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).

翻译：针对卷积神经网络（CNN）的方形图像数据增强问题，我们提出了一种新方法：将原始图像通过共形映射投影到圆盘上，绕圆盘中心旋转后，再经过保持圆盘不变的莫比乌斯变换进行处理，最后通过逆映射恢复为原始方形。与传统CNN数据增强中常用的变换不同，这一过程不会因移除原始图像边缘区域而导致信息丢失。本文给出了所有所需映射的数学公式，以及图像变换代码编写的详细说明。通过模拟数据测试，将10张训练图像通过该方法增强为40张后，CNN对160张测试图像的预测误差在统计上显著降低（p值=0.0360）。