This paper presents convex filter networks that are obtained from extensions of morphological filters. The proposed filter network consists of a convex and concave filter that are extensions of the dilation and erosion of mathematical morphology with the maxout activation function. Maxout can approximate arbitrary convex functions as piecewise linear functions, including the max function. The class of the convex function hence includes the morphological dilation and can be trained for specific image processing tasks. In this paper, the closing filter is extended to a convex-concave filter network with maxout. The convex-concave filter is trained by the stochastic gradient method for noise and mask removal. The examples of noise and mask removal show that the convex-concave filter can obtain a recovered image, whose quality is comparable to inpainting by using the total variation minimization with reduced computational cost without mask information of the corrupted pixels.