A Fast and Efficient Shape Blending by Stable and Analytically Invertible Finite Descriptors

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.

I. Introduction

In the field of computer graphics, shape morphing, also termed shape blending, is an active research topic due to its usefulness in numerous applications e.g., animation, modeling, medicine, microbiology, entertainment, industrial design, and robotics. Shape blending [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] aims at creating an aesthetic intuitive and continuous transition between two shapes. In addition to that, the intermediate shapes should preserve the properties of the input shapes such as global appearance, angles, closure, smoothness, topology, etc. In this work, we focus on planar shape morphing, and more particularly, on the case of simple closed planar curves. Numerous approaches of the state-of-the-art have been introduced in this context [2], [3], [5], [6], [7], [8], [12], [13], [14], [15], [16], [17].