Nicolas Brodu

Nicolas Brodu, James P. Crutchfield

We merge computational mechanics' definition of causal states (predictively-equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely-applicable method that infers causal structure directly from observations of a system's behaviors whether they are over discrete or continuous events or time. A structural representation -- a finite- or infinite-state kernel $ε$-machine -- is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker-Plank equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably-infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally-driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high dimensional data.

Nicolas Brodu

A new resolution enhancement method is presented for multispectral and multi-resolution images, such as these provided by the Sentinel-2 satellites. Starting from the highest resolution bands, band-dependent information (reflectance) is separated from information that is common to all bands (geometry of scene elements). This model is then applied to unmix low-resolution bands, preserving their reflectance, while propagating band-independent information to preserve the sub-pixel details. A reference implementation is provided, with an application example for super-resolving Sentinel-2 data.

Nicolas Brodu, Hussein Yahia

This article introduces the Stochastic Texture Difference method for analyzing data at prescribed spatial and value scales. This method relies on constrained random walks around each pixel, describing how nearby image values typically evolve on each side of this pixel. Textures are represented as probability distributions of such random walks, so a texture difference operator is statistically defined as a distance between these distributions in a suitable reproducing kernel Hilbert space. The method is thus not limited to scalar pixel values: any data type for which a kernel is available may be considered, from color triplets and multispectral vector data to strings, graphs, and more. By adjusting the size of the neighborhoods that are compared, the method is implicitly scale-dependent. It is also able to focus on either small changes or large gradients. We demonstrate how it can be used to infer spatial and data value characteristic scales in measured signals and natural images.