When I saw the picture in the newspaper, I also thought that U.S. President Barack Obama bowed too deeply to the Emperor. I believe that, before visiting Japan, he probably learned that the more deeply one bows, the more polite it is considered in Japan. I think Obama was showing his respect to the Emperor rather than his humility.
(As a Japanese citizen) I wish the Emperor himself would have bowed more deeply in the Japanese way to the American leader. I'm afraid that, in general, Imperial family members, including cousins, do not bow deeply even to other Japanese. We are not in the feudal age and we are not Imperial subjects. The Imperial family lives off the taxes we pay. Japan's Constitution says the Imperial family symbolizes the nation's unity. I wonder.