On the Taut String Interpretation of the One-dimensional Rudin-Osher-Fatemi Model: A New Proof, a Fundamental Estimate and Some Applications
This work provides incremental theoretical insights for researchers in signal processing and mathematical imaging, clarifying foundational denoising models.
The paper tackles the equivalence between the Taut String Algorithm and the one-dimensional Rudin-Osher-Fatemi model for denoising, presenting a new elementary proof based on duality and Hilbert space projection, and derives a fundamental bound that ensures strong convergence of the denoised signal to the input as regularization vanishes.
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 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 bound on the denoised signal is derived. This bound implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the insignal as the regularization parameter vanishes. The methods developed in the paper can be modified to cover other interesting applications such as isotonic regression.