Component Tree Loss Function: Definition and Optimization

In this article, we propose a method to design loss functions based on component trees which can be optimized by gradient descent algorithms and which are therefore usable in conjunction with recent machine learning approaches such as neural networks. We show how the altitudes associated to the nodes of such hierarchical image representations can be differentiated with respect to the image pixel values. This feature is used to design a generic loss function that can select or discard image maxima based on various attributes such as extinction values. The possibilities of the proposed method are demonstrated on simulated and real image filtering.