  Accurate segmentation of long and thin tubular structures is required in a
wide variety of areas such as biology, medicine, and remote sensing. The
complex topology and geometry of such structures often pose significant
technical challenges. A fundamental property of such structures is their
topological self-similarity, which can be quantified by fractal features such
as fractal dimension (FD). In this study, we incorporate fractal features into
a deep learning model by extending FD to the pixel-level using a sliding window
technique. The resulting fractal feature maps (FFMs) are then incorporated as
additional input to the model and additional weight in the loss function to
enhance segmentation performance by utilizing the topological self-similarity.
Moreover, we extend the U-Net architecture by incorporating an edge decoder and
a skeleton decoder to improve boundary accuracy and skeletal continuity of
segmentation, respectively. Extensive experiments on five tubular structure
datasets validate the effectiveness and robustness of our approach.
Furthermore, the integration of FFMs with other popular segmentation models
such as HR-Net also yields performance enhancement, suggesting FFM can be
incorporated as a plug-in module with different model architectures. Code and
data are openly accessible at
https://github.com/cbmi-group/FFM-Multi-Decoder-Network.
