Image thresholding using fuzzy entropies

An image can be regarded as a fuzzy subset of a plane. A fuzzy entropy measuring the blur in an image is a functional which increases when the sharpness of its argument image decreases. We generalize and extend the relation "sharper than" between fuzzy sets in view of implementing the properties of a relation "sharper than" between images. We show that there are infinitely many implementations of this relation into an ordering between fuzzy sets (equivalently, images). Relying upon these orderings, we construct classes of fuzzy entropies which are useful for image thresholding by cost minimization. Assuming the image to be a degraded version of an ideal two level image (object/background), a fuzzy entropy can be introduced in a cost functional to force the fitting function to be as close as possible to a two-valued function. The minimization problem is numerically solved, and the results obtained on a synthetic image are reported.