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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, computed tomography, magnetic-resonance positron emission tomography, and electron tomography.
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A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation, and beyond. 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. 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. Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. A real-time, memory e cient version of the decoder was implemented, and is available as a plug-in for popular web browsers. A typical magazine page in color at 300dpi can be compressed down to between 40 to 60 KB, approximately 5 to 10 times better than JPEG for a similar level of subjective quality.
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