According to the manifold hypothesis, real data can be compressed to lie on a low-dimensional manifold. This paper explores the estimation of the dimensionality of this manifold with an interest in identifying independent degrees of freedom and possibly identifying state variables that would govern materials systems. The challenges identified that are specific to materials science are (i) accurate estimation of the number of dimensions of the data, (ii) coping with the intrinsic random and low-bit-depth nature of microstructure samples, and (iii) linking noncompressed domains such as processing to microstructure. Dimensionality estimates are made with the maximum-likelihood-estimation method with the Minkowski p-norms being used as a measure of the distance between microstructural images. It is found that, where dimensionality estimates are required to be accurate, it is necessary to use the Minkowski 1-norm (also known as the L_{1}-norm or Manhattan distance). This effect is found to be due to image quantification and proofs are given regarding the distortion produced by quantization. It is also found that homogenization is an effective way of estimating the dimension of random microstructure image sets. An estimate of 40 dimensions for the fibers of a SiC/SiC fiber composite is obtained. It is also found that, with images generated from a sparse domain (surrogate to the process domain), it is possible to infer the nature of the process manifold from images alone.