Abstract. The roto-translation group [Formula: see text] has been of active interest in image analysis due to methods that lift the image data to multiorientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image denoising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on [Formula: see text]. We make several theoretical contributions (generalizable to matrix Lie groups) such as the nonoptimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn-like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on [Formula: see text]. We observe that our framework of lifting images to [Formula: see text] and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on [Formula: see text].

Isometric feature mapping is an established time-honored algorithm in manifold learning and non-linear dimensionality reduction. Its prominence can be attributed to the output of a coherent global low-dimensional representation of data by preserving intrinsic distances. In order to enable an efficient and more applicable isometric feature mapping, a diverse set of sophisticated advancements have been proposed to the original algorithm to incorporate important factors like sparsity of computation, conformality, topological constraints and spectral geometry. However, a significant shortcoming of most approaches is the dependence on large-scale dense-spectral decompositions and the inability to generalize to points far away from the sampling of the manifold. In this paper, we explore an unsupervised deep learning approach for computing distance-preserving maps for non-linear dimensionality reduction. We demonstrate that our framework is general enough to incorporate all previous advancements and show a significantly improved local and non-local generalization of the isometric mapping. Our approach involves training with only a few landmark points and avoids the need for population of dense matrices as well as computing their spectral decomposition.

We consider the problem of computing dense correspondences between non-rigid shapes with potentially significant partiality. Existing formulations tackle this problem through heavy manifold optimization in the spectral domain, given hand-crafted shape descriptors. In this paper, we propose the first learning method aimed directly at partial non-rigid shape correspondence. Our approach uses the functional map framework, can be trained in a supervised or unsupervised manner, and learns descriptors directly from the data, thus both improving robustness and accuracy in challenging cases. Furthermore, unlike existing techniques, our method is also applicable to partial-to-partial non-rigid matching, in which the common regions on both shapes are unknown a priori. We demonstrate that the resulting method is data-efficient, and achieves state-of-the-art results on several benchmark datasets. Our code and data can be found online: https://github.com/pvnieo/DPFM.

In this paper, we provide a theoretical foundation for pointwise map recovery from functional maps and highlight its relation to a range of shape correspondence methods based on spectral alignment. With this analysis in hand, we develop a novel spectral registration technique: Fast Sinkhorn Filters, which allows for the recovery of accurate and bijective pointwise correspondences with a superior time and memory complexity in comparison to existing approaches. Our method combines the simple and concise representation of correspondence using functional maps with the matrix scaling schemes from computational optimal transport. By exploiting the sparse structure of the kernel matrices involved in the transport map computation, we provide an efficient trade-off between acceptable accuracy and complexity for the problem of dense shape correspondence, while promoting bijectivity.1

This paper explores a fully unsupervised deep learning approach for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We use the Siamese configuration to train a neural network to solve the problem of least squares multidimensional scaling for generating maps that approximately preserve geodesic distances. By training with only a few landmarks, we show a significantly improved local and nonlocal generalization of the isometric mapping as compared to analogous non-parametric counterparts. Importantly, the combination of a deep-learning framework with a multidimensional scaling objective enables a numerical analysis of network architectures to aid in understanding their representation power. This provides a geometric perspective to the generalizability of deep learning.

Many patch-based image denoising methods can be viewed as data-dependent smoothing filters that carry out a weighted averaging of similar pixels. It has recently been argued that these averaging filters can be improved using their doubly stochastic approximation, which are symmetric and stable smoothing operators. In this paper, we introduce a simple principle of consistency that argues that the relative similarities between pixels as imputed by the averaging matrix should be preserved in the filtered output. The resultant consistency filter has the theoretically desirable properties of being symmetric and stable, and is a generalized doubly stochastic matrix. In addition, we can also interpret our consistency filter as a specific form of Laplacian regularization. Thus, our approach unifies two strands of image denoising methods, i.e., symmetric smoothing filters and spectral graph theory. Our consistency filter provides high-quality image denoising and significantly outperforms the doubly stochastic version. We present a thorough analysis of the properties of our proposed consistency filter and compare its performance with that of other significant methods for image denoising in the literature.