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Each of these three matrices can be written as the product of three extended 2D shearing matrices. Thus, nine shears can be used to perform 3D rotation. In fact, by using the combination rules for shear matrices the expression can be somewhat simplified.

It is possible to write the 3D rotation matrix as the product of only four shear matrices as

The values of the constants a, b, c, d, e, f, g, h are determined by solving the equations obtained by multiplying out the right-hand side of Eq. (5) and equating each entry to the corresponding entry on the left-hand side of Eq. (5) (obtained by multiplying out the three matrices on the right-hand side of Eq. (4). A formal proof of this decomposition is given by [5]. We believe that this product of four shears does not have the information-conserving property.

3.4 Conservation of Information

Fourier interpolation is information-conserving in the sense defined in Section 2.1. It is well known that the Fourier transform itself is information-conserving in the sense of invertibility, i.e., the Fourier transform of data which is itself the inverse Fourier transform of some original data reproduces that original data. That is,

Consequently, translations and rotations implemented by Fourier interpolation are information-conserving, unless details of the implementation, i.e., what happens between forward and inverse transformation, destroy this property. For example, translations implemented as phases-shifts in fc-space, utilizing Eq. (1) are information-conserving. In shorthand, we can write

Thus,

FIGURE 2 The intermediate steps in the rotation of a two-dimensional image by three successive shears. Panel A is the original image. Panel B is the result after shearing parallel to the x-axis. Panel C is the result after shearing parallel to the y-axis. Panel D is the final rotated image after the third shear parallel to the x-axis.

FIGURE 2 The intermediate steps in the rotation of a two-dimensional image by three successive shears. Panel A is the original image. Panel B is the result after shearing parallel to the x-axis. Panel C is the result after shearing parallel to the y-axis. Panel D is the final rotated image after the third shear parallel to the x-axis.

Similarly, rotations are information-conserving. From Eq. (2) we know that

Consequently, since RT = R1 we have that

Remember that, for example, linear interpolation does not have this information-conserving property.

Implementation details matter. Implementation details matter a great deal. For example, implementation of the Fourier interpolation by use of sinc interpolation directly on the image will not yield the exact result because of "edge effects." In Panel A of Fig. 1 the Fourier interpolation and the 4-point windowed sinc interpolation both appear to give high-fidelity results. However, Panel B of Figure 1 shows that while the Fourier interpolation produced an exact result, the windowed sinc interpolation had not only errors, but also a systematic spatial pattern of errors. This stems from two sources: First, the windowed sinc interpolator only used a four-point approximation to the true sinc interpolator and second, the actual implementation did not deal with the edge effects (or the wrap-around) correctly. This point is discussed extensively in Unser et al. [18], although these authors do not address the issue of Fourier interpolation, directly.

Implementation details matter. This can be seen most clearly in Table 1. There are two different Fourier interpolation methods and two different linear interpolation methods given in the table, and the results are different within each pair of methods.

The crux of the matter seems to be to use a Fourier formulation that does not truncate the spectrum or otherwise degenerate into a lower-fidelity interpolator. The periodicity of the full Fourier operator accurately conserves all the information inherent in the data. With respect to implementation, the accuracy and efficiency, and simplicity of substituting shift for rotation (by shearing), make this an attractive choice.

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