Adaptive Filtering Method Based On Convex Sets Of Filters

In scientific research and application fields, signals are collected from the realworld. They are analyzed and explained for discovering new scientific laws. Owing tothe random fluctuations of physical objects, random noises always exist in scientificobservations, and they would obey certain probability distributions. random noisescorrupt the signals which are desired, and increase the uncertainties and difficulties ofthe analysis on signals. Thus, the core of signal processing is denoising when noisyobservations have been received.In practical signal processing, there are always aliases of desired signals andrandom noises in time-spatial and frequency domains. Also, subtle structures of thesignals and statistical characteristics of the noises are always unknown. Thus, one canhardly reconstruct signals completely from noises. Usually, when filtering a noisysignal, the reduction of the noises and losses of the signals occur simultaneously, andthey are positively related. The degree of this correlation is due to the length and theshape of the filter which is used. Besides, the rate of changes in signals is not even ateach sample index. Thus, the losses of the signals caused by the filter are not the sameon the whole indices. In the fields such as seismic prospecting and explanation ormedical diagnosis, accurate signal processing are needed, and denoising has beenrequired to attenuate more noises and to preserve more local information of thesignals.Recently, the adaptive filtering based on time-invariant low-pass filters receivedmore and more attentions in signal processing. Unlike the traditional adaptive filteringalgorithms such as the LMS algorithm and Kalman filter, they focus on the local riskof filtering, and they are more suitable for those high-accuracy-desired fields. Therepresentative of these algorithms is the “intersection of the confidence intervals rule”(ICI rule). The ICI rule selects a “proper” filter from a finite set of given filters at eachsample index to minimize some local risk function. The only parameter of this filterset is the filter length. The main strategy of the ICI rule is to select a larger filterlength where the signal changes smoothly or the local SNR is relatively low, and itwill select a smaller filter length conversely. However, the ICI rule does not take theinfluence of the filter shape into account, and the filter length in the filter set isdiscrete definitely. Thus, the optimal estimation in the ICI rule can be hardlyguaranteed. Besides, if the filter shape changes in the filter set, ICI rule may fail towork.By the theory of convex optimization, this paper presents a novel framework of adaptive filtering based on the convex hull of a given filter set. It can choose theoptimal filter from the filter set adaptively at each sample index. Based on theadaptive framework, two novel adaptive algorithms have been presented in this paper，i.e., the adaptive time-frequency peak filtering and the adaptive spatial-temporalfiltering for array signals. The mainbody of this paper can be summarized as:1) The framework of adaptive filtering method based on convex sets of filters.a) Construct the convex hull of the filter set. In the convex hull, every filtersatisfies certain constraints on the filter set and the length and shape of the filters canbe both changed. Thus, unlike the ICI rule, the influence of the filter shape has beentaken into account. Besides, if choose some appropriate convex risk function definedon this convex filter set, unique optimal selection of the filters can be achieveddefinitely.b) By the assumption that the signals are in the Sobolev space, the penalizedleast squares criterion (PLS) is introduced as the objective function. This can result infar more smooth estimations than those of the ICI rule.c) Based on the minimax rule, novel method of adaptive parameters selection ispresented. The rule is to make them track the amplitude attenuations of the signal andnoise within the boundaries of the feasible region, respectively. This allows the PLSselect the optimal filter at each sample index, and prevents the failure of the ICI rulewhen the filter shapes change.2) Adaptive time-frequency peak filtering.Recently, the time-frequency peak filtering (TFPF) becomes popular in seismicevent recovering. It encodes a signal as the instantaneous frequency (IF) of an analyticsignal, and estimates the signal as the IF by using the Wigner-Ville distribution. Whenthe noise is white and the signal is linear in time, TFPF will give the unbiasedestimation of the signal. However, since the signals and noises are always complicatedin practice, TFPF cannot control the tradeoff between the noise attenuation and signalpreservation, the key problem is to select the shape and length of the windowfunctions which are used in Wigner-Ville distribution. In this paper, an importantproperty of TFPF has been proven, that is, the TFPF is equivalent to a low-passtime-invariant filter. Thus, the presented adaptive framework can be applied in TFPFto select the optimal window function at each sample index adaptively. This algorithmis so-called the adaptive TFPF. Applications of denoising to both synthetic and fieldseismic data have shown that the adaptive TFPF can attenuate more noise andpreserve more signal information in comparison with the traditional TFPF and ICIrule.3) Adaptive spatial-temporal filtering for array signals 2-dimensional filter design for array signals has been shown to be difficult. Arraysignals run into specific directions, and they have correlations in both time-andspace-domain. Considering this fact, a novel adaptive spatial-temporal algorithmbased on the presented framework is shown in this paper. With a given set of filters,the directional derivates is taken as the penalty to least squares. Thus, thespatial-temporal correlations between array signals are taken into account in thealgorithm. Denoising on both synthetic and field seismic data demonstrates thevalidity of the proposed method, and it shows that this adaptive spatial-temporalfiltering method is better than adaptive TFPF.Theoretical analysis and numerical results indicate that the adaptive algorithmspresented in this paper do not need much training. It overcomes the shortcoming ofICI rule and can achieve the optimal solution of the estimation. When the signalchanges fast or the input SNR is low, the algorithm performs better than traditionalmethods.