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Structural adaptive smoothing procedures. (English) Zbl 1272.94006

Dahlhaus, Rainer (ed.) et al., Mathematical methods in signal processing and digital image analysis. Berlin: Springer (ISBN 978-3-540-75631-6/hbk). Springer Complexity, 183-229 (2008).
Summary: An important problem in image and signal analysis is denoising. Given data \(y_j\) at locations \(x_j\), \(j=1,\dots ,N\), in space or time, the goal is to recover the original image or signal \(m_j\), \(j=1,\dots , N\), from the noisy observations \(y_j\), \(j=1,\dots ,N\). Denoising is a special case of a function estimation problem: If \(m_j=m(x_j)\) for some function \(m(x)\), we may model the data \(y_j\) as real-valued random variables \(Y_j\) satisfying the regression relation \[ Y_j=m\left(x_j\right)+\varepsilon _j,\quad j=1,\dots ,N, \] where the additive noise \(\varepsilon_j\), \(j=1,\dots ,N\), is independent, identically distributed (i.i.d.) with mean \(\mathbb E\,\varepsilon_j=0\). The original denoising problem is solved by finding an estimate \( \hat m\left( x \right) \) of the regression function \(m(x)\) on some subset containing all the \(x_j\). More generally, we may allow the function arguments to be random variables \(X_j\in\mathbb R^d\) themselves ending up with a regression model with stochastic design \[ Y_j=m\left(X_j\right)+\varepsilon_j,\quad j=1,\dots ,N, \] where \(X_j\), \(Y_j\) are identically distributed, and \(\mathbb E\left\{\varepsilon _j| X_j\right\}=0\). In this case, the function \(m(x)\) to be estimated is the conditional expectation of \(Y_j\) given \(X_j=x\).
For the entire collection see [Zbl 1130.68006].

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
62H35 Image analysis in multivariate analysis

Software:

AWS; KernSmooth; R; fmri; adimpro
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Full Text: DOI

References:

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