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Boundary

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The image.
Boundaries of the objects are in red and green.

Suppose we are given an object in a digital image (see Objects in binary 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.

In image analysis, some prefer to think of boundary as a collection of pixels. We instead rely on cell decomposition of images. Then every (binary) image is a 2-dimensional cubical complex and the boundary is made of edges of pixels, i.e., a 1-dimensional subcomplex. (When it is displayed by Pixcavator, it is, of course, a collection of pixels. To experiment with the concepts, download the free Pixcavator Student Edition.)

Definition. 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. An important fact is the Jordan theorem.

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

Note: Boundary = border.

In 3D objects are made of voxels, as a result their boundaries are made of the 2D faces of voxels, i.e. squares. Boundaries are still "cycles" but can't be treated as sequences anymore.

Mathematically, the concept of boundary in in image analysis is identical to the concept of frontier in point-set topology. But the word "boundary" usuallay means the boundary of a manifold.