1 research outputs found
Higher-order total variation approaches and generalisations
Over the last decades, the total variation (TV) evolved to one of the most
broadly-used regularisation functionals for inverse problems, in particular for
imaging applications. When first introduced as a regulariser, higher-order
generalisations of TV were soon proposed and studied with increasing interest,
which led to a variety of different approaches being available today. We review
several of these approaches, discussing aspects ranging from
functional-analytic foundations to regularisation theory for linear inverse
problems in Banach space, and provide a unified framework concerning
well-posedness and convergence for vanishing noise level for respective
Tikhonov regularisation. This includes general higher orders of TV, additive
and infimal-convolution multi-order total variation, total generalised
variation (TGV), and beyond. Further, numerical optimisation algorithms are
developed and discussed that are suitable for solving the Tikhonov minimisation
problem for all presented models. Focus is laid in particular on covering the
whole pipeline starting at the discretisation of the problem and ending at
concrete, implementable iterative procedures. A major part of this review is
finally concerned with presenting examples and applications where higher-order
TV approaches turned out to be beneficial. These applications range from
classical inverse problems in imaging such as denoising, deconvolution,
compressed sensing, optical-flow estimation and decompression, to image
reconstruction in medical imaging and beyond, including magnetic resonance
imaging (MRI), computed tomography (CT), magnetic-resonance positron emission
tomography (MR-PET), and electron tomography