FUNDAMENTALS OF TEXTURE PROCESSING FOR BIOMEDICAL IMAGE ANALYSIS

sampled lattices) is not straightforward and raises several chal-lenges related to translation, scaling and rotation invariances and covariances that are becoming more complex in 3-D. 1.2. Related work and scope of this survey Depending on research communities, various taxonomies are used to refer to 3-D texture information. A clarification of the tax-onomy is proposed in this section to accurately define the scope of this survey. It is partly based on Toriwaki and Yoshida, 2009. Three-dimensional texture and volumetric texture are both general and identical terms designing a texture defined in R3 and include: (1) Volumetric textures existing in ‘‘filled’ ’ objects fV: x; y; z 2 Vx;y;z! R3g that are generated by a volumetric data acquisition device (e.g., tomography, confocal imaging). (2) 2.5D textures existing on surfaces of ‘‘hollow’ ’ objects as fC: x; y; z 2 Cu;v! R3g, (3) Dynamic textures in two–dimensional time sequences as fS: x; y; t 2 Sx;y;t! R3g, Solid texture refers to category (1) and accounts for textures de-fined in a volume Vx,y,z indexed by three coordinates. Solid textures have an intrinsic dimension of 3, which means that a number of variables equal to the dimensionality of the Euclidean space is needed to represent the signal (Bennett, 1965; Foncubierta-Rodrí-guez et al., 2013a). Category (2) is designed as textured surface in Dana and Nayar (1999) and Cula and Dana (2004), or 2.5-dimen-sional textures in Lu et al. (2006) and Aguet et al. (2008), where textures C are existing on the surface of 3-D objects and can be in-dexed uniquely by two coordinates (u,v). (2) is also used in Kajiy