  Unsupervised domain adaptation approaches have recently succeeded in various
medical image segmentation tasks. The reported works often tackle the domain
shift problem by aligning the domain-invariant features and minimizing the
domain-specific discrepancies. That strategy works well when the difference
between a specific domain and between different domains is slight. However, the
generalization ability of these models on diverse imaging modalities remains a
significant challenge. This paper introduces UDA-VAE++, an unsupervised domain
adaptation framework for cardiac segmentation with a compact loss function
lower bound. To estimate this new lower bound, we develop a novel Structure
Mutual Information Estimation (SMIE) block with a global estimator, a local
estimator, and a prior information matching estimator to maximize the mutual
information between the reconstruction and segmentation tasks. Specifically, we
design a novel sequential reparameterization scheme that enables information
flow and variance correction from the low-resolution latent space to the
high-resolution latent space. Comprehensive experiments on benchmark cardiac
segmentation datasets demonstrate that our model outperforms previous
state-of-the-art qualitatively and quantitatively. The code is available at
https://github.com/LOUEY233/Toward-Mutual-Information}{https://github.com/LOUEY233/Toward-Mutual-Information
