  Convolutional neural networks (CNN) have recently achieved state-of-the-art
results in various applications. In the case of image recognition, an ideal
model has to learn independently of the training data, both local dependencies
between the three components (R,G,B) of a pixel, and the global relations
describing edges or shapes, making it efficient with small or heterogeneous
datasets. Quaternion-valued convolutional neural networks (QCNN) solved this
problematic by introducing multidimensional algebra to CNN. This paper proposes
to explore the fundamental reason of the success of QCNN over CNN, by
investigating the impact of the Hamilton product on a color image
reconstruction task performed from a gray-scale only training. By learning
independently both internal and external relations and with less parameters
than real valued convolutional encoder-decoder (CAE), quaternion convolutional
encoder-decoders (QCAE) perfectly reconstructed unseen color images while CAE
produced worst and gray-scale versions.
