A smoothing operator polynomial in the Hilbert spaceH λ

1995 ◽  
Vol 77 (5) ◽  
pp. 3415-3419
Author(s):  
V. V. Khlobystov
Keyword(s):  
Operator Polynomial ◽  
Smoothing Operator ◽  
Hilbert Spaceh

scholarly journals On the functional Hodrick–Prescott filter with non-compact operators

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Boualem Djehiche ◽  
Astrid Hilbert ◽  
Hiba Nassar
Keyword(s):  
Compact Operators ◽  
Closed Range ◽  
Underlying Distribution ◽  
Smoothing Operator ◽  
Optimal Smoothing ◽  
Compact Case ◽  
Densely Defined

AbstractWe study a version of the functional Hodrick–Prescott filter in the case when the associated operator is not necessarily compact but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case when the underlying distribution of the data is Gaussian.

scholarly journals A functional Hodrick–Prescott filter

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Boualem Djehiche ◽  
Hiba Nassar
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Functional Data ◽  
Separable Hilbert Space ◽  
Underlying Distribution ◽  
Smoothing Operator ◽  
Infinite Dimensional ◽  
Optimal Smoothing ◽  
Functional Version

AbstractWe propose a functional version of the Hodrick–Prescott filter for functional data which take values in an infinite-dimensional separable Hilbert space. We further characterize the associated optimal smoothing operator when the associated linear operator is compact and the underlying distribution of the data is Gaussian.

Smoothing imaging condition for shot-profile migration

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S149-S154 ◽  
Author(s):  
Antoine Guitton ◽  
Alejandro Valenciano ◽  
Dimitri Bevc ◽  
Jon Claerbout
Keyword(s):  
Time Domain ◽  
Signal To Noise Ratio ◽  
Wiener Filter ◽  
Window Size ◽  
Trial And Error ◽  
Data Set ◽  
Smoothing Operator ◽  
Imaging Condition ◽  
Damping Parameter ◽  
The Time Domain

Amplitudes in shot-profile migration can be improved if the imaging condition incorporates a division (deconvolution in the time domain) of the upgoing wavefield by the downgoing wavefield. This division can be enhanced by introducing an optimal Wiener filter which assumes that the noise present in the data has a white spectrum. This assumption requires a damping parameter, related to the signal-to-noise ratio, often chosen by trial and error. In practice, the damping parameter replaces the small values of the spectrum of the downgoing wavefield and avoids division by zero. The migration results can be quite sensitive to the damping parameter, and in most applications, the upgoing and downgoing wavefields are simply multiplied. Alternatively, the division can be made stable by filling the small values of thespectrum with an average of the neighboring points. This averaging is obtained by running a smoothing operator on the spectrum of the downgoing wavefield. This operation called the smoothing imaging condition. Our results show that where the spectrum of the downgoing wavefield is high, the imaging condition with damping and smoothing yields similar results, thus correcting for illumination effects. Where the spectrum is low, the smoothing imaging condition tends to be more robust to the noise level present in the data, thus giving better images than the imaging condition with damping. In addition, our experiments indicate that the parameterization of the smoothing imaging condition, i.e., choice of window size for the smoothing operator, is easy and repeatable from one data set to another, making it a valuable addition to our imaging toolbox.

scholarly journals A Generalized Framework for Edge-Preserving and Structure-Preserving Image Smoothing

2020 ◽  
Vol 34 (07) ◽  
pp. 11620-11628
Author(s):  
Wei Liu ◽  
Pingping Zhang ◽  
Yinjie Lei ◽  
Xiaolin Huang ◽  
Jie Yang ◽  
...  
Keyword(s):  
Penalty Function ◽  
State Of The Art ◽  
Superior Performance ◽  
Image Smoothing ◽  
Edge Preserving ◽  
Structure Preserving ◽  
Smoothing Operator ◽  
Optimization Framework ◽  
Generalized Framework ◽  
Smooth Optimization

Image smoothing is a fundamental procedure in applications of both computer vision and graphics. The required smoothing properties can be different or even contradictive among different tasks. Nevertheless, the inherent smoothing nature of one smoothing operator is usually fixed and thus cannot meet the various requirements of different applications. In this paper, a non-convex non-smooth optimization framework is proposed to achieve diverse smoothing natures where even contradictive smoothing behaviors can be achieved. To this end, we first introduce the truncated Huber penalty function which has seldom been used in image smoothing. A robust framework is then proposed. When combined with the strong flexibility of the truncated Huber penalty function, our framework is capable of a range of applications and can outperform the state-of-the-art approaches in several tasks. In addition, an efficient numerical solution is provided and its convergence is theoretically guaranteed even the optimization framework is non-convex and non-smooth. The effectiveness and superior performance of our approach are validated through comprehensive experimental results in a range of applications.

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