Due to inherent physical and hardware limitations, 3D MR images are often acquired in the form of orthogonal thick slices, resulting in highly anisotropic voxels. This causes the partial volume effect, which introduces blurring of image details, appearance of staircase artifacts and significantly decreases the diagnostic value of images. To restore high resolution isotropic volumes, we propose to use a convolutional neural network (CNN) driven by patches taken from three orthogonal thick-slice images. To assess the validity and efficiency of this postprocessing approach, we used 1x1x1 mm3-voxel brain images of different modalities, available via the well known BrainWeb database. They served as a high resolution reference and were numerically preprocessed to create input images of different slice thickness and anatomical orientation, for CNN training, validation and testing. The visual quality of reconstructed images was indeed superior, compared to images obtained by fusion of interpolated thick-slice images, or to images reconstructed with the CNN using a single input MR scan. The significant increase of objectively computed figures of merit, e.g. the Structural Similarity Image Metric, was also noticed. Keeping in mind that any single value of such quality metrics represents a number of psychophysical effects, we applied the CNN trained on brain images for superresolution reconstruction of synthetic and acquired blood vessel tree images. We then used the restored superresolution volumes for estimation of vessel radii. It was demonstrated that vessel radius values derived from superresolution images of simulated vessel trees are significantly more accurate than those obtained from a standard fusion of interpolated thick-slice orthogonal scans. Superiority of the CNN-based superresolution images was also observed for scanner-acquired MR scans according to the evaluated parameters. These three experiments show the efficiency of CNN-based image reconstruction for qualitative and quantitative improvement of its diagnostic quality, as well as illustrates the practical usefulness of transfer learning - networks trained on example images of one kind can be used to restore superresolution images of physically different objects.