Core Concepts

Sparse signals can be recovered from a limited number of measurements, proportional to the signal sparsity, by leveraging compressed sensing techniques in the context of ill-posed inverse problems. This is demonstrated for the sparse Radon transform, which models computed tomography, where stable recovery is achieved under the condition that the number of angles is proportional to the signal sparsity.

Abstract

The paper develops a general theory of infinite-dimensional compressed sensing for abstract ill-posed inverse problems, involving an arbitrary forward operator. The key ideas are:
Introducing a generalized restricted isometry property (g-RIP) and a quasi-diagonalization property of the forward map to handle ill-posedness.
Providing recovery guarantees for the ℓ1-minimization problem, showing that the number of samples required is proportional to the signal sparsity, up to logarithmic factors.
As a notable application, the authors obtain rigorous recovery estimates for the sparse Radon transform, in both the parallel-beam and fan-beam settings. Assuming the unknown signal is s-sparse with respect to a wavelet basis, they prove stable recovery under the condition that the number of angles m satisfies m ≳ s, up to logarithmic factors.
The authors also discuss how to further optimize the recovery estimates to depend only on the noise level and the number of samples, under suitable assumptions on the signal regularity. For instance, for cartoon-like images, the reconstruction error decays as β^(1/2), where β is the noise level, provided that the number of samples is proportional to β^(-2).

Stats

The number of angles m required for stable recovery of an s-sparse signal satisfies m ≳ s, up to logarithmic factors.

Quotes

"Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors."
"For the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles θ1, . . . , θm), which models computed tomography, in both the parallel-beam and the fan-beam settings."

Key Insights Distilled From

by Giovanni S. ... at **arxiv.org** 05-06-2024

Deeper Inquiries

In order to extend the proposed framework to handle nonlinear inverse problems, several modifications and adaptations would be necessary. Here are some key steps that could be taken:
Nonlinear Measurement Operators: The framework would need to accommodate nonlinear measurement operators, as linear operators are not sufficient for modeling nonlinear phenomena. This would involve redefining the forward map to incorporate nonlinear transformations of the input signal.
Non-Convex Optimization: Nonlinear inverse problems often involve non-convex optimization due to the presence of nonlinearities. Techniques from non-convex optimization theory would need to be integrated into the framework to handle such optimization challenges.
Regularization Techniques: Nonlinear inverse problems are prone to overfitting and instability. Regularization techniques such as Tikhonov regularization, total variation regularization, or sparsity-promoting regularization could be incorporated to stabilize the reconstruction process.
Statistical Learning Theory: Leveraging concepts from statistical learning theory could enhance the framework's ability to handle nonlinearities and uncertainties in the data. This would involve incorporating probabilistic models and Bayesian inference methods.
Adaptive Sampling Strategies: Nonlinear inverse problems may benefit from adaptive sampling strategies that intelligently select measurement points based on the current estimate of the signal. This adaptive approach can improve reconstruction accuracy and efficiency.
By incorporating these elements and adapting the framework to accommodate the complexities of nonlinear inverse problems, it can be extended to handle a broader range of real-world applications.

Considering statistical noise models instead of deterministic noise has several implications for the analysis and interpretation of the results in the context of compressed sensing for inverse problems:
Robustness to Uncertainty: Statistical noise models capture the inherent uncertainty and variability in real-world data. By accounting for stochastic variations in the measurements, the framework becomes more robust to noise and outliers in the data.
Probabilistic Guarantees: Statistical noise models enable the derivation of probabilistic guarantees for the reconstruction process. Instead of deterministic error bounds, the framework can provide confidence intervals and probabilistic statements about the accuracy of the recovered signal.
Bayesian Inference: Statistical noise models open up the possibility of applying Bayesian inference techniques to the inverse problem. Bayesian methods allow for the incorporation of prior knowledge, regularization, and uncertainty quantification in the reconstruction process.
Modeling Complex Noise Structures: Statistical noise models can capture complex noise structures such as correlated noise, heteroscedastic noise, or non-Gaussian noise distributions. This allows for more realistic modeling of noise in practical scenarios.
Optimization under Uncertainty: Optimization algorithms can be adapted to optimize the reconstruction process under uncertainty from statistical noise models. This leads to more robust and adaptive reconstruction methods.
By considering statistical noise models, the framework gains a more realistic and data-driven perspective, leading to improved performance and reliability in solving inverse problems.

Considering more general dictionaries beyond orthonormal wavelet bases can indeed lead to further improvements in the results of the framework for compressed sensing in inverse problems. Here are some ways in which the results can be enhanced by incorporating more general dictionaries:
Adaptive Signal Representations: General dictionaries allow for adaptive signal representations tailored to the specific characteristics of the data. By choosing dictionaries that better capture the structure of the signal, the reconstruction accuracy can be significantly improved.
Sparse Signal Recovery: More general dictionaries, such as redundant dictionaries or learned dictionaries, can better promote sparsity in the signal representation. This can lead to more accurate and stable recovery of sparse signals from limited measurements.
Incorporating Redundancy: Redundant dictionaries can capture redundancies in the signal that may not be captured by orthonormal bases. This redundancy can improve the stability of the reconstruction process and enhance the robustness to noise.
Handling Nonlinearities: General dictionaries can better handle nonlinear relationships in the data, allowing for more flexible modeling of complex signals. This can be particularly beneficial in scenarios where the signal exhibits nonlinear behavior.
Domain-Specific Features: Tailoring the dictionary to domain-specific features of the signal can lead to more interpretable and meaningful reconstructions. By incorporating domain knowledge into the dictionary design, the framework can achieve more insightful results.
Incorporating more general dictionaries beyond orthonormal wavelet bases can therefore enhance the flexibility, accuracy, and robustness of the framework for compressed sensing in inverse problems, opening up new possibilities for solving challenging real-world problems.

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