The terms Smoothing and Filtering are used for four concepts that may initially be confusing: Smoothing (in two senses: estimation and convolution), and Filtering (again in two senses: estimation and convolution). Smoothing (estimation) and smoothing (convolution), despite being labelled with the same name in the English language, can mean totally different mathematical procedures. The requirements of the problems they solve are different. These concepts are distinguished by the context (signal processing versus estimation of stochastic processes). The historical reason for this confusion is that initially, Wiener's suggested "smoothing" filter was just a convolution. Later on, his proposed solutions for obtaining a smoother estimation separate developments into two distinct concepts. One was about attaining a smoother estimation by taking into account past observations, and the other was smoothing using a filter design (design of a convolution filter). Both the smoothing problem (in the sense of estimation) and the filtering problem (in the sense of estimation) are often confused with smoothing and filtering in other contexts (especially in non-stochastic signal processing, often referring a various types of convolution). These names are used in the context of World War 2 with problems framed by people like
Norbert Wiener. One source of confusion is that the
Wiener Filter is in the form of a simple convolution. However, in Wiener's filter, two time series are given. When the filter is defined, a straightforward convolution is the answer. However, in later developments, such as Kalman filtering, the nature of filtering differs from convolution and warrants a different name. The distinction is described in the following two senses: 1. Convolution: The smoothing in the sense of convolution is simpler. For example, moving average, low-pass filtering, convolution with a kernel, or blurring using Laplace filters in
image processing. It is often a
filter design problem. Especially non-stochastic and non-Bayesian signal processing, without any hidden variables. 2. Estimation: The smoothing problem (or Smoothing in the sense of estimation) uses Bayesian and state-space models to estimate the hidden state variables. This is used in the context of World War 2, defined by people like Norbert Wiener, in (stochastic) control theory, radar, signal detection, tracking, etc. The most common use is the Kalman Smoother used with the Kalman Filter, which was actually developed by Rauch. The procedure is called Kalman-Rauch recursion. It is one of the main problems solved by
Norbert Wiener. Most importantly, in the Filtering problem (sense 2), the information from observation up to the time of the current sample is used. In smoothing (also sense 2), all observation samples (from the future) are used. Filtering is causal, but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on serial incremental observations. But the usual and more common smoothing and filtering (in the sense of 1.) do not have such a distinction because there is no distinction between hidden and observable. The distinction between Smoothing (estimation) and Filtering (estimation): In smoothing, all observation samples are used (from the future). Filtering is causal, whereas smoothing is batch processing of the given data. Filtering is the estimation of a (hidden) time-series process based on serial incremental observations. == See also ==