Bright objects on a dark background, such as cells in microscopy images, can sometimes be modeled as maxima of sufficient dynamic, called h-maxima. Such a model could be sufficient to count these objects in images, provided we know the dynamic threshold that tells apart actual objects from irrelevant maxima. In a previous study we introduced a neural architecture that includes a morphological pipeline counting the number of h-maxima in an image, preceded by a classical CNN which estimates the dynamic h yielding the right number of objects. This was made possible by geodesic reconstruction layers, and a new module counting connected components. The resulting architecture could be trained end-to-end to count cells by minimizing a loss function focusing mainly on an intermediate image and very marginally on the counting errors. In the present paper, we extend that work with a detailed theoretical analysis of the derivability of such pipeline, and with more cell counting experiments on different datasets. Our analysis explains why it is difficult to train the proposed pipeline when penalizing counting errors, and ends up with a solution to that challenge. Our experiments show that our approach can obtain results comparable to the state of the art with much fewer parameters and an increased interpretability of the model.
