Reparameterization Group Research Articles

RNE-DSNet: A Re-parameterization Neighborhood Enhancement-based Dual-Stream Network for CT image recognition

Deep learning models based on Transformer and CNN are current research hotspots. However, there are more time complexity and space complexity in extracting global–local features, the feature extraction ability is insufficient for small lesions in CT images. A Re-parameterization Neighborhood Enhancement-based Dual-Stream Network (RNE-DSNet) is proposed for CT image recognition in this paper. This method improves the ability to recognize small lesions while reducing the Params number and FLOPs, which gives a trade-off between performance and cost. The main works are as following: Firstly, a Re-parameterization Neighborhood Enhancement-based Dual-Stream Network is designed to improve the global–local feature extraction ability. Secondly, the Re-parameterization Group Residual Block is designed in the CNN branch, which improves the local feature extraction ability through multi-scale and multi-branch structures, and the number of the Params and FLOPs are reduced effectively. The Neighborhood Enhancement Transformer Block is designed in the Transformer branch, in which the global attention is calculated by parallel Hadamard product and Dot product, and the global attention maps are enhanced by deep convolution operation to embed local features in global features. Thirdly, the Adaptive-random Mask Cropping method is designed to improve the features extraction ability for small lesions in CT images through heat map and random Mask techniques. RNE-DSNet is compared with other SOTA models. The Accuracy (ACC), Precision (PRE), Recall (REC), F1 score (F1), and Area Under Curve (AUC) values of RNE-DSNet are reached 99.22%, 99.23%, 99.08%, 99.23% and 99.38%, respectively, which verified the effectiveness of the RNE-DSNet model. The number Params and FLOPs are reduced by 62.78% and 84.20%, respectively. The effectiveness of RNE-DSNet is verified by heat map visualization. The performance of RNE-DSNet is generally better than the others methods. There is positive significance for computer aided diagnosis.

Rate-Invariant Analysis of Covariance Trajectories

Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action recognition, a natural mathematical representation of activity videos is as parameterized trajectories on the set of symmetric, positive-definite matrices (SPDMs). The execution rates of actions, implying arbitrary parameterizations of trajectories, complicate their analysis. To handle this challenge, we represent covariance trajectories using transported square-root vector fields, constructed by parallel translating scaled-velocity vectors of trajectories to their starting points. The space of such representations forms a vector bundle on the SPDM manifold. Using a natural Riemannian metric on this vector bundle, we approximate geodesic paths and geodesic distances between trajectories in the space of this vector bundle. This metric is invariant to the action of the re-parameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories. We demonstrate this framework in multiple contexts, using both generative statistical models and discriminative data analysis. The latter is illustrated using several applications involving video-based action recognition and dynamic functional connectivity analysis.

Kähler structures on spaces of framed curves

We consider the space $${\mathcal {M}}$$ of Euclidean similarity classes of framed loops in $${\mathbb {R}}^3$$ . Framed loop space is shown to be an infinite-dimensional Kahler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of $${\mathcal {M}}$$ by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of $$\mathcal {M}$$ explicitly. Using this description, we show that $${\mathcal {M}}$$ and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in $${\mathcal {M}}$$ as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on $${\mathcal {M}}$$ is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional.

Elastic Geodesic Paths in Shape Space of Parameterized Surfaces

This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces.

Parameterization-Invariant Shape Comparisons of Anatomical Surfaces

We consider 3-D brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized. Past comparisons of such surfaces involve either volume deformations or nonrigid matching under fixed parameterizations of surfaces. We propose a new mathematical representation of surfaces, called q-maps, such that L² distances between such maps are invariant to re-parameterizations. This property allows for removing the parameterization variability by optimizing over the re-parameterization group, resulting in a proper parameterization-invariant distance between shapes of surfaces. We demonstrate this method in shape analysis of multiple brain structures, for 34 subjects in the Detroit Fetal Alcohol and Drug Exposure Cohort study, which results in a 91% classification rate for attention deficit hyperactivity disorder cases and controls. This method outperforms some existing techniques such as spherical harmonic point distribution model (SPHARM-PDM) or iterative closest point (ICP).

Shape Analysis of Elastic Curves in Euclidean Spaces

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL(2) metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.