Avoiding subtraction and division of stochastic signals using normalizing flows: NFdeconvolve

Across the scientific realm, we find ourselves subtracting or dividing stochastic signals. For instance, consider a stochastic realization, $x$, generated from the addition or multiplication of two stochastic signals $a$ and $b$, namely $x=a+b$ or $x = ab$. For the $x=a+b$ example, $a$ can be fluorescence background and $b$ the signal of interest whose statistics are to be learned from the measured $x$. Similarly, when writing $x=ab$, $a$ can be thought of as the illumination intensity and $b$ the density of fluorescent molecules of interest. Yet dividing or subtracting stochastic signals amplifies noise, and we ask instead whether, using the statistics of $a$ and the measurement of $x$ as input, we can recover the statistics of $b$. Here, we show how normalizing flows can generate an approximation of the probability distribution over $b$, thereby avoiding subtraction or division altogether. This method is implemented in our software package, NFdeconvolve, available on GitHub with a tutorial linked in the main text.

Re-thinking Richardson-Lucy without Iteration Cutoffs: Physically Motivated Bayesian Deconvolution

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.