Three Unfolded cube

A cube of data sized (mxnxp) can be unfolded in three different directions along the row space, along the column space, and along the third direction of the cube, also called the tube space. The three unfolding procedures give a row-wise... 

From this definition we see that each ensuing spatial dimension is structured in a direction perpendicular to the one preceding it. (The word perpendicular refers to a line or plane at right angles to another line or plane.) This unfolding sequence is a consistent principle, an equation for the erection of higher dimensional structures. Hence, a one-dimensional line is perpendicular to a zero-dimensional point a two-dimensional plane is perpendicular to a one-dimensional line, and a three-dimensional cube is perpendicular to a two-dimensional plane. If the principle is constant, it should point the way to four-dimensional space. The logical question is which "direction" is perpendicular to a cube ... 

So a ideal 3D display mode shows the complete colonic surface without image distortion (so polyps can be recognized as such) in a time efficient way. A techniques that satisfy these three requirements has a less colonoscopy -like representation than the conventional 3D fly-through the unfolded cube display. 

Vos F, Van Gelder R, Serlie I et al. (2003) Three dimensional display modes for CT colonography conventional 3D virtual colonoscopy versus unfolded cube projection. Radiology 228 878-885... 

Weak Multiway Methods Figure 7 shows the three steps in weak N-way analysis Unfold the data cube, perform the selected chemometric methods, and refold the matrix in order to display distribution maps. Weak N-way analysis comprises two main variants ... 

Though there is a clear gain in the quality and quantity of information when going from two- to three-way data sets, the mathematical complexity associated with the treatment of three-way data sets can seem, at first sight, a drawback. To avoid this problem, most of the three-way data analysis methods transform the original cube of data into a stack of matrices, where simpler mathematical methods can be applied. This process is often known as unfolding (see Figure 11.10). 

FIGURE 11.10 Three-way data array (cube) unfolding ormatricization. Two-way datamatrix augmentation. 

Both PCA and MCR-ALS can be easily extended to complex data arrays ordered in more than two ways or modes, giving three-way data arrays (data cubes or parallelepipeds) or multiway data arrays. In PCA and MCR-ALS, the multiway data set is unfolded prior to data analysis to give an augmented two-way data matrix. After analysis is complete, the resolved two-way profiles can be regrouped to recover the profiles in the three modes. The current state of the art in multiway data analysis includes, however, other methods where the structure of the multiway data array is explicitly built into the model and fixed during the resolution process. Among these... 

Fig. 3. Two dimensional lattice with iciest neighbor interaction. The equilibrium constants for going from Idt to ri are indicated. In the first case, an unfolded site with no folded neighbors bectmies folded in the second, a site with three folded neighbors becomes folded. The statistical weights, with respect to all unfolded conformation, are indicated below. These are calculated with p qf where v is the number of folded sites, ft the number of contacts between fok sites. The three dimensional case is treated similarly to this now the sites are cubes and sevCTal layers of cubes are placed on top of each other...

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