  Objective: Overlapping measures are often utilized to quantify the similarity
between two binary regions. However, modern segmentation algorithms output a
probability or confidence map with continuous values in the zero-to-one
interval. Moreover, these binary overlapping measures are biased to structure
size. Addressing these challenges is the objective of this work. Methods: We
extend the definition of the classical Dice coefficient (DC) overlap to
facilitate the direct comparison of a ground truth binary image with a
probabilistic map. We call the extended method continuous Dice coefficient
(cDC) and show that 1) cDC is less or equal to 1 and cDC = 1 if-and-only-if the
structures overlap is complete, and, 2) cDC is monotonically decreasing with
the amount of overlap. We compare the classical DC and the cDC in a simulation
of partial volume effects that incorporates segmentations of common targets for
deep-brainstimulation. Lastly, we investigate the cDC for an automatic
segmentation of the subthalamic-nucleus. Results: Partial volume effect
simulation on thalamus (large structure) resulted with DC and cDC averages (SD)
of 0.98 (0.006) and 0.99 (0.001), respectively. For subthalamic-nucleus (small
structure) DC and cDC were 0.86 (0.025) and 0.97 (0.006), respectively. The DC
and cDC for automatic STN segmentation were 0.66 and 0.80, respectively.
Conclusion: The cDC is well defined for probabilistic segmentation, less biased
to structure size and more robust to partial volume effects in comparison to
DC. Significance: The proposed method facilitates a better evaluation of
segmentation algorithms. As a better measurement tool, it opens the door for
the development of better segmentation methods.
