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Chapter 1:
The book is devoted to signal and image reconstruction based on two independent ideas: local approximation for the design of linear and nonlinear filters (estimators) and adaptation of these filters to unknown smoothness of the signal of interest. As flexible universal tools we use local polynomial approximation (LPA) for approximation and intersection of confidence intervals (ICI) for adaptation. The LPA is applied for linear filter design using a polynomial fit in a sliding window. The window as well as the order of the polynomial defines a desirable filter. The window size is considered as a varying adaptation parameter of the filter. The ICI is an adaptation algorithm. It searches for a largest local window size where LPA assumptions fit well to the observations. It is shown that the ICI adaptive LPA is efficient and allows for a nearly optimal quality of estimation. To deal with anisotropic signals, in particular those that are typical in imaging, narrowed nonsymmetric directional windows are used. The corresponding directional (multidirectional) LPA filters equipped with the ICI adaptive window sizes demonstrate an advanced performance. The LPA combined with the maximum likelihood and quasi-likelihood are used for the design of nonlinear filters (estimators). The ICI rule used for the window size selection of these filters defines nonlinear filters as adaptive to the unknown smoothness of the reconstructed signal. LPA-ICI algorithms are powerful recent tools in signal and image processing. These algorithms demonstrate the state-of-art performance and can visually and quantitatively outperform some of the best existing methods. The ICI rule is universal and can be applied with many existing linear and nonlinear filters (not only with the LPA-based filters) where the bias-variance trade-off is valid for selecting tunable parameters. The LPA-ICI techniques define a wide class of pointwise spatially/time adaptive filters applicable in many scientific and engineering fields. In this introduction we start with motivation, ideas, and constructive comments to review basic elements of the techniques presented in the book. These comments are presented for univariate signals while the book is addressed mainly to multidimensional problems.
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Nonlinear filtering

Digital filtering

Linear filtering

Electronic filtering

Image filtering

Image processing

Image restoration

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