Re-thinking Richardson-Lucy without Iteration Cutoffs: Physically Motivated Bayesian Deconvolution (2411.00991v1)

Abstract: Richardson-Lucy deconvolution is widely used to restore images from degradation caused by the broadening effects of a point spread function and corruption by photon shot noise, in order to recover an underlying object. In practice, this is achieved by iteratively maximizing a Poisson emission likelihood. However, the RL algorithm is known to prefer sparse solutions and overfit noise, leading to high-frequency artifacts. The structure of these artifacts is sensitive to the number of RL iterations, and this parameter is typically hand-tuned to achieve reasonable perceptual quality of the inferred object. Overfitting can be mitigated by introducing tunable regularizers or other ad hoc iteration cutoffs in the optimization as otherwise incorporating fully realistic models can introduce computational bottlenecks. To resolve these problems, we present Bayesian deconvolution, a rigorous deconvolution framework that combines a physically accurate image formation model avoiding the challenges inherent to the RL approach. Our approach achieves deconvolution while satisfying the following desiderata: I deconvolution is performed in the spatial domain (as opposed to the frequency domain) where all known noise sources are accurately modeled and integrated in the spirit of providing full probability distributions over the density of the putative object recovered; II the probability distribution is estimated without making assumptions on the sparsity or continuity of the underlying object; III unsupervised inference is performed and converges to a stable solution with no user-dependent parameter tuning or iteration cutoff; IV deconvolution produces strictly positive solutions; and V implementation is amenable to fast, parallelizable computation.

• The paper introduces a Bayesian deconvolution framework that integrates a physically motivated noise model to overcome the limitations of traditional Richardson-Lucy deconvolution.
• It employs Monte Carlo sampling with a tailored prior to achieve parameter-free, stable image restoration while mitigating noise amplification and artifacts.
• Simulated and experimental results demonstrate superior contrast and artifact reduction, enhancing imaging quality in scientific applications.

Bayesian Deconvolution in Image Restoration: A Physically Motivated Approach

The paper "Re-thinking Richardson-Lucy without Iteration Cutoffs: Physically Motivated Bayesian Deconvolution" presents a novel approach to image deconvolution by addressing the limitations of the Richardson-Lucy (RL) algorithm. The authors propose a Bayesian deconvolution framework that integrates a physically motivated noise model and avoids traditional regularization techniques. This paper highlights key theoretical advancements and the practical implications of achieving more stable and parameter-free deconvolution results in the field of scientific imaging.

Overview of Classical Deconvolution Challenges

The RL algorithm, developed in the 1970s, provides an iterative method to restore images degraded by optical systems, especially where Poisson noise predominates. However, RL's propensity for overfitting, driven by its preference for sparse solutions, often results in amplified noise and image artifacts. Attempts to mitigate these issues through regularizers or iteration cutoffs have necessitated heuristics that complicate implementation, often requiring parameter tuning and potentially introducing computational bottlenecks.

Bayesian Deconvolution Framework

Responding to RL's limitations, the authors introduce a Bayesian framework that incorporates a comprehensive noise model without the need for tuning hyperparameters or relying on ad hoc iteration cutoffs. The key features of their approach are:

1. Integration of all known noise sources to provide accurate probability distributions over the object density being recovered.
2. Estimation of probability distributions without assumptions on sparsity or continuity of the underlying object, avoiding biases inherent in prior deconvolution methods.
3. Unsupervised inference that converges autonomously to stable solutions.
4. Ensuring strictly positive deconvolution outcomes, aligning with physical expectations.
5. A framework conducive to fast, parallelizable computation, enhancing its suitability for real-time applications.

Methodological Insights and Results

The Bayesian deconvolution strategy leverages Monte Carlo sampling to estimate the posterior probability distribution of the object being imaged, incorporating both the likelihood and prior probability distributions. The prior is designed to penalize frequencies beyond the system's bandpass while promoting a frequency spectrum anticipated by the optical transfer function (OTF). This physically motivated prior facilitates robust noise handling and reduces overfitting to high-frequency noise artifacts.

Critically, the authors demonstrate the efficacy of their approach through simulations and applications to experimental data. In simulated data, the Bayesian method converges to the diffraction-limited ground truth without the artifacts typical of RL results under similar conditions. For experimental images of mitochondria in HeLa cells, the approach circumvents RL's deterioration at later iterations, instead delivering high-contrast reconstructions without resorting to regularization.

Implications and Future Developments

The paper's contributions lie in its elimination of user-dependent tuning requirements, paving the way for more consistent application across varied imaging contexts. The method's adaptability to alternative noise models, potentially including EMCCD camera characteristics, is suggested as a future enhancement. The fusion of Bayesian inferential logic with parallelized computational strategies signifies a step forward in imaging technologies, reducing the computational overhead while delivering superior image restoration quality.

Looking ahead, extensions of this methodology could positively impact other computational imaging modalities relying on PSFs with limited dimensions. The ability to generalize this framework promises substantial improvements in fields like structured illumination microscopy and Fourier ptychography, which can benefit from the robust noise modeling and artifact reduction highlighted in this work.

Overall, the proposed Bayesian deconvolution framework stands as an innovative advancement in image restoration, addressing longstanding challenges associated with RL and promoting a rigorous, physically grounded approach that holds promise for enhancing diverse scientific imaging applications.