Author:
Niels Chr Overgaard
Affiliation:
Lund University, Sweden
Keyword(s):
Total Variation Minimization, Taut String, Denoising, Regression Splines, Lewy–Stampacchia Inequality, Sub-modularity, Moreau-Yosida Approximation, Isotonic Regression.
Related
Ontology
Subjects/Areas/Topics:
Applications
;
Cardiovascular Imaging and Cardiography
;
Cardiovascular Technologies
;
Health Engineering and Technology Applications
;
Pattern Recognition
;
Regression
;
Signal Processing
;
Software Engineering
;
Sparsity
;
Theory and Methods
Abstract:
A new proof of the equivalence of the Taut String Algorithm and the one-dimensional Rudin–Osher–Fatemi
model is presented. Based on duality and the projection theorem in Hilbert space, the proof is strictly elementary.
Existence and uniqueness of solutions (in the continuous case) to both denoising models follow as
by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends
to zero, are recalled and efficient proofs provided. Moreover, a new and fundamental estimate on the denoised
signal is derived. It implies, among other things, the strong convergence (in the space of functions of bounded
variation) of the denoised signal to the in-signal as the regularization parameter vanishes.