Blind Deconvolution for Jump‐Preserving Curve Estimation

Abstract

In many applications, observed signals are contaminated by both random noise and blur. This paper proposes a blind deconvolution procedure for estimating a regression function with possible jumps preserved, by removing both noise and blur when recovering the signals. Our procedure is based on three local linear kernel estimates of the regression function, constructed from observations in a left-side, a right-side, and a two-side neighborhood of a given point, respectively. The estimated function at the given point is then defined by one of the three estimates with the smallest weighted residual sum of squares. To better remove the noise and blur, this estimate can also be updated iteratively. Performance of this procedure is investigated by both simulation and real data examples, from which it can be seen that our procedure performs well in various cases.

Change-point regression with a smooth additive disturbance

“…all derivatives may change abruptly at a change-point location. Both models were considered in various papers by several approaches, mostly from the perspective of extending smoothing approaches, such as kernel regression (Müller, 1992;Qiu, 1994;Eubank and Speckman, 1994;Qiu, 2003;Gijbels and Goderniaux, 2004), local polynomial regression (Loader, 1996;Spokoiny, 1998;Gijbels et al, , 2007Mboup et al, 2008;Xia and Qiu, 2015;Zhang, 2016), (smoothing) splines (Koo, 1997;Lee, 2002;Miyata and Shen, 2003;Huang and Qiu, 2010;Yang and Song, 2014;Liu et al, 2018), wavelet (Raimondo, 1998;Antoniadis and Gijbels, 2002;Abramovich et al, 2007) and Bayesian (Denison et al, 1998;Ogden, 1999;DiMatteo et al, 2001;Punskaya et al, 2002;Fearnhead, 2005;Moreno et al, 2013; approaches. A helpful review is given in (Qiu, 2005).…”

Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

“…Consequently, δ ( t ) is taken as the asymptotic identity in ( L 1 ; ∗) in the domain of generalized functions. Accordingly, the inverse of the conventional convolution discussed by, for instance, Mikusinski [112], Bracewell [115], Huang and Qiu [116], Abutaleb et al [117], Rhoads and Ekstrom [118], Todoeschuck and Jensen [119], and Moreau et al [120], exists because the necessary and sufficient condition that the inverse of an operation exists is that there exists the identity in that system, see, for example, Korn and Korn [121], Zhang [122], Riley et al [123], Bronshtein et al [124], and Stillwell [125], but it should be in the sense of generalized functions. As a matter of fact, the conventional convolution itself is in that sense, see, for example, Smith [126].…”

Deconvolution When Classifying Noisy Data Involving Transformations

“…This work includes contributions by Kundur and Hatzinakos (1998), Cannon (1976), Carasso (2001), Galatsanos et al (2002), Figueiredo and Nowak (2003), Joshi and Chaudhuri (2005), Hall and Qiu (2007a, b), Qiu (2008), Huang and Qiu (2010), and Popescu and Hellicar (2010). However, the problems of deconvolution and point spread function estimation are very different from those of classification, to such an extent that, even if T were known, the methods suggested in this article would still be recommended.…”