1. An image processing apparatus comprising: edge-direction detecting means for detecting an edge direction at a target pixel being considered in an original image; confidence detecting means for detecting a confidence of the edge direction; contrast detecting means for detecting a contrast intensity of the target pixel, the contrast intensity indicating an intensity of contrast in a first region including and neighboring the target pixel; texture-contrast-weight setting means for setting a texture-contrast weight for the target pixel, the texture-contrast weight being a weight that is based on a degree of closeness of the contrast intensity of the target pixel to a predetermined contrast intensity that occurs with a high frequency of occurrence in a texture region, the texture region being a region where pixel values vary by larger amounts than in a flat region and by smaller amounts than in an edge region, the flat region being a region where pixel values are substantially constant, and the edge region being a region where pixel values vary sharply; first edge-contrast-weight setting means for setting a first edge-contrast weight for the target pixel, the first edge-contrast weight being a weight that is based on a degree of closeness of the contrast intensity of the target pixel to a predetermined contrast intensity that occurs with a high frequency of occurrence in the edge region; texture-weight setting means for setting a texture weight, the texture weight being a weight that is based on edge directions of individual pixels in a second region including and neighboring the target pixel, confidences of the edge directions of the individual pixels, and texture-contrast weights for the individual pixels; edge-weight setting means for setting an edge weight, the edge weight being a weight that is based on edge directions of individual pixels in a third region including and neighboring the target pixel, confidences of the edge directions of the individual pixels, and first edge-contrast weights for the individual pixels; texture filtering means for performing texture filtering on the original image to generate a first texture-filter image, the texture filtering being directed to processing involving the texture region; edge filtering means for performing edge filtering on the original image to generate a first edge-filter image, the edge filtering being directed to processing involving the edge region; texture combining means for adding together pixel values at corresponding positions of the original image and the first texture-filter image, using weights that are based on the texture weight, to generate a first texture-combination image; and edge combining means for adding together pixel values of pixels at corresponding positions of the first texture-combination image and the first edge-filter image, using weights that are based on the edge weight, to generate a first edge-combination image.