In this paper, we present a new mathematical framework for modeling texture images using independent Basic Gray Level Aura Matrices (BGLAMs). We prove that independent BGLAMs are the basis of Gray Level Aura Matrices (GLAMs), and that an image can be uniquely represented by its independent BGLAMs. We propose a new BGLAM distance measure for automatically evaluating synthesis results w.r.t. input textures to determine if the output is a successful synthesis of the input. For the application to texture synthesis, we present a new algorithm to synthesize textures by sampling only the independent BGLAMs of an input texture. With respect to synthesis of textures and evaluation of the results, the performance of our approach is extensively evaluated and compared with symmetric GLAMs that are used in existing techniques and with Gray Level Cooccurrence Matrices (GLCMs). Experimental results have shown that (1) our approach significantly outperforms both symmetric GLAMs and GLCMs; (2) the new BGLAM distance measure has the ability to evaluate synthesis results, which can be used to automate the conventional visual inspection process for determining whether or not the output texture is a successful synthesis of the input; and (3) a broad range of textures can be faithfully synthesized using independent BGLAMs and the synthesis results are comparable to existing techniques.