In a previous work, the equivariant Three-polar representation was introduced, and its effectiveness for face recognition was demonstrated. This approach involves representing 3D objects through a set of closed space curves derived from three reference points extracted from the surface of a 3D model. However, the reconstruction of models from the equivariant representation has not yet been addressed. In this article, we present an almost complete extension of the Threepolar representation for 3D model reconstruction. We propose an optimal reparameterization of the level curves based on the nearest neighbor notion. Finally, we perform the reconstruction of 3D models from the BU-3DFE dataset.

In this article, we introduce a new technique for augmenting 2D shape datasets based on a planar blending. In particular, a recently introduced shape blending method is applied to numerous pairs of shapes extracted from a given class. Several in-between data belonging to the same category are therefore generated. While traditional data augmentation approaches mainly apply simple transformations to the original data, the proposed technique allows the generation of non-linear variations of the input shapes by covering significantly the shape space. To demonstrate its interest, our augmentation technique is applied to the task of 2D shape classification. Experiments are performed on two benchmarks, namely KIMIA’99 and MPEG-7 CE using two different Convolutional Neural Network (CNN) architectures. The results show the superiority of our method over traditional augmentation techniques.

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.

In this paper, a novel method for blending planar shapes is introduced. This approach is based on the Fined-Fourier-based Invariant Descriptor (Fined-FID) that is invertible, invariant under Euclidean transformations and stable. Our approach extracts the Fined-FID from the two shapes of interest (the source and the target ones). Then, the extracted descriptors are averaged enabling the calculation of intermediate descriptors. Finally, thanks to the inversion criterion, the intermediate shapes are easily recovered by applying the inverse analytical expression to these intermediate descriptors. Compared to previous works, the Fined-FID-based morphing avoid the usual registration step, generates naturally closed intermediate contours and ensure invariance under Euclidean transformations and invariance to the starting point, while being computationally efficient (almost-linear complexity). The performed experiments show the performance of the proposed blending approach with respect to curvature-based methods.