On the Hidden Waves of Image
Yinpeng Chen Dongdong Chen Xiyang Dai Mengchen Liu Lu Yuan Zicheng Liu Youzuo Lin
In this paper, we introduce an intriguing phenomenon-the successful
reconstruction of images using a set of one-way wave equations with hidden and
learnable speeds. Each individual image corresponds to a solution with a unique
initial condition, which can be computed from the original image using a visual
encoder (e.g., a convolutional neural network). Furthermore, the solution for
each image exhibits two noteworthy mathematical properties: (a) it can be
decomposed into a collection of special solutions of the same one-way wave
equations that are first-order autoregressive, with shared coefficient matrices
for autoregression, and (b) the product of these coefficient matrices forms a
diagonal matrix with the speeds of the wave equations as its diagonal elements.
We term this phenomenon hidden waves, as it reveals that, although the speeds
of the set of wave equations and autoregressive coefficient matrices are
latent, they are both learnable and shared across images. This represents a
mathematical invariance across images, providing a new mathematical perspective
to understand images.