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

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 second part, we focus on our algorithmic contributions. We provide an algorithm for functional inverse diffusion that solves the variational problem we posed in Part I. As part of the derivation of this algorithm, we present the proximal operator for the non-negative group-sparsity regularizer, which is a novel result that is of interest in itself, also in comparison to previous results on the proximal operator of a sum of functions. We then present a discretized approximated implementation of our algorithm and evaluate it both in terms of operational cell-detection metrics and in terms of distributional optimal-transport metrics.

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

“…which is convex and suited to iterative non-smooth convex optimization methods, as we show in Part II [13]. Here, for each r ∈ R 2 , a r : [0, σ max ] → R + is such that a r (σ) = a(r, σ) for any σ ∈ [0, σ max ], λ > 0 is the regularization parameter, and ξ ∈ L ∞ + [0, σ max ] is a non-negative bounded weighting function in σ that can be used to incorporate further prior knowledge.…”

“…5], we present an optimization problem to address inverse diffusion on a functional (infinitedimensional) setting and discretize the problem only after that. Similarly, in Part II [13], we propose first the functional version of the algorithm, and use the simple discretization in Section IV only after that to provide an implementable algorithm and empirical results.…”

“…Finally, if one wanted to impose the restrictions of the model only on a certain range of σs, say σ ∈ ℵ ⊂ [0, σ max ], and relax them for its complement ℵ c = [0, σ max ] \ ℵ, one could choose ξ such that supp (ξ) = ℵ. The case in which ξ is simply the (0, 1)-indicator function of a set ℵ is of special relevance in Part II [13] due to its tractability, and is useful, for example, to use the values of a for σ ∈ ℵ c to account for a low-frequency background that could not be explained by cell secretion alone.…”

“…Then, we present a discretization scheme that allows both for the synthesis of realistic data and for numerical solutions to the proposed optimization problem. Part II of this paper [13] is devoted to algorithmic solutions to solve this optimization problem.…”

“…In finalizing this first part, we provide results on real data by comparing our solution to the labeling of a human expert. In Part II of this paper [13], we provide a thorough evaluation of the proposed methodology using synthetic data. SL on 2D or 3D data from linear observation models has been widely studied for biologic [14]- [22], astronomic [23], [24], acoustic [25], [26], heat conduction [27], [28], and environmental applications [29]- [31], as well as in more generic settings [32]- [36].…”