On the Hidden Waves of Image

TL;DR — Reconstructing images from a set of one-way wave equations with hidden speed

Abstract

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.