Conclusion

FIGURE 21 Summary of the main experimental results. Triangles: interpolating functions. Circles: Noninterpolating functions. The hollow circles correspond to an accelerated implementation.

We have presented two important methods for the exact interpolation of data given by regular samples: in classical interpolation, the synthesis functions must be interpolants, while noninterpolating synthesis functions are allowed in generalized interpolation. We have tried to dispel the too commonly held belief according to which noninterpolating functions (typically, cubic B-splines) should be avoided. This misconception, present in many a book or report on interpolation, arises because practitioners failed to recognize the difference between classical and generalized interpolation and attempted to use in the former setting synthesis functions that are suited to the latter only. We have provided a unified framework for the theoretical analysis of the performance of both interpolating and noninterpolating methods. We have applied this analysis to specific cases that involve piecewise polynomial functions as well as sinc-based interpolants. We have performed 2D experiments that support the 1D theory.

We conclude from both theoretical and practical concerns that the most important index of quality is the approximation order of the synthesis function, its support being the most important parameter with respect to efficiency. Thus, the class of functions with maximal order and minimal support, or Moms, stands apart as the best achievable compromise between quality and speed. We have observed that many formerly proposed synthesis functions, such as Dodgson, Keys, and any of the apodized versions of a sinc, do not belong to this class. Experiments have confirmed that these synthesis functions do indeed perform poorly. In particular, no sinc-based interpolation results in an acceptable quality with regard to its computational demand. In addition, this family of synthesis functions is difficult to handle analytically, which leads to unnecessary complications for simple operations such as differentiation or integration.

The more favorable class of Moms functions can be further divided into subclasses, the most relevant being B-splines,

Schaum, and o-Moms. Of those three, the Schaum's functions are the only representatives that are interpolating. Nevertheless, experiments have shown that this strong constraint is detrimental to the performance; we observe that the time spent in computing the interpolation coefficients required by B-splines and o-Moms is a small, almost negligible investment that offers a high payoff in terms of quality. For this reason, we discourage the use of Schaum and promote generalized interpolation instead, with noninter-polating synthesis functions such as B-splines and o-Moms. For a better impact, we include in the Appendix an efficient routine for the computation of the required interpolation coefficients.

Finally, comparing B-splines to o-Moms, we conclude that the lack of continuity of the latter makes them less suitable than B-splines for imaging problems that require the computation of derivatives — for example, to perform operations such as edge detection or image-based optimization of some sort (e.g., snake contouring, registration). These operations are very common in medical imaging. Thus, despite a poorer objective result than o-Moms, B-splines are very good candidates for the construction of an image model. Moreover, they enjoy additional properties, such as easy analytical manipulation, several recursion relations, the m-scale relation (of great importance for wavelets, a domain that has strong links with interpolation [44,45]), minimal curvature for cubic B-splines, easy extension to inexact interpolation (smoothing splines, least-squares [6]), simplicity of their parametrization (a single number — their degree — is enough to describe them), and possible generalization to irregular sampling, to cite a few.

A property that has high relevance in the context of interpolation is the convergence of the cardinal spline to the non-truncated sinc for high degrees [40,41]. Throughout the paper, we have given numerous reasons why it is more profitable to approximate a true sinc by the use of noninter-polating synthesis functions rather than by apodization. Even for moderate degrees, the spline-based approximations offers a sensible quality improvement over apodization for a given computational budget.

None of the other synthesis functions, such as Schaum, Dodgson, Keys or sinc-based, offers enough gain in quality to be considered. We note, however, that the study presented in Table 2 and Fig. 21 relies on two images only; moreover, these images are not truly representative of genuine biomedical data. In addition, the comparison criterion is mean-square, which is precisely the form for which o-Moms are optimal. Perhaps other conclusions would be obtained by the use of more varied images or a different criterion, for example, a psycho-visual one.

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