On the Schur Stability of Some Image Reconstruction Operators
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| Format: | Preprint |
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2025
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| author | Banerjee, Debraj Chaudhury, Kunal Narayan |
| author_facet | Banerjee, Debraj Chaudhury, Kunal Narayan |
| contents | We investigate an open problem arising in iterative image reconstruction. In its general form, the problem is to determine the stability of the parametric family of operators $P(t) = W (I-t B)$ and $R(t) = I-W + (I+tB)^{-1} (2W-I)$, where $W$ is a stochastic matrix and $B$ is a nonzero, nonnegative matrix. We prove that if $W$ is primitive, then there exists $T > 0$ such that the spectral radii $\varrho(P(t))$ and $\varrho(R(t))$ are strictly less than $1$ for all $0 < t < T$. The proof combines standard perturbation theory for eigenvalues and an observation about the analyticity of the spectral radius. This argument, however, does not provide an estimate of $T$. To this end, we compute $T$ explicitly for specific classes of $W$ and $B$. Building on these partial results and supporting numerical evidence, we conjecture that if $B$ is positive semidefinite and satisfies certain technical conditions, then $\varrho(P(t)), \, \varrho(R(t))<1$ for all $0 < t < 2/\varrho(B)$. As an application, we show how these results can be applied to establish the convergence of certain iterative imaging algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Schur Stability of Some Image Reconstruction Operators Banerjee, Debraj Chaudhury, Kunal Narayan Spectral Theory 15A45, 15B51, 47A55, 68U10, 94A08 We investigate an open problem arising in iterative image reconstruction. In its general form, the problem is to determine the stability of the parametric family of operators $P(t) = W (I-t B)$ and $R(t) = I-W + (I+tB)^{-1} (2W-I)$, where $W$ is a stochastic matrix and $B$ is a nonzero, nonnegative matrix. We prove that if $W$ is primitive, then there exists $T > 0$ such that the spectral radii $\varrho(P(t))$ and $\varrho(R(t))$ are strictly less than $1$ for all $0 < t < T$. The proof combines standard perturbation theory for eigenvalues and an observation about the analyticity of the spectral radius. This argument, however, does not provide an estimate of $T$. To this end, we compute $T$ explicitly for specific classes of $W$ and $B$. Building on these partial results and supporting numerical evidence, we conjecture that if $B$ is positive semidefinite and satisfies certain technical conditions, then $\varrho(P(t)), \, \varrho(R(t))<1$ for all $0 < t < 2/\varrho(B)$. As an application, we show how these results can be applied to establish the convergence of certain iterative imaging algorithms. |
| title | On the Schur Stability of Some Image Reconstruction Operators |
| topic | Spectral Theory 15A45, 15B51, 47A55, 68U10, 94A08 |
| url | https://arxiv.org/abs/2509.06715 |