Image-based immunoassays are designed to estimate the proportion of biological cells in a sample that generate a specific kind of particles. These assays are instrumental in biochemical, pharmacological and medical research, and have applications in disease diagnosis. In this talk, I describe the model, inverse problem, functional optimization framework, and algorithmic solution to analyze image-based immunoassays that we presented in  and . In particular, I will delve into 1) the radiation-diffusion-adsorption-desorption partial differential equation and a re-parametrization of its solution in terms of convolutional operators, 2) the set up, analysis and algorithmic solution of an optimization problem in Hilbert spaces to recover spatio-temporal information from a single image observation, and 3) the derivation of the proximal operator in function spaces for the non-negative group-sparsity regularizer. After discretization, our work results in a convergent, high-performing algorithm with 25 million optimization variables that requires the entire engineering toolbox of tips and tricks, and was recently incorporated in a commercial product . If time allows, I will introduce our work in , in which we use the structure of our algorithm to learn a faster, approximated solver for our optimization problem. : Pol del Aguila Pla and Joakim Jaldén, "Cell detection by functional inverse diffusion and non-negative group sparsity—Part I: Modeling and Inverse Problems", IEEE Transactions on Signal Processing, vol. 66, no. 20, pp. 5407–5421, 2018. Access at: https://doi.org/10.1109/TSP.2018.2868258 : Pol del Aguila Pla and Joakim Jaldén, "Cell detection by functional inverse diffusion and non-negative group sparsity—Part II: Proximal optimization and Performance Evaluation", IEEE Transactions on Signal Processing, vol. 66, no. 20, pp. 5422–5437, 2018. Access at: https://doi.org/10.1109/TSP.2018.2868256 : Mabtech Iris reader. See product page: https://www.mabtech.com/iris : Pol del Aguila Pla, Vidit Saxena, and Joakim Jaldén, "SpotNet – Learned iterations for cell detection in image-based immunoassays", 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019). Access at: https://arxiv.org/abs/1810.06132
Deep learning has emerged as a competitive new tool in image reconstruction. However, recent results demonstrate such methods are typically highly unstable – tiny, almost undetectable perturbations cause severe artefacts in the reconstruction, a major concern in practice. This is paradoxical given the existence of stable state-of-the-art methods for these problems. Thus, approximation theoretical results non-constructively imply the existence of stable and accurate neural networks. Hence the fundamental question: Can we explicitly construct/train stable and accurate neural networks for image reconstruction? I will discuss two results in this direction. The first is a negative result, saying such constructions are in general impossible, even given access to the solutions of common optimisation algorithms such as basis pursuit. The second is a positive result, saying that under sparsity assumptions, such neural networks can be constructed. These neural networks are stable and theoretically competitive with state-of-the-art results from other methods. Numerical examples of competitive performance are also provided.
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