Bulletin of the
Korean Mathematical Society BKMS

We define the average of a set of continuous functions of two variables (surfaces) using the structure of the two-parameter Wiener space that constitutes a probability space. The average of a sample set in the two-parameter Wiener space is defined employing the two-parameter Wiener process, which provides the concept of distribution over the two-parameter Wiener space. The average defined in our work, called an average function, also turns out to be a continuous function which is very desirable. It is proved that the average function also lies within the range of the sample set. ~The average function can be applied to model 3D shapes, which are regarded as their boundaries (surfaces), and serve as the average shape of them.

Keywords: Average of the set of two-variable functions, two-parameter Wiener space, two-parameter Wiener process, average of surfaces

MSC numbers: 28C20, 46G12, 60G15

Supported by: This work was supported by the Hongik University new faculty research support fund.