1. A mobile positioning apparatus comprising: a position storage device which holds a position information group indicating a current position of a mobile body and likelihood information thereof as a nonparametric distribution; a movement vector estimation device which outputs a movement vector indicating a moving distance and a moving direction of the mobile body at a discrete time T and likelihood information thereof; an image database which holds a peripheral map of a range of movement of the mobile body as an image group of a resolution according to a moving resolution of the mobile body; and a pixel crossing determination device which performs map matching using the position information group and the movement vector based on the peripheral map; wherein the image database accumulates the peripheral map as a pixel image group which has a different pixel value depending on a region where the mobile body can move and the other regions and which has a known correspondence to position information; and the pixel crossing determination device generates a position candidate group at a discrete time T+1 based on the position information group at a discrete time T held by the position storage device, the movement vector outputted by the movement vector estimation device, and likelihood information of the movement vector; performs a pixel crossing determination on each position candidate taken from the position candidate group to determine whether a moving path connecting between a point before movement and a point after movement thereof crosses a pixel inhibiting movement and passage on an image; filters the generated position candidate group and generates the remaining position candidate group; normalizes the result, generates the position information group of a nonparametric distribution of a position at a discrete time T+1, and updates a content of the position storage device.