Cell Detection by Functional Inverse Diffusion and Non-negative Group Sparsity$-$Part I: Modeling and Inverse Problems

TL;DR

This paper introduces a novel PDE-based framework with non-negative group sparsity regularization for localizing biological cells in biochemical assay images, combining modeling, inverse problems, and algorithm development.

Contribution

It presents a new PDE model, a parametrization approach, and a regularized optimization formulation for cell detection in biochemical images, along with discretization and preliminary validation.

Findings

Preliminary comparison shows promising results against human labels.

Proposed discretization enables synthetic data generation and algorithm implementation.

Framework effectively addresses inverse diffusion in biochemical assays.

Abstract

In this two-part paper, we present a novel framework and methodology to analyze data from certain image-based biochemical assays, e.g., ELISPOT and Fluorospot assays. In this first part, we start by presenting a physical partial differential equations (PDE) model up to image acquisition for these biochemical assays. Then, we use the PDEs' Green function to derive a novel parametrization of the acquired images. This parametrization allows us to propose a functional optimization problem to address inverse diffusion. In particular, we propose a non-negative group-sparsity regularized optimization problem with the goal of localizing and characterizing the biological cells involved in the said assays. We continue by proposing a suitable discretization scheme that enables both the generation of synthetic data and implementable algorithms to address inverse diffusion. We end Part I by providing a preliminary comparison between the results of our methodology and an expert human labeler on real data. Part II is devoted to providing an accelerated proximal gradient algorithm to solve the proposed problem and to the empirical validation of our methodology.