1. An ultrasonic image processing apparatus comprising: a rendering unit which sets a plurality of rays with respect to volume data acquired from a three-dimensional space within a living organism and executes a rendering operation along each ray, to thereby generate an original image in the form of a three-dimensional image; and a directional interpolation processing unit which generates a display image based on the original image, the directional interpolation processing unit applying directional interpolation processing on the basis of the original image to each of noted pixels forming the display image to generate an interpolated pixel value for the noted pixel, wherein in the directional interpolation processing, a pixel reference direction for computing the interpolated pixel value is determined in a direction of flow of a tissue which is three-dimensionally represented in the original image.