Translation-rotation transformation

WT has been proposed as a new method for compressing spectra for storage and library searching in our study. In this kind of work, spectra are reconstructed from time to time from the compressed data. In order to maintain the quality of the reconstructed spectra, we have introduced another technique called the translation-rotation transformation (TRT) method [23] in the wavelet computation. In the FWT operation, the spectral data vector Cj needs to be extended periodically at the two extremes in the following manner ... 

It must be emphasized that Procrustes analysis is not a regression technique. It only involves the allowed operations of translation, rotation and reflection which preserve distances between objects. Regression allows any linear transformation there is no normality or orthogonality restriction to the columns of the matrix B transforming X. Because such restrictions are released in a regression setting Y = XB will fit Y more closely than the Procrustes match Y = XR (see Section 35.3). 

One physical restriction, translated into a mathematical requirement, must be satisfied that is that the simple fluid relation must be objective, which means that its predictions should not depend on whether the fluid rotates as a rigid body or deforms. This can be achieved by casting the constitutive equation (expressing its terms) in special frames. One is the co-rotational frame, which follows (translates with) each particle and rotates with it. The other is the co-deformational frame, which translates, rotates, and deforms with the flowing particles. In either frame, the observer is oblivious to rigid-body rotation. Thus, a constitutive equation cast in either frame is objective or, as it is commonly expressed, obeys the principle of material objectivity . Both can be transformed into fixed (laboratory) frame in which the balance equations appear and where experimental results are obtained. The transformations are similar to, but more complex than, those from the substantial frame to the fixed (see Chapter 2). Finally, a co-rotational constitutive equation can be transformed to a co-deformational one. 

In order to discuss the spectroscopic properties of diatomic molecules it is useful to transform the kinetic energy operators (2.5) or (2.6) so that the translational, rotational, vibrational, and electronic motions are separated, or at least partly separated. In this section we shall discuss transformations of the origin of the space-fixed axis system the following choices of origin have been discussed by various authors (see figure 2.1) ... 

The many-electron wave function in a crystal forms a basis for some irreducible representation of the space group. This means that the wave function, with a wave vector k, is left invariant under the symmetry elements of the crystal class (e.g. translations, rotations, reflections) or transformed into a new wave function with the same wave vector k. 

We will use transformation matrices to determine the symmetry of all nine motions and then assign them to translation, rotation, and vibration. Fortunately, it is only necessary to determine the characters of the transformation matrices, not the individual matrix elements. 

All S5mimetiy operations (e.g., translation, rotation, reflection in a plane) are isometric i.e., W = U and U does not change the distance between points of the transformed object (Figs. 2.2 and 2.3). 

Objects are reoriented and repositioned in the model space using simple transformations such as translation, rotation, and their combinations. In Figure 1-14 point P is defined by x, y, and z global coordinates. Following this, P is translated to point P and the object is rotated in two steps around the y and z axes of the model coordinate system. If a series of transformations of an object is time programmed, the object is animated. 

Transformations in the model space are explained in Figure 7-25. Shapes are repositioned, copied, and re-dimensioned by translation, rotation, mirroring, and scaling transformations and their combinations. 

This is done by identifying characteristic features on each image as comers, edges or objects that are matched and caUbrated by correspondence, to obtain the rigid transformation (translation + rotation) that permit to switch from one reference to another. Subsequently, data fusion can be performed by grouping all points in a single file. This operation is usually accompanied by a... 

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