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# Boundary

The image.
Boundaries of the objects are in red.

Suppose we are given an object in a digital image (see Objects in binary images and Objects in gray scale images). The idea of boundary is quite simple. It is what separates the object from the rest of the image (the complement). It is what "bounds" the object, the contour.

Cell decomposition of an image with 8 pixels arranged in a square.

Some prefer to think of boundary as a collection of pixels. We instead rely on cell decomposition of images. As a result the boundary is made of edges of pixels. When it is displayed by Pixcavator, it is, of course, a collection of pixels. (Download the free Pixcavator Student Edition here.)

The boundary of an object is the collection of all edges that are adjacent to both the object and its complement.

The edges of the boundary form a contour around the object. It is possible to represent it as a sequence of edges, or a cycle.

The perimeter of the object is the length of the boundary of the object.

Boundary = border.

In 3D objects are made of voxels, as a result their boundaries are made of the 3D faces of voxels. Boundaries are still "cycles" but they aren't sequences.