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)正方形图像数据的增强问题，本文提出了一种新方法：将原始图像通过共形映射映射至圆盘，绕圆盘中心旋转后，在保持圆盘不变的默比乌斯变换下进行映射，最后再映射回原始正方形形状。与CNN数据增强中使用的典型变换不同，该过程不会因移除原始图像边缘区域而导致信息丢失。本文给出了所有必要映射的公式，以及编写图像变换代码的详细说明。通过模拟数据测试表明，将10张训练图像扩展为40张的训练集增强方法，能使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）。