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

Abstract

In this two-part paper, we present a novel framework and methodology to analyze data from certain imagebased 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.

Cell Detection by Functional Inverse Diffusion and Non-negative Group Sparsity—Part II: Proximal Optimization and Performance Evaluation

“…Digital Object Identifier 10.1109/TSP.2018.2868256 is measured. For more details on the exact setup, its biological application or the physics involved, see Part I [19,Section II].…”

“…In Part I [19,Section III], we proposed the following optimization problem to detect particle-generating (active) cells in this setting,…”

“…and w ∈ L ∞ + R 2 is a non-negative bounded weighting function that penalizes errors at different locations according to sensor properties. The bounded linear operator A ∈ L (A, D) represents the forward diffusion process, that maps an a onto d obs , and was derived in [19, as the mapping a → σ m a x 0 G σ a σ dσ, where G σ is the convolutional operator with 2D rotationally invariant Gaussian kernel of standard deviation σ. The diffusion operator A was extensively characterized in Part I [19,.…”

“…The bounded linear operator A ∈ L (A, D) represents the forward diffusion process, that maps an a onto d obs , and was derived in [19, as the mapping a → σ m a x 0 G σ a σ dσ, where G σ is the convolutional operator with 2D rotationally invariant Gaussian kernel of standard deviation σ. The diffusion operator A was extensively characterized in Part I [19,. Finally, the group-sparsity regularizer includes a non-negative bounded weighting function ξ ∈ L ∞ + [0, σ max ] that allows incorporating further prior knowledge in terms of the relative importance of each value of σ.…”

“…In this paper, we derive an accelerated proximal gradient (APG) algorithm to solve (1). Namely, we combine the characterization of the diffusion operator A we presented in Part I [19, with the derivation of the proximal operator of the non-negative group-sparsity regularizer, i.e., of f in (1). Furthermore, we present an efficient implementation of a discretization of the resulting algorithm and provide thorough performance evaluation on synthetic data, complementing the real data example in Part I [19,.…”