The Sunflower Samurai is Kasumi Seizō, a Christian samurai who played a leading role in the Shimabara rebellion. His death is sought by the Shogunate as a result of this betrayal.
The objective of the trio's quest, he is revealed later on in the series to actually be Fuu's father. He left Fuu and her mother to save them from anti-Christian persecution (he was a major proponent of Christianity, considered heretical at the time). Fuu, however, thinks that his abandoning the family was unforgivable. Her only memory of him is a departing figure shifting through a field of sunflowers.
He was relentlessly pursued by the Shogunate, but succeeded in evading them. When Fuu began to search for him, the Shogunate had her followed. She led them right to him, sick and dying in a hut on Ikitsuki Island. Kariya Kagetoki arrives just after Fuu's tearful reconciliation with her father and kills him right in front of her.