scholarly journals Error Estimations for Total Variation Type Regularization

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1373
Author(s):  
Kuan Li ◽  
Chun Huang ◽  
Ziyang Yuan
Keyword(s):  
Machine Learning ◽  
Convergence Rate ◽  
Total Variation ◽  
A Priori ◽  
A Posteriori ◽  
Imaging Processing ◽  
Mild Conditions ◽  
Tv Regularization ◽  
Regularization Problem ◽  
Error Estimations

This paper provides several error estimations for total variation (TV) type regularization, which arises in a series of areas, for instance, signal and imaging processing, machine learning, etc. In this paper, some basic properties of the minimizer for the TV regularization problem such as stability, consistency and convergence rate are fully investigated. Both a priori and a posteriori rules are considered in this paper. Furthermore, an improved convergence rate is given based on the sparsity assumption. The problem under the condition of non-sparsity, which is common in practice, is also discussed; the results of the corresponding convergence rate are also presented under certain mild conditions.

scholarly journals Tube-Based Taut String Algorithms for Total Variation Regularization

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1141
Author(s):  
Artyom Makovetskii ◽  
Sergei Voronin ◽  
Vitaly Kober ◽  
Aleksei Voronin
Keyword(s):  
Exact Solutions ◽  
Total Variation ◽  
Total Variation Regularization ◽  
Geometric Description ◽  
String Method ◽  
String Algorithms ◽  
Practical Applications ◽  
Taut String ◽  
Tv Regularization ◽  
Regularization Problem

Removing noise from signals using total variation regularization is a challenging signal processing problem arising in many practical applications. The taut string method is one of the most efficient approaches for solving the 1D TV regularization problem. In this paper we propose a geometric description of the linearized taut string method. This geometric description leads to the notion of the “tube”. We propose three tube-based taut string algorithms for total variation regularization. Different weight functionals can be used in the 1D TV regularization that lead to different types of tubes. We consider uniform, vertically nonuniform, vertically and horizontally nonuniform tubes. The proposed geometric approach is used to speed-up TV regularization processing by dividing the tubes into subtubes and using parallel processing. We introduce the concept of a relatively convex tube and describe the relationship between the geometric characteristics of tubes and exact solutions to the TV regularization. The properties of exact solutions can also be used to design efficient algorithms for solving the TV regularization problem. The performance of the proposed algorithms is discussed and illustrated by computer simulation.

Error Estimations in Linear Inverse Problems With a Priori Information

2011 ◽  
Author(s):  
Anatoly G. Yagola ◽  
Yury M. Korolev
Keyword(s):  
Exact Solution ◽  
A Priori ◽  
Posteriori Error ◽  
Posteriori Error Estimate ◽  
Compact Set ◽  
A Posteriori ◽  
A Priori Information ◽  
A Posteriori Error ◽  
Error Estimations ◽  
Priori Information

We consider an inverse problem for an operator equation Az = u. The exact operator A and the exact right-hand side u are unknown. Only their upper and lower estimations are available. We provide techniques of calculating upper and lower estimations for the exact solution belonging to a compact set in this case, as well as a posteriori error estimations. We obtain approximate solutions with an optimal a posteriori error estimate. We also make use of a priori information about the exact solution, e.g. its monotonicity and convexity. The developed software package was applied to solving practical ill-posed problems.

A priori/a posteriori

2017 ◽  
Author(s):  
Heinrich Schepers ◽  
Giorgio Tonelli ◽  
Rudolf Eisler
Keyword(s):  
A Priori ◽  
A Posteriori

A Critique of Modernist Synthesis in Islamic Thought

1994 ◽  
Vol 11 (4) ◽  
pp. 475-503
Author(s):  
Masudul Alum Choudhury
Keyword(s):  
Scientific Inquiry ◽  
A Priori ◽  
Scientific Investigation ◽  
Muslim World ◽  
A Posteriori ◽  
Islamic Thought ◽  
And Behavior ◽  
Muslim Scholars

Is it the realm of theoretical constructs or positive applications thatdefines the essence of scientific inquiry? Is there unison between thenormative and the positive, between the inductive and deductivecontents, between perception and reality, between the micro- andmacro-phenomena of reality as technically understood? In short, isthere a possibility for unification of knowledge in modernist epistemologicalcomprehension? Is knowledge perceived in conceptionand application as systemic dichotomy between the purely epistemic(in the metaphysically a priori sense) and the purely ontic (in thepurely positivistically a posteriori sense) at all a reflection of reality?Is knowledge possible in such a dichotomy or plurality?Answers to these foundational questions are primal in order tounderstand a critique of modernist synthesis in Islamic thought thathas been raging among Muslim scholars for some time now. Theconsequences emanating from the modernist approach underlie muchof the nature of development in methodology, thinking, institutions,and behavior in the Muslim world throughout its history. They arefound to pervade more intensively, I will argue here, as the consequenceof a taqlid of modernism among Islamic thinkers. I will thenargue that this debility has arisen not because of a comparativemodem scientific investigation, but due to a failure to fathom theuniqueness of a truly Qur'anic epistemological inquiry in the understandingof the nature of the Islamic socioscientific worldview ...

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