1. An image-processing apparatus comprising: a memory; and a processor coupled to the memory, wherein the processor executes a process comprising: identifying for each gradation value in a set of gradation values a connected component of pixels of not less than or not more than the gradation value neighboring and connected to each other in an input image; generating hierarchical structure data of a hierarchical structure including identified connected component; determining based on the hierarchical structure data whether the identified connected component satisfies a feature of character likelihood; extracting the identified connected component satisfying the feature of character likelihood as a character-like region; acquiring based on a maximum gradation value and a minimum gradation value of pixels included in the character-like region a threshold of binarization used exclusively for the character-like region; acquiring based on the threshold of binarization a corrected region where the character-like region is binarized; acquiring a background region where a gradation value of a pixel included in a region of the input image other than the corrected region is changed to a gradation value for a background; and acquiring binary image data of a binary image composed of the corrected region and the background region.