Image Transformation matrix

The image registration process produces a description of the difference in spatial location between the two images. In the simplest case this description will contain details of the movements (translations) required in the three ordinal directions (anterior-posterior, right left and superior-inferior) and of the rotations required in the same three planes (pitch, roll and yaw). This mathematical description is called the transformation matrix and can be used to guide resampling of the test data set so that it matches the base images. 

Fig. 6 shows the result of an in vivo experiment in which the carotid is reconstructed with the US probe. The US worked in power Doppler mode, 9cm imaging depth. The US probe was calibrated with the proposed method. The carotid was scanned in freehand along its length. 2D Power Doppler images and location data of probe were acquired simultaneously. Colored points in 2D US images were extracted. The 3D corresponding positions of these colored points were computed by using the transformation matrix in (5). The 3D points were reconstructed and the surface of carotid was rendered with VTK (www.vtk.org). 

The representation of cooccurrence matrix as an image in 256 levels of gray necessitates a law of coefficients values transformation. In order that this law is common to all images, there will be no recodage on the maximum coefficient but on a theoretical maximal value. Thus the rule of conversion is the following ... 

Fig. 12.8 Electron microscopy study of a PDMS-Si02-CaO ormosil (A) Original HRTEM image of the amorphous matrix, (B) filtered HRTEM image and (C) Fourier transform pattern. Distances up to 0.53 nmfor (Si04)4-can be observed in the filtered image, indicating the presence of Ca2+ between the tetrahedra.

If the ends of the rows wrap around so that the leftmost element of each row is the rightmost element of the row above, and so on, the matrix is called circulant. The foregoing properties have significance when noniterative solutions to the matrix-posed problem are sought by discrete Fourier transform. Andrews and Hunt (1977) provide numerous details and references, especially for the case in which these linear methods are applied to the restoration of two-dimensional images. 

Why This kind of computation, using two different bases at once, can be confusing. We will use two typefaces to distinguish expressions in the standard basis from expressions in the new basis. Thus the new basis, written in the new basis, is ((1, 0), (0,1) )- Notice that our favorite rotation takes (1, 0), otherwise known as (2, 0), to the vector (0, 2), otherwise known as (0, 2). Similarly, the rotation takes (0, 1) to (-2, 0). Since the columns of any matrix are the images of the basis vectors under the linear transformation represented by the matrix, these calculations show that the matrix given above is correct. See Figure 2.5. 

So, for example, we can use the matrix of Formrrla 2.3 to see that the determinant of our favorite rotation is (0)(0) — (1)(—1) = 1. Note that we could just as well have used the matrix of Formula 2.4 to calculate the same answer (0) (0) — (2)(— ) = 1. No one familiar with the geometric interpretation of the determinant will be surprised by this result the determinant of a matrix with real entries is always the signed volume of the image of the unit square (or cube, or higher-dimensional cube), with a negative sign if the linear transformation changes the orientation. For more on this topic, see Lax [La, Chapter 5]. 

For the special case of S2 s i, the mirror image is produced by the inversion operation, but must be rotated by 180° to bring it into an exact reflective relationship to the original. This can be seen in Figure 3.4 and is conveniently expressed by using the matrices for the coordinate transformations. (Readers unfamiliar with matrix algebra may consult Appendix I.) Thus, we represent the operation S2 35 i hy the first matrix shown below and a rotation by n... 

A set of training vectors was assembled from this data. Let ns be the number of training samples. Each input vector vc, with i e 1,..., ns] corresponded to a horizontal line from the illuminated Mondrian. Since they worked with simulated data, the corresponding reflectances vr, were known. Let nx be the width of the Mondrian image. Therefore, each vector vc, and vr, contained nx data samples. The training samples were collected in a matrix of size ns x nx. This resulted in two matrices R and C. They assumed that the transform from input to output, i.e. from measured intensities to reflectances, is linear. 

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