1. An object detection apparatus, comprising: an image input unit configured to receive an image; an image feature quantity extraction unit configured to extract a predetermined image feature quantity from the image to generate a feature-quantity extraction image; a symmetry evaluation unit configured to evaluate, for every processing target pixel in the feature-quantity extraction image, first-direction symmetry of an image area around the processing target pixel based on an image feature quantity of pixels around the processing target pixel, and calculate a symmetry width in a first direction, the first-direction symmetry being symmetry in the first direction; a center detection unit configured to detect the target pixel as a symmetry center pixel when the evaluated symmetry for the target pixel exceeds a predetermined reference; and an object area detection unit configured to detect an object width in the first direction for an object having symmetry included in the image based on the symmetry width of each symmetry center pixel, and detect an object width in a second direction orthogonal to the first direction based on distribution of each symmetry center pixel in the second direction, wherein the symmetry evaluation unit calculates a symmetry evaluation value SYM w (i, j) for the target pixel by defining a value based on a value of (Pmax−abs(P i−k −P i+k )), adding up the defined value for k from 1 to w, and dividing the added-up value by w, while varying a value of 2w+1, where w is a natural number, and where the symmetry evaluation value is larger when a pixel value in the feature-quantity extraction image is 0 or a positive value and the image feature quantity is larger, Pmax is a predetermined value equal to or greater than a maximum value of pixel values in the feature-quantity extraction image, P i is a pixel value of the target pixel located at coordinates (i, j) in the feature-quantity extraction image, P i−k is a pixel value of a pixel located distant from the target pixel by k pixels to a first detection side that is one side in the first direction, where k is a natural number, and P i+k is a pixel value of a pixel located distant from the target pixel by k pixels to a second detection side that is opposite to the first detection side in the first direction, 2w+1 is a width of the symmetry evaluation area in the first direction, and abs(x) denotes an absolute value of x.