The processing of compressed and encrypted imagery can exhibit advantages of computational efficiency as well as data security. Due to the data reduction inherent in compression, the computational speedup achieved via compressive processing can equal or exceed the compression ratio. Encryptive transformations which yield compressed ciphertext can similarly facilitate computational speedup, and are well known. However, due to analytical difficulties inherent in the derivation of operations which compute over the range space of commonly employed compressive transforms, reports of such processing paradigms are not evident in the open literature. In this introductory paper, we describe compressive and encryptive transformations in terms of functional mappings derived from abstract mathematics and image algebra (IA). An emerging technology, IA is a rigorous, concise notation which unifies linear and nonlinear mathematics in the image domain, and has been implemented on a variety of serial and parallel computers. Additionally, we derive a taxonomy of image transformations. Each taxonomic class is analyzed in terms of computational complexity and applicability to image and signal processing. We further present decompositions specific to each transformational class, which facilitate the design of operations over a given transform's range space. Examples and analysis are given for several image operations.