Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page

Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)

In this work we study the formulation of convection/diffusion equations on the 3D motion group SE(3) in terms of the irreducible representations of SO(3). Therefore, the left-invariant vector-fields on SE(3) are expressed as linear operators, that are differential forms in the translation coordinate and algebraic in the rotation. In the context of 3D image processing this approach avoids the explicit discretization of SO(3) or $S_2$, respectively. This is particular important for SO(3), where a direct discretization is infeasible due to the enormous memory consumption. We show two applications of the framework: one in the context of diffusion-weighted magnetic resonance imaging and one in the context of object detection

GP-SUM. Gaussian Processes Filtering of non-Gaussian Beliefs

This work studies the problem of stochastic dynamic filtering and state propagation with complex beliefs. The main contribution is GP-SUM, a filtering algorithm tailored to dynamic systems and observation models expressed as Gaussian Processes (GP), and to states represented as a weighted sum of Gaussians. The key attribute of GP-SUM is that it does not rely on linearizations of the dynamic or observation models, or on unimodal Gaussian approximations of the belief, hence enables tracking complex state distributions. The algorithm can be seen as a combination of a sampling-based filter with a probabilistic Bayes filter. On the one hand, GP-SUM operates by sampling the state distribution and propagating each sample through the dynamic system and observation models. On the other hand, it achieves effective sampling and accurate probabilistic propagation by relying on the GP form of the system, and the sum-of-Gaussian form of the belief. We show that GP-SUM outperforms several GP-Bayes and Particle Filters on a standard benchmark. We also demonstrate its use in a pushing task, predicting with experimental accuracy the naturally occurring non-Gaussian distributions.Comment: WAFR 2018, 16 pages, 7 figure

From Multiview Image Curves to 3D Drawings

Reconstructing 3D scenes from multiple views has made impressive strides in recent years, chiefly by correlating isolated feature points, intensity patterns, or curvilinear structures. In the general setting - without controlled acquisition, abundant texture, curves and surfaces following specific models or limiting scene complexity - most methods produce unorganized point clouds, meshes, or voxel representations, with some exceptions producing unorganized clouds of 3D curve fragments. Ideally, many applications require structured representations of curves, surfaces and their spatial relationships. This paper presents a step in this direction by formulating an approach that combines 2D image curves into a collection of 3D curves, with topological connectivity between them represented as a 3D graph. This results in a 3D drawing, which is complementary to surface representations in the same sense as a 3D scaffold complements a tent taut over it. We evaluate our results against truth on synthetic and real datasets.Comment: Expanded ECCV 2016 version with tweaked figures and including an overview of the supplementary material available at multiview-3d-drawing.sourceforge.ne

Physics-Informed Computer Vision: A Review and Perspectives

Incorporation of physical information in machine learning frameworks are opening and transforming many application domains. Here the learning process is augmented through the induction of fundamental knowledge and governing physical laws. In this work we explore their utility for computer vision tasks in interpreting and understanding visual data. We present a systematic literature review of formulation and approaches to computer vision tasks guided by physical laws. We begin by decomposing the popular computer vision pipeline into a taxonomy of stages and investigate approaches to incorporate governing physical equations in each stage. Existing approaches in each task are analyzed with regard to what governing physical processes are modeled, formulated and how they are incorporated, i.e. modify data (observation bias), modify networks (inductive bias), and modify losses (learning bias). The taxonomy offers a unified view of the application of the physics-informed capability, highlighting where physics-informed learning has been conducted and where the gaps and opportunities are. Finally, we highlight open problems and challenges to inform future research. While still in its early days, the study of physics-informed computer vision has the promise to develop better computer vision models that can improve physical plausibility, accuracy, data efficiency and generalization in increasingly realistic applications