The identification of the actual capillary network in the liver is significant for advancing medical science because it can help elucidate how the liver grows and develops its internal complexity. Although the exact mechanism behind the formation of the liver’s unique 3D capillary network is not fully understood, it is crucial to accurately identify the actual capillary network in the liver to better understand this process. Previous research has involved the manual observation of a 3D skeletonized capillary network in very localized areas. However, wide areas have not been visualized and surveyed quantitatively due to the enormous time and effort required for the manual method. In this paper, we design and implement an algorithm to visualize a 3D skeletonized capillary network over a wide area of the liver.

The 3-D projective space is represented as a 3-D Euclidean space extended to infinity. From a projective geometric perspective, it is expected that all 3-D geometric elements derived from 4-D homogeneous processing will provide clearer understanding infinity within a 3-D projective space. In this paper, I present a unified representation of 3-D geometric element definitions and interferences using 4-D homogeneous processing, with a focus on understanding the concept of infinity in 3-D geometric elements. By investigating the geometric relations between geometric element interferences via planes in 3-D space, I explore the nature of geometric elements and infinity within 3-D projective space. Consequently, it was confirmed that 3-D geometric elements can be represented as projective lines and projective planes using 4-D homogeneous processing, allowing a consistent expression of generality, including infinity.