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Plug and Play Algorithms for Imaging

Hughes, Alexander LU (2024) In Bachelor’s Theses in Mathematical Sciences MATK11 20232
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
In this thesis, we explore the Alternating Direction Method of Multipliers (ADMM) algorithm, the Plug & Play (PnP) algorithm and consensus equilbrium in order to understand their importance for solving complex imaging problems. We begin by motivating the invention of ADMM which combines the strengths of dual ascent and the method of multipliers. A proof of convergence for ADMM is then provided. Afterwards we go on to show how ADMM can be used to solve the global consensus optimisation problem and how PnP builds upon ADMM and extends its capabilities. However, using PnP means we no longer have guaranteed convergence. To remedy this the Consensus Equilibrium (CE) framework is introduced, which is an optimisation free framework that... (More)
In this thesis, we explore the Alternating Direction Method of Multipliers (ADMM) algorithm, the Plug & Play (PnP) algorithm and consensus equilbrium in order to understand their importance for solving complex imaging problems. We begin by motivating the invention of ADMM which combines the strengths of dual ascent and the method of multipliers. A proof of convergence for ADMM is then provided. Afterwards we go on to show how ADMM can be used to solve the global consensus optimisation problem and how PnP builds upon ADMM and extends its capabilities. However, using PnP means we no longer have guaranteed convergence. To remedy this the Consensus Equilibrium (CE) framework is introduced, which is an optimisation free framework that generalises the global consensus optimisation problem. The thesis concludes by showing ADMM as a special case of the CE equations. (Less)
Popular Abstract (Swedish)
I den digitala tidsåldern spelar invecklad matematik ofta en undanskymd, men ändå avgörande roll. När vi löser bildproblem, som att avpixla foton och förbättra upplösningen, använder vi ofta komplex matematik. I den här avhandlingen vill vi lyfta fram den genom att beskriva tre viktiga algoritmer - ADMM, Plug and Play och Consensus Equilibrium. Denna avhandling utforskar hur dessa algoritmer bygger på varandra, deras sammankopplingar och deras underliggande betydelse.
Please use this url to cite or link to this publication:
author
Hughes, Alexander LU
supervisor
organization
alternative title
Plug and Play-algoritmer för bildbehandling
course
MATK11 20232
year
type
M2 - Bachelor Degree
subject
keywords
ADMM, Plug & Play algorithm, consensus equilibrium, consensus optimisation
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4154-2024
ISSN
1654-6229
other publication id
2024:K1
language
English
id
9144589
date added to LUP
2025-06-27 15:57:06
date last changed
2025-06-27 15:57:06
@misc{9144589,
  abstract     = {{In this thesis, we explore the Alternating Direction Method of Multipliers (ADMM) algorithm, the Plug & Play (PnP) algorithm and consensus equilbrium in order to understand their importance for solving complex imaging problems. We begin by motivating the invention of ADMM which combines the strengths of dual ascent and the method of multipliers. A proof of convergence for ADMM is then provided. Afterwards we go on to show how ADMM can be used to solve the global consensus optimisation problem and how PnP builds upon ADMM and extends its capabilities. However, using PnP means we no longer have guaranteed convergence. To remedy this the Consensus Equilibrium (CE) framework is introduced, which is an optimisation free framework that generalises the global consensus optimisation problem. The thesis concludes by showing ADMM as a special case of the CE equations.}},
  author       = {{Hughes, Alexander}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{Plug and Play Algorithms for Imaging}},
  year         = {{2024}},
}