A linear framework for region-based image segmentation and inpainting involving curvature penalization

We present the first method to handle curvature regularity in region-based image segmentation and inpainting that is independent of initialization. To this end we start from a new formulation of length-based optimization schemes, based on surface continuation constraints, and discuss the connections to existing schemes. The formulation is based on a \emph{cell complex} and considers basic regions and boundary elements. The corresponding optimization problem is cast as an integer linear program. We then show how the method can be extended to include curvature regularity, again cast as an integer linear program. Here, we are considering pairs of boundary elements to reflect curvature. Moreover, a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables. We show that by solving the linear programming relaxation one gets quite close to the global optimum, and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity.

💡 Research Summary

The paper introduces a novel globally optimal framework for region‑based image segmentation and inpainting that incorporates curvature regularization without dependence on an initial labeling. Traditional region‑based methods rely heavily on length regularization, which penalizes the total perimeter of the segmentation. While effective for many tasks, length regularization suffers from a well‑known shrinking bias that often eliminates thin, elongated structures. Human visual perception, however, is known to be sensitive to curvature, suggesting that curvature‑based regularizers could better preserve perceptually important details.

The authors start by discretizing the image domain into a cell complex. The domain is partitioned into a set of non‑overlapping basic regions (cells) (F). Each cell is assigned a binary label indicating foreground or background. The boundaries between cells are represented by boundary elements (E); each boundary element is considered in both orientations, forming the set of oriented boundary elements (E^{O}). This orientation is essential for defining a surface continuation constraint that enforces consistency between region labels and boundary activation: for every boundary element the signed sum of incident region variables must equal the signed sum of incident oriented boundary variables. Mathematically, for each (e\in E),

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