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Persistence

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An important characteristic of a cycle, or a homology class, in a filtration of complexes is its “lifespan”, aka persistence.

In the image analysis context, the filtration comes from thresholding of a gray scale image. Its elements (called frames) are the lower level sets of the gray scale function of the image.

Then an object (i.e., connected component) emerges at frame B (birth) and merges with the surrounding area at frame D=255 (death); or a hole emerges in an object at frame B and is filled at frame D. The total lifespan is D - B. In this case, persistence is simply the contrast of the cycle.

See further persistent homology.