This paper proposes a three-dimensional(3D) segmentation algorithm using hyper-complex edge detection operator
and applies the new algorithm to three-dimensional hepatic vessel segmentation from computed tomography (CT)
volumetric data. A 3D hyper-complex edge detection operator is constructed by combining octonion and gradient
operator. We replace every voxel of the volumetric data by one octonion which consist of its gray-level and its 6
neighborhoods' gray-level. Via this the original volumetric data is defined as octonion volumetric data. Similar to the
Sobel operator, there are three principal directions (coordinate axes) in 3D hyper-complex edge detection operator, and
each element in this operator is a octonion. The operator is circularly convoluted with octonion volumetric data to get the
value of matching response. If matched, this voxel is the edge of vessel. Experimental results show that the algorithm can
effectively segment small vascular tree branches.
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