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On the optimality of convergence conditions for multiscale decompositions in imaging and inverse problems

Simone Rebegoldi, Luca Rondi

TL;DR

The paper investigates when the classical multiscale decomposition for inverse problems converges in the unknowns space, revealing that linear problems with Hilbert-norm regularization guarantee subsequence convergence and improved stability, while purely Banach-norm regularization may fail even in linear settings. It introduces a Parseval-like energy decomposition and dual-characterizations of minimizers in the linear case, connecting to BV–L^2 decompositions. Under Hilbert-norm regularization, convergence to a unique optimal solution is achieved as the scale parameter grows, and numerical experiments on image deblurring confirm stability and accuracy advantages over single-step regularization. However, the authors provide several counterexamples showing that, without Hilbert regularization or a tighter multiscale scheme, the classical procedure can fail to converge, underscoring the practical value of the Hilbert-norm approach for robust convergence in imaging inverse problems.

Abstract

We consider the multiscale procedure developed by Modin, Nachman and Rondi, Adv. Math. (2019), for inverse problems, which was inspired by the multiscale decomposition of images by Tadmor, Nezzar and Vese, Multiscale Model. Simul. (2004). We investigate under which assumptions this classical procedure is enough to have convergence in the unknowns space without resorting to use the tighter multiscale procedure from the same paper. We show that this is the case for linear inverse problems when the regularization is given by the norm of a Hilbert space. Moreover, in this setting the multiscale procedure improves the stability of the reconstruction. On the other hand, we show that, for the classical multiscale procedure, convergence in the unknowns space might fail even for the linear case with a Banach norm as regularization.

On the optimality of convergence conditions for multiscale decompositions in imaging and inverse problems

TL;DR

The paper investigates when the classical multiscale decomposition for inverse problems converges in the unknowns space, revealing that linear problems with Hilbert-norm regularization guarantee subsequence convergence and improved stability, while purely Banach-norm regularization may fail even in linear settings. It introduces a Parseval-like energy decomposition and dual-characterizations of minimizers in the linear case, connecting to BV–L^2 decompositions. Under Hilbert-norm regularization, convergence to a unique optimal solution is achieved as the scale parameter grows, and numerical experiments on image deblurring confirm stability and accuracy advantages over single-step regularization. However, the authors provide several counterexamples showing that, without Hilbert regularization or a tighter multiscale scheme, the classical procedure can fail to converge, underscoring the practical value of the Hilbert-norm approach for robust convergence in imaging inverse problems.

Abstract

We consider the multiscale procedure developed by Modin, Nachman and Rondi, Adv. Math. (2019), for inverse problems, which was inspired by the multiscale decomposition of images by Tadmor, Nezzar and Vese, Multiscale Model. Simul. (2004). We investigate under which assumptions this classical procedure is enough to have convergence in the unknowns space without resorting to use the tighter multiscale procedure from the same paper. We show that this is the case for linear inverse problems when the regularization is given by the norm of a Hilbert space. Moreover, in this setting the multiscale procedure improves the stability of the reconstruction. On the other hand, we show that, for the classical multiscale procedure, convergence in the unknowns space might fail even for the linear case with a Banach norm as regularization.

Paper Structure

This paper contains 14 sections, 14 theorems, 215 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The linear case
  3. Parseval-like identity
  4. Characterization of minimizers
  5. Applications to the classical ($BV$-$L^2$) decompositions
  6. Regularization by Hilbert norms
  7. Convergence results
  8. Improvement of the stability of the reconstruction
  9. Image deblurring with Hilbert regularization: numerical illustration
  10. Counterexamples
  11. First counterexample for sequences spaces
  12. Second counterexample for sequences spaces
  13. A not converging multiscale decomposition for a bounded linear blurring operator on the line
  14. Conclusions

Key Result

Theorem 1.1

Assume that Assume also that and that Assumption 1 holds. If then for the multiscale sequence $\{\sigma_n\}_{n\geq 0}$ given by (regularizedpbm), (inductiveconstr) and (partialsum1) we have $\varepsilon_0=\delta_0$, that is,

Figures (5)

  • Figure 1: Test images. Crab Nebula NGC 1952 (left) and planetary Nebula NGC 7027 (right).
  • Figure 2: Noiseless tests. Zoomed details of the Crab Nebula NGC 1952 (first row) and the planetary Nebula NGC 7027 (second row). From left to right: blurred image, best deblurred image with single-step procedure, and best deblurred image with multiscale procedure.
  • Figure 3: Noisy tests. Zoomed details of the Crab Nebula NGC 1952 (first row) and the planetary Nebula NGC 7027 (second row). From left to right: blurred image, best deblurred image with single-step procedure, and best deblurred image with multiscale procedure.
  • Figure 4: Noiseless tests. Decrease of the relative error $\|\sigma_{h,n}-\hat{\sigma}_{h}\|/\|\hat{\sigma}_{h}\|$ for the multiscale method (blue, dashed line) and $\|\tilde{\sigma}_{h,n}-\hat{\sigma}_{h}\|/\|\hat{\sigma}_{h}\|$ for the single-step regularization method (red, solid line), with respect to the iteration number. Left: Nebula NGC 1952. Right: Nebula NGC 7027.
  • Figure 5: Noisy tests. Decrease of the relative error $\|\sigma_{h,n}-\hat{\sigma}_{h}\|/\|\hat{\sigma}_{h}\|$ for the multiscale method (blue, dashed line) and $\|\tilde{\sigma}_{h,n}-\hat{\sigma}_{h}\|/\|\hat{\sigma}_{h}\|$ for the single-step regularization method (red, solid line), with respect to the iteration number. Left: Nebula NGC 1952. Right: Nebula NGC 7027.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • ...and 29 more