Deep convolutional neural networks meet variational shape compactness priors for image segmentation

Integrating spatial priors into image segmentation methods is essential for enhancing prediction accuracy and robustness. In this work, we focus on the shape compactness prior, which measures the circularity of objects, for variational image segmentation tasks. Segmentation of circular objects has significant applications across various fields. We propose utilizing shape compactness as a tool to address these applications. Existing algorithms face challenges such as computational inefficiency, model nonconvexity, and the need for extensive hyperparameter fine-tuning. To overcome these issues, we introduce two novel optimization algorithms coupled with the primal–dual formulation of the original problem: the primal–dual threshold dynamics (PD-TD) and the primal–dual soft threshold-dynamics algorithm (PD-STD). The PD-STD is a relaxed version of the PD-TD and offers superior computational efficiency. Building on the variational explanation of the sigmoid function, the proposed PD-STD algorithm can be integrated into deep neural networks (DNNs) to enhance segmentation results for circular objects. Conventional DNNs often act as black boxes, making them susceptible to noise and adversarial attacks, which leads to inconsistencies in segmentation when input images are corrupted. Our approach improves the robustness and stability of DNNs by enforcing the shape compactness prior information. Extensive experiments demonstrate that the proposed algorithms outperform state-of-the-art methods in both numerical efficiency and effectiveness. This is particularly evident when the algorithms are tested on Iris datasets with high levels of noise. Our findings highlight the potential of incorporating shape compactness priors to enhance image segmentation in challenging scenarios.