Measures of Fuzziness and Image Information

If we understand an image (or its segments) as fuzzy sets, then we have to answer the question how fuzzy is the image. The increase or decrease of image fuzziness can be used in processing tasks such as enhancment, segmentation and classification. Arnold Kaufmann introduced the (linear) index of fuzziness as follows:

where we have an image of size MxN, and calculate the fuzziness regarding to the difference between the membership values and their complements. The quadratic index of fuzziness can be defined in a similar way:

The amount of fuzziness is zero if all memberships are 0 or 1 (ordinary set: e.g. binary image). The fuzziness reaches its maximum if all membership are equal to 0.5 (see Fig.1).

Of course, there are other ways to calculate the image fuzziness. DeLuca and Termin introduced the (logarithmic) fuzzy entropy:

where

Besides of these definitions, there are other measures of image information which are defined for fuzzy sets, for example:

• Fuzzy correlation

• Fuzzy expected value

• Weighted fuzzy expected value

• Fuzzy divergence

• Hybrid entropy

• ...