  Image segmentation is one of the most fundamental tasks of computer vision.
In many practical applications, it is essential to properly evaluate the
reliability of individual segmentation results. In this study, we propose a
novel framework to provide the statistical significance of segmentation results
in the form of p-values. Specifically, we consider a statistical hypothesis
test for determining the difference between the object and the background
regions. This problem is challenging because the difference can be deceptively
large (called segmentation bias) due to the adaptation of the segmentation
algorithm to the data. To overcome this difficulty, we introduce a statistical
approach called selective inference, and develop a framework to compute valid
p-values in which the segmentation bias is properly accounted for. Although the
proposed framework is potentially applicable to various segmentation
algorithms, we focus in this paper on graph cut-based and threshold-based
segmentation algorithms, and develop two specific methods to compute valid
p-values for the segmentation results obtained by these algorithms. We prove
the theoretical validity of these two methods and demonstrate their
practicality by applying them to segmentation problems for medical images.
