Adaptive Varying Window Size Selection Based on Intersection of Confidence Intervals Rule

We propose an approach to resolve the problem of varying window size selection for a wide class of linear and nonlinear lters. A window size of a lter is considered as a parameter of estimation and a number of estimates are calculated for a set of values of this parameter. Then, the intersection of con dence intervals (ICI) rule is used for selection of the estimate with the best window size. The ICI rule is applied in a point-wise manner. Thus, we obtain a varying adaptive window size selection optimizing the local (point-wise) accuracy of estimation. Originally, the ICI rule has been developed for linear lters. We present a number of applications which demonstrate how the ICI rule can be use for many other signal processing applications. 1. INTRODUCTION In this paper we present an overview of a recently developed method, called the Intersection of Con dence Intervals (ICI) rule [7], [9]. Originally, this rule was proposed and theoretically justi ed for local polynomial approximation (LPA) lters. Later, it was shown that this rule can be adapted and ef cient for highly nonlinear problems such as instantaneous frequency estimation for timefrequency analysis [11], [18], estimation of a directionof-arrival (DOA) on the base of sensor-array observations [10], estimation of the probability density [12], and orderstatistics ltering using median and stack lters with varying window sizes [14], [19]. Generalization of the ICI rule to two dimensional signals was shown to be ef cient for image de-noising. In particular, in [13] and [4], the image de-noising problem is considered for quite a general observation model including image dependent noise. The de-noising and window size selection are combined into one algorithm where the adaptive windows and signal de-noising are produced on the basis of the same observation model. The main original results of this overview paper are concerned with the development of the following algorithm. Firstly, the ICI rule for signal independent noise is given. Secondly, we separate the window size selection and denoising into a two-stage algorithm. At the rst stage, the sliding sample mean (simplest lter) is used for rough point-wise segmentation of the signal. At the second stage, more sophisticated de-noising methods (e.g. DCT, wavelet transform) are employed with varying size windows found in the rst stage. Simulation experiments con rm that these new methods can be much more effective. The ICI based algorithms are modi ed for the case of signal dependent noise. The window size in the ICI rule depends on the variance of a signal estimate and this variance in turn depends on the unknown signal if the noise is signal dependent. Thus, in order to obtain the window size we need to have some estimate of the signal in advance. The plug-in estimation assumes that these auxiliary estimates, used for window size selection, to some extent have to be independent with respect to the main estimates of the signal. Actually, these two types of estimates are calculated with different window sizes and/or for different observation models [5]. It is emphasized that the approach does not require estimation of the bias and differs from the usual quality-oft statistics (e.g. [5]). Filters (estimators) considered in this paper equipped with the rule for the window size selection possess simultaneously many attractive asymptotic properties, namely, 1) they are nearly optimal in the point-wise ! risk for estimating both the function and its derivatives" 2) they are spatially adaptive over a wide range of signal classes in the sense that their quality is close to that which one could achieve if smoothness of the signal was known in advance. These sorts of results can be produced as a generalization of the ones given in [7] for the #%$ & estimators. 2. VARYING WINDOW SIZE SELECTION BASED ON THE ICI RULE We consider an observed signal ')(+*-, modeled as .)/+0-132547698-:<;>=7?+@BA!CEDGFIHBJLK>M7N9O-PRQGS!T9UWV9X (1) where Y7Z9[-\ is the noise-free signal and ]G^I_B` is a noise with mean 0 and variance a!bcBd Note that in the case of different values of e this model will coincide with: an additive noise model, fhgji , a signaldependent additive noise model, k lnmoljp (e.g. q lm-grain noise, if r sutnvxwzy { ), a multiplicative speckle noise model, | }~ € Let us introduce a  nite set of window sizes: ‚„ƒ<…‡†ˆW‰‹ŠEŒŒŽŒŒ-Šj<’‘”“ starting with some small •—– and the corresponding estimates ˜ ™)šI›-œž ŸG¡ of the true signal ¢7£9¤-¥+¦ Let §‹ ̈9©«až¬’­G® and ̄!°+±W2ž3 ́Gμ be the bias and the standard deviation of these estimates. Denote by ¶)·1 ̧IoB» the ideal window size corresponding to the minimum value of the mean squared error 1⁄43⁄41⁄2À¿ Á Â)Ã+ÄWÅžÆ ÇGÈ3ÉËÊGÌ+Í-Î9ÎRτР. For a wide class of Ñ lters, the asymptotic estimation error has the following properties [9]: A) For the ideal window size the ratio Ò ÓÕÔ+ÖWמØ)Ù1Ú+Û-Ü9Ü+ݍÞàß!á9â«ãžä)åžæ9çWèRè is constant independent of the signalé B) For êoëìê)ížî9ïWð‡ñ òôóIõ-öG÷-øIùLúEû!üIý-þGÿ and for "! Then the #%$'&"( ) gives the optimal bias-variance balance and the estimate of this *%+', . can be obtained as follows. We determine a sequence of con/ dence intervals 0214365 of the biased estimate 7 8:9 ; <'=6> ? as follows [7], [9] @ A4B6C DFEHG IKJML%N'OQP RTSVUXW Y Z"[ \']6^ _M`ba cKdMe f'g6h ikjml2n o pMq%r's6t'u4vxw (2) where y is a threshold parameter of the conz dence interval. The following describes the ICI rule that is used in order to obtain the adaptive window size: Consider the intersection of the intervals {X|4}6~M€ƒ‚…„‡† ˆ ‰ with increasing Š , and let ‹Œ be the largest of those Ž for which the intervals  4‘6’ “ ”X•—–™˜—š› have a point in common. This œ dež nes the adaptive bandwidth and the adaptive estimate as follows Ÿ K¡£¢"¤ ¥§¦© ̈ a:« ¬ ­ ® ̄±°32M ́ μMμ ¶ (3) Thus, the ICI rule gives both the optimal estimate and the corresponding adaptive window size. This algorithm can be justi· ed, at least in the asymptotic sense with quite general assumptions and for a rather general class of estimates. It is emphasized that for the implementation of the ICI rule we need the estimates ̧ 1:o » 1⁄4 1⁄2Q3⁄4 ¿ and the corresponding standard deviations À ÁMÂ%Ã'Ä6Å Æ obtained for different window sizes. 3. APPLICATION OF THE ICI RULE TO A VARIETY OF SIGNAL PROCESSING PROBLEMS 3.1. LPA Ç lters The ÈÊÉÌË presents a sliding window polynomial Í ltering (transform). The linear Î rst degree ÏÑÐÌÒ treats the discrete time ÓÕÔ signal as sampled from an underlying continuous function within the selected window and uses the following loss function [8, 3]