Landmark Registration

The first experiment compares the inverse consistency error associated with the traditional unidirectional landmark thin-plate spline (UL-TPS) algorithm to that of the consistent landmark thin-plate spline (CL-TPS) algorithm. This simple experiment is designed to show that the UL-TPS algorithm can have significant inverse consistency error while this error is minimized using the CL-TPS algorithm. The experiment shown in Fig. 4

Figure 4: The location of local displacements at the landmarks points for the forward, and reverse transformations of images with $100 \times 100$ pixels. Application of the thin-plate spline deformation fields to uniformly spaced grids for the forward and reverse transformations.

consisted of matching eight landmarks in one image to their corresponding landmarks in a second image using both the UL-TPS and the CL-TPS algorithm. The arrows in the first and second panels show the displacement between the corresponding landmarks in the forward and reverse directions, respectively. The four landmarks in the corners of the images were fixed. The forward transformation $h$ maps the four inner points to the four outer points and the reverse transformation $g$ maps the outer points to the inner points. Applying the CL-TPS transformations to a rectangular grid shows that the forward transformation--defined with respect to a Eulerian frame of reference--causes the center of the image to expand (third panel of Fig. 4) while the reverse transformation causes a contraction of the central portion of the image (fourth panel of Fig. 4).

Figure 5: The left and center panels show the inverse consistency errors of the forward and reverse transformations, respectively. The tables in the right columns list the landmark errors associated with selected image points. The top and bottom rows are the inverse consistency errors associated with the unidirectional (UL-TPS) and consistent (CL-TPS) landmark thin-plate spline algorithms, respectively.

The top row of Fig. 5 shows the spatial locations and magnitudes of the inverse consistency errors of the forward and reverse transformations generated by the UL-TPS algorithm. The images in the left column were computed by taking the Euclidean norm of the difference between the forward transformation $h$ and the inverse of the reverse transformation $g^{-1}$. The images in the center column were computed in a similar fashion with $g$ and $h^{-1}$. The CL-TPS result was created using AUL-TPS initialization and minimizing for 100 iterations with $\alpha=0.5$ and $\beta=0.012$. This registration took approximately 3 minutes on a single 667MHz alpha processor.

The tables in Fig. 5 tabulate the inverse consistency error at four representative points in the images. The points $A$ and $C$ are located at points away from landmarks while the points $B$ and $D$ are located at landmark locations. The inverse consistency error at the landmark points is small for both algorithms. However, the landmark error is quite large away from the landmark locations in the UL-TPS algorithm. The range of intensities on the color bar for each method shows that the range of inverse consistency errors for the UL-TPS algorithm was in the range of 0.002 to 4.9 pixels while this same error for the CL-TPS algorithm ranged from 0.00 to 0.009. This shows that the CL-TPS algorithm reduced the inverse consistency error by over 500 times that of the UL-TPS algorithm for this example.

Figure 6: Deformed grids showing the error between the forward and reverse transformations estimated with the landmark-based thin-plate spline algorithm(left panel) and the CL-TPS algorithm(right panel). The grids were deformed by the transformation constructed by composing the forward and reverse transformations together, i.e., $g(h(x))$. Ideally, the composition of the forward and reverse transformations is the identity mapping which produces no distortion of the grid as in the right panel. The fuzziness associated with the grids are due to the bilinear interpolation.

A pair of transformations are point-wise consistent if the composite function $h(g(x))$ maps a point $x$ to itself. Spatial deviations from the identity mapping can be visualized by applying the composite mapping to a uniformly spaced grid. The grid is deformed by the composite transformation in regions where the forward and reverse transformations have inverse consistency errors. The composite transformation does not deform the grid for a perfectly inverse consistent set of forward and reverse transformations. Fig. 6 shows the composite mapping produced by the UL-TPS (left) and the CL-TPS (right) applied to a rectangular grid for this experiment. Notice that there is a considerable amount of inverse consistency error in the UL-TPS algorithm while there is no visually detectable inverse consistency error produced by the CL-TPS algorithm. The blurring of the grid is due to bilinear interpolation used to deform the grid images with the error displacements. Both images are created with the same technique, but the inverse consistent image needs very little interpolation since there is nearly zero displacement error.

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Table I: Comparison between the unidirectional (UL-TPS), averaged unidirectional (AUL-TPS), and consistent (CL-TPS) thin-plate spline image registration algorithms. The table columns are the Experiment, (ICC), transformation Direction (TD), average landmark error (ALE) in pixels, maximum landmark error (MLE), maximum inverse error (MIE) in pixels, average inverse error (AIE) in pixels, minimum Jacobian value (MJ), inverse of the maximum Jacobian value (IJ), and the Jacobian error (JE).

Experiment	ICC	TD	ALE	MLE	AIE	MIE	MJ	IJ	JE
UL-TPS	No	Forward	0.010	0.016	2.2	4.1	2.4	4.8	1.4
		Reverse	0.0056	0.010	2.0	4.9	2.9	3.2
AUL-TPS	No	Forward	0.0074	0.013	0.091	0.20	0.28	0.47	0.011
		Reverse	0.0072	0.012	0.082	0.29	0.45	0.27
CL-TPS	Yes	Forward	0.00055	0.0011	0.0028	0.0078	0.28	0.48	0.0012
(100 iter.)		Reverse	0.00046	0.00094	0.0024	0.0088	0.48	0.28

The minimum and maximum Jacobian values of the forward (reverse) transformation specify the maximum expansion and contraction of the transformation, respectively. The Jacobian error, calculated as $\frac{1}{2}\vert min \{Jac(h)\}-1/max\{Jac(g)\}\vert+ \frac{1}{2}\vert min \{Jac(g)\}-1/max\{Jac(h)\}\vert$, provides an indirect measure of the inconsistency between the forward and reverse transformations. The Jacobian error is zero if the forward and reverse transformations are inverses of one another, but the converse is not true. Table I shows that the Jacobian error was 1000 times smaller for the CL-TPS algorithm compared to the UL-TPS algorithm.

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