  For augmentation of the square-shaped image data of a convolutional neural
network (CNN), we introduce a new method, in which the original images are
mapped onto a disk with a conformal mapping, rotated around the center of this
disk and mapped under such a M\"obius transformation that preserves the disk,
and then mapped back onto their original square shape. This process does not
result the loss of information caused by removing areas from near the edges of
the original images unlike the typical transformations used in the data
augmentation for a CNN. We offer here the formulas of all the mappings needed
together with detailed instructions how to write a code for transforming the
images. The new method is also tested with simulated data and, according the
results, using this method to augment the training data of 10 images into 40
images decreases the amount of the error in the predictions by a CNN for a test
set of 160 images in a statistically significant way (p-value=0.0360).
