In a previous work, we have proposed a morphing method based on invertible and stable descriptors that are invariant to Euclidean transformations and to the starting point. The stability guarantees the closeness in the shape sense of the reconstructed intermediate contours. However, this set of descriptors is not defined by a general expression. Here, we propose several sets of stable and invertible finite descriptors expressed by the same formula. Its stability is proven for a subset of this family thanks to the finite-dimension of the invariant space. Such finite dimension results from the Discrete Fourier Transform model that is used instead of Fourier coefficients. Moreover, an analytical general inverse formula is established. The use of the inverse analytical formula and the double utilization of the Fast Fourier Transform ensure an effective blending while being computationally efficient. Finally, we propose a new quantitative criterion for shape morphing. The latter is based on the Euclidean distances between successive curves in a morphing sequence after applying a given registration. Therefore, it allows us to compare the blending results of each set of descriptors. Several experiments are conducted on KIMIA’99 and MPEG-7 datasets. The results highlight the concordance of this criterion with the morphing visual quality and indicate which set of descriptors generates the most appropriate blending.

中文翻译：

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通过稳定且解析可逆的有限描述符实现快速高效的形状融合

在之前的工作中，我们提出了一种基于对欧几里得变换和起点不变的可逆和稳定描述符的变形方法。稳定性保证了重建的中间轮廓在形状感上的接近性。但是，这组描述符不是由通用表达式定义的。在这里，我们提出了几组由相同公式表示的稳定且可逆的有限描述符。由于不变空间的有限维，它的稳定性得到了这个族的一个子集的证明。这种有限维度来自离散傅里叶变换模型，该模型用于代替傅里叶系数。此外，还建立了解析通用逆公式。逆解析公式的使用和快速傅里叶变换的双重利用确保了有效的混合，同时具有计算效率。最后，我们提出了一种新的形状变形量化标准。后者基于应用给定配准后变形序列中连续曲线之间的欧几里得距离。因此，它允许我们比较每组描述符的混合结果。在 KIMIA'99 和 MPEG-7 数据集上进行了几个实验。结果突出了该标准与变形视觉质量的一致性，并指出了哪一组描述符生成了最合适的混合。后者基于应用给定配准后变形序列中连续曲线之间的欧几里得距离。因此，它允许我们比较每组描述符的混合结果。在 KIMIA'99 和 MPEG-7 数据集上进行了几个实验。结果突出了该标准与变形视觉质量的一致性，并指出了哪一组描述符生成了最合适的混合。后者基于应用给定配准后变形序列中连续曲线之间的欧几里得距离。因此，它允许我们比较每组描述符的混合结果。在 KIMIA'99 和 MPEG-7 数据集上进行了几个实验。结果突出了该标准与变形视觉质量的一致性，并指出了哪一组描述符生成了最合适的混合。