  Many media are divided into elementary units with irregular shape and size,
as exemplified by domains in magnetic materials, bubbles in foams, or cells in
biological tissues. Such media are essentially characterized by geometrical
disorder of their elementary units, which we term cells. Cells set a reference
scale at which are often assessed parameters and fields reflecting material
properties and state. Here, we consider the spectral analysis of spatially
varying fields. Such analysis is difficult in geometrically disordered media,
because space discretization based on standard coordinate systems is not
commensurate with the natural discretization into geometrically disordered
cells. Indeed, we found that two classical spectral methods, the Fast Fourier
Transform and the Graph Fourier transform, fail to reproduce all expected
properties of spectra of plane waves and of white noise. We therefore built a
method, which we call Cellular Fourier Transform (CFT), to analyze cell-scale
fields, which comprise both discrete fields defined only at cell level and
continuous fields smoothed out from their sub-cell variations. Our approach is
based on the construction of a discrete operator suited to the disordered
geometry and on the computation of its eigenvectors, which, respectively, play
the same role as the Laplace operator and sine waves in Euclidean coordinate
systems. We show that CFT has the expected behavior for sinusoidal fields and
for random fields with long-range correlations. Our approach for spectral
analysis is suited to any geometrically disordered material, such as a
biological tissue with complex geometry, opening the path to systematic
multiscale analyses of material behavior.
