This paper presents a scale method for developing high dimensional scale functions to blend implicitly defined objects. Scale functions are differentiable on the entire domain except the origin, provide blending range control, and behave like Min/Max operators everywhere, so even a successive composition of blending operations containing overlapped blending regions can be generated smoothly. Because the scale method is a generalized method, implicit or parametric curves, such as cubic Bezier curves, rational conic curves, and implicit conics and hyper‐ellipsoids, can be used to develop scale functions. As a result, it can enhance the flexibility of generating the implicitly blending surfaces in Ricci's constructive geometry, soft objects modeling, and implicit sweep objects. ACM CSS: I.3.5 Computer Graphics—Computational Geometry and Object Modeling ‐ Curve, surface, solid and object representations

Motion-related artifacts are still a major problem in data analysis of functional magnetic resonance imaging (FMRI) studies of brain activation. However, the traditional image registration algorithm is prone to inaccuracy when there are residual variations owing to counting statistics, partial volume effects or biological variation. In particular, susceptibility artifacts usually result in remarkable signal intensity variance, and they can mislead the estimation of motion parameters. In this study, Two robust estimation algorithms for the registration of FMRI images are described. The first estimation algorithm was based on the Newton method and used Tukey's biweight objective function. The second estimation algorithm was based on the Levenberg-Marquardt technique and used a skipped mean objective function. The robust M-estimators can suppress the effects of the outliers by scaling down their error magnitudes or completely rejecting outliers using a weighting function. The proposed registration methods consisted of the following steps: fast segmentation of the brain region from noisy background as a preprocessing step; pre-registration of the volume centroids to provide a good initial estimation; and two robust estimation algorithms and a voxel sampling technique to find the affine transformation parameters. The accuracy of the algorithms was within 0.5 mm in translation and within 0.5 degrees in rotation. For the FMRI data sets, the performance of the algorithms was visually compared with the AIR 2.0 software, which is a software for image registration, using colour-coded statistical mapping by the Kolmogorov-Smirov method. Experimental results showed, that the algorithms provided significant improvement in correcting motion-related artifacts and can enhance the detection of real brain activation.

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