This paper presents nonlinear filters that are obtained from extensions of morphological filters. The proposed nonlinear filter consists of a convex and concave filter that are extensions of the dilation and erosion of morphological filter with the maxout activation function. Maxout can approximate arbitrary convex functions as piecewise linear functions, including the max function of the morphological filters. 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 with maxout. The convex-concave filter is trained for noise and mask removal with a training set. The examples of noise and mask removal show that the convex-concave filter can obtain a recovered image, whose quality is comparable to in-painting by using the total variation minimization with reduced computational cost without mask information of the corrupted pixels.