properly the smooths used to reconstruct the details. Subtracting smooths of very different scales, such as using the difference between x and Sλ3 x in Fig- 2 x1 x 2 2 1 1 z1 0 2 0 0 0.5 1 0 0.5 1 1 x2 Sλ x 2 0 2 2 0 0.5 1 1 1 z2 0 0 2 0 0.5 1 0 0.5 1 1 x3 Sλ x 3 0 2 2 0 0.5 1 1 1 z3 0 0 2 0 0.5 1 0 0.5 1 1 x4 Sλ x 4 0 2 2 0 0.5 1 1 1 0 0 0 0.5 1 0 0.5 1 Figure 1: The curve x = Sλ1 x in the uppermost panel in the middle column is constructed as the sum of the curves x1, x2, x3 and x4 on the left. Differences of the smooths of Sλi x in the right hand column reveal its underlying components. ure 1, could easily miss the mid-scale features, whereas differences of smooths of very similar scales probably will not reveal anything interesting. We there- fore also try to provide tools for identifying a sequence of smooths that, when differenced, would reveal interesting features underlying the data. The scale space multiresolution procedure we propose has three steps: (1) Bayesian signal synthesis or reconstruction, (2) forming of scale-dependent de- tail components of the reconstructed or synthesized signal using differences of smooths at neighboring scales, and (3) posterior credibility analysis of the fea- tures in these details. In the second step, methods for finding useful smoothing levels are needed. The paper is organized as follows. In Section 2 we explain the idea of us- ing differences of smooths in more detail and also discuss ideas for the selec- tion of suitable sets of smoothing levels. Section 3 demonstrates the appli- cation of the proposed method to the analysis of digital images and random fields. In Section 4 we consider alternative approaches to resolving a signal into meaningful details, concentrating on wavelet image decomposition. Sec- tion 5 is a short summary of the main points made in the paper and the Ap- 3 pendices describe some technicalities associated with inference and the partic- ular smoothers used in the analyses. The Matlab code used for all compu- tations, together with instructions for its use and examples can be found at http:/cc.oulu.fi/∼lpasanen/MRBSiZer/. The Matlab functions are also avail- able with the electronic version of the paper. 2. Smoothing-based multiresolution analysis 2.1. The detail decomposition Let x be a signal, e.g. a curve or an image or, more generally, a random field, represented as an n-dimensional random vector. Also, let Sλ be a smoother represented as an n × n matrix. Here λ ≥ 0 is a smoothing parameter that controls the level of smoothing, a small λ corresponding to little smoothing and a large λ corresponding to heavy smoothing. For example, Sλ could be a kernel smoother and λ the kernel bandwidth or Sλ could be a spline smoother with λ controlling the size of the roughness penalty. Now, let 0= λ1 < λ2 < ··· < λL−1 < λL = ∞ (1) be a set of smoothing levels. We assume that S0 is the identity mapping, S0x = Sλ1 x = x. Theeffect of S∞ depends on the particular smoother used. For the Nadaraya-Watson kernel smoother used in Figure 1, L = 4 and the smooth S∞x = S4x is the mean of the curve x, whereas for a local linear regression smoother S∞x is a linear fit to the data x. In all applications presented in this paper S∞x is in fact the mean of x. We now have trivially that L−1 L−1 x = (Sλi − Sλi+1 )x + SλL x ≡ zi + zL, (2) Xi=1 Xi=1 where zi = (Sλi − Sλi+1 )x for i =1,...,L − 1 and zL = S∞x. (To save space, the mean z4 was not included in the right hand column of Figure 1.) Here zi for i = 1,...,L − 1 is the difference between two consecutive smooths and it can be interpreted as the detail lost when smoothing is increased from λi to λi+1.
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