## Conservation of Information

We adopt as a basic premise that whatever information has been captured in the MR data acquisition process should be conserved (and, ideally, enhanced) in any subsequent data processing, including resampling. Every method of pairwise rigid image registration has two parts: (a) a method for determination (i.e., estimation) of the movement required to register one image to another; and (b) a method for relocation (registration) of the one image to the other.

The essential problem in relocating one image ("the image") to align with another ("the target") is that almost certainly the (relocated) image will no longer be on the pixel grid determined by the target image. Thus, some method will have to be found for "moving" the image to align with the target. That is, a method is needed to assign new values to pixels of the image at the location of the pixels of the target (based on the current values of the pixels of the image).

First, it should be noted that there are certain special values of the translations and rotation for which the resampling can be exact. Specifically, for translations that are integral multiples of the pixel size in each coordinate and for rotations that are integral multiples of 90° (re/2 radians) there is no error in resampling. For these specific relocations we know the optimal method for resampling: Move the image the exact number of pixels required. We should state, unequivocally, that without some additional information, beyond what is contained in the data itself, there is no "correct" method for registration of one image to another at distances other than these special values noted. The additional information would typically be an assumption about the images, such as: Each image is a discretization of an underlying periodic function with period equal to the image size. Many assumptions other than periodic are possible, each leading to a different optimal method of resampling the data. In the absence of such an additional assumption there is no correct answer to the registration problem. As a consequence it is very difficult to evaluate methods for registration.

### 2.1 Mathematical Framework

We denote by TM a relocation transformation by a method M. We say that the method is information-conserving if, for every image I, and every sequence of movements {(R, 5)} (we denote rotations by R and translations by 5) that returns the image to its original location, the final image is identical to the original image.

For example, a sequence of rotations of an image that results in a net rotation of zero because the sum of the clockwise and counterclockwise rotations is some multiple of 2re radians satisfies the condition of the definition. Likewise, a sequence of translations whose net result is no translation at all satisfies this condition. And, obviously, so does a combination of the two. The point of this definition is that none of the statistical information in the image I is destroyed by repeated application of the transformation TM.

A particular special sequence that satisfies the condition of the definition is the sequence of two transformations:

We will focus on this special case.

Let T be a registration transformation expressed, for example, in homogeneous coordinates. For our purposes here we can think of a two-dimensional homogeneous transformation T as simply the representation of a two-dimensional rotation matrix Re and a two-dimensional translation vector 5 by a single three-dimensional matrix

where 0 is a 1x2 vector of 0's. Now let M be a method for implementing T, often referred to as an interpolation method

(but it can also be thought of as a reconstruction method) denoted by TM. We can write tm (I) = TRe (TS(I))

where Tg (I) represents a shift in location by the vector 5 and TRg (I) represents a rotation around the origin by an angle 6. Notice that, using the inverse transforms

and tR61(i) = tR 6 (I), we can define TM1 by tm1(i ) = T5-1 (tR61(i ))

Note that translations and rotations do not commute. Thus, the composition of two translation-rotation pairs expressed as a single translation-rotation pair is

where Rr = R62 R61 = R61+62 and 5* = R^ R61 ¿1 + R62 ¿2.

More generally, the composition of a sequence of successive translation-rotation pairs expressed as a single translationrotation pair is

TR6t (T5t (•••(tR61 (T51 (I )))•••)) = TR6*(T5.(I))

A simple example of a transformation that is not information-conserving is the method of linear interpolation. To see this, let TR be the transformation that shifts an image to the right by exactly one pixel. Let TH be the transformation that shifts an image to the right by one-half pixel using the method of linear interpolation (not worrying about edge effects). So we define

except in the very unusual circumstance that tR (I) + tRm1(i) = 2I.

This can only happen when there are two alternating columns of values across the image. Thus, linear interpolation is not information-conserving, as defined earlier.

An easy empirical way to check whether any particular method TM is information-conserving is to simply compute s = I M TM1 ( tM (I)).

If s is larger than the computer rounding error the method is «of information-conserving.

### 2.2 Empirical Demonstration

In a typical fMRI study there is no absolute reference against which to compare a resampled image; i.e., there is no "gold standard." Nonetheless, in the act of resampling and regrid-ding our data we want to avoid introducing interpolation artifacts and false correlations due to loss of information.

Here, we have performed some simple motion-correction experiments with actual brain images using various algorithms from widely recognized fMRI processing systems. In one of these experiments, we investigated the "round-trip" error associated with taking a 64 x 64 pixel image and (1) rotating it tc/64 radians, (2) translating it 1/4 pixel diagonally (3) translating it back 1/4 pixel diagonally, and, finally, (4) rotating it back tc/64 radians — all using the same algorithm.

The results from this experiment are shown in Table 1. Selected cases are highlighted in Fig. 1. Eleven different interpolation methods were compared as follows: Fourier, Fourier method as described in [7]; WS16, windowed sinc, 16-point operator; WS8, windowed sinc, 8-point operator; Fourier (2), alternative implementation of Fourier method; WS4, windowed sinc, 4-point operator; NN, nearest neighbor interpolation; Quintic, fifth-degree, polynomial; Cubic, third-degree polynomial; WS2, windowed sinc, 2-point operator; Linear, linear interpolation; Linear(2), second implementation of linear interpolation. The implementations were all taken from publicly available systems that are currently in wide use in fMRI studies.

The table presents a comparison of the relative accuracy of different methods as measured by the mean-squared difference between the moved image and the original. We have also

FIGURE 1 (A) Comparison of an image before and after the motion described in Table 1. (B) The difference images produced by subtracting each moved image in A from the original image.

 Method Min Max Mean
0 0