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Histogram equalization (HE) is a simple and effective image enhancing technique, however, it tends to change the mean brightness of the image to the middle level of the permitted range, and hence is not very suitable for consumer electronic products, where preserving the original brightness is essential to avoid annoying artifacts. This paper proposes a novel extension of histogram equalization, actually histogram specification, to overcome such drawback as HE. To maximize the entropy is the essential idea of HE to make the histogram as flat as possible. Following that, the essence of the proposed algorithm, named brightness preserving histogram equalization with maximum entropy (BPHEME), tries to find, by the variational approach, the target histogram that maximizes the entropy, under the constraints that the mean brightness is fixed, then transforms the original histogram to that target one using histogram specification. Comparing to the existing methods including HE, brightness preserving bi-histogram equalization (BBHE), equal area dualistic sub-image histogram equalization (DSIHE), and minimum mean brightness error bi-histogram equalization (MMBEBHE), experimental results show that BPHEME can not only enhance the image effectively, but also preserve the original brightness quite well, so that it is possible to be utilized in consumer electronic products.