A Deep Neural Network Framework for Multivalued Mapping Problems with Varying Cardinality and Its Applications to Imaging

This paper addresses the challenging problem of modeling and computing multivalued mappings with varying cardinality, where a single input can correspond to multiple valid outputs, and the number of possible outputs may vary for different inputs. Such scenarios are prevalent in various imaging applications where multiple plausible solutions exist for a given input. We introduce a deep neural network framework to model multivalued mappings with varying cardinality. The framework integrates a discrete codebook with a generative network to produce valid outputs for each input. The discrete codebook variables are combined with the input to guide the generator in producing different valid solutions. The discrete nature of the codebook enables the framework to efficiently estimate the conditional probability distribution of possible outputs through a fixed equiangular tight frame classifier. By jointly optimizing the discrete codebook and its uncertainty estimation during training using a specially designed covariance loss function, an accurate computation of multiple solution candidates with reliable confidence measures can be achieved. We demonstrate the effectiveness of the proposed framework on various imaging applications, using both synthetic and real datasets. Experimental results show the efficacy of our proposed model to generate multiple high-quality outputs while providing meaningful uncertainty estimates for each solution.