Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Smets, Bart M. N.; Portegies, Jacobus W.; St-Onge, Etienne; Duits, Remco

doi:10.1007/s10851-020-00991-4

Cited by 7 publications

(6 citation statements)

References 46 publications

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“…This allows intuition and techniques from geometric PDE-based image analysis to be carried over to neural networks. In geometric PDE-based image processing, it can be beneficial to include mean curvature or other geometric flows [32][33][34][35] as regularization and our framework provides a nat-ural way for such flows to be included into neural networks. In the PDE layer from Fig.…”

Section: Drawing Inspiration From Pde-based Image Analysismentioning

confidence: 99%

PDE-Based Group Equivariant Convolutional Neural Networks

et al. 2022

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We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer’s trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and nonlinear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers; we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch. Just like for linear convolution, a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs, we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning-based imaging applications with far fewer parameters than traditional CNNs.

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Section: Drawing Inspiration From Pde-based Image Analysismentioning

confidence: 99%

PDE-Based Group Equivariant Convolutional Neural Networks

et al. 2022

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“…The diagonalization of the new data-driven left-invariant models G 𝑈 and F 𝑈 provides locally adaptive frames that are beneőcial over previous approaches to locally adaptive frames in M 2 [38,56,68] in the sense that:…”

Section: Discussionmentioning

confidence: 99%

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

Berg

Smets

Pai

et al. 2022

Preprint

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We introduce a data-driven version of the plus Cartan connection on the homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$ that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented \emph{Mathematica} notebooks available at \cite{githubNicky}.

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“…In our tracking, we integrate PDE-enhancements, like crossing-preserving total variation flow (TV-flow) enhancement in M 2 [18]. We will show this improves the results.…”

Section: Geodesic Tracking Of Retinal Vascular Trees With Optical And...mentioning

confidence: 92%

“…In our experiments, we use cake wavelets [8,18] as they do not tamper with data evidence and allow for fast reconstruction by integration over θ.…”

Section: Definition 2 (Orientation Score) the Orientation Score Trans...mentioning

confidence: 99%

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Geodesic Tracking of Retinal Vascular Trees with Optical and TV-Flow Enhancement in SE(2)

Berg

Zhang

Smets

et al. 2023

Lecture Notes in Computer Science

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Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Cited by 7 publications

References 46 publications

PDE-Based Group Equivariant Convolutional Neural Networks

PDE-Based Group Equivariant Convolutional Neural Networks

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

Geodesic Tracking of Retinal Vascular Trees with Optical and TV-Flow Enhancement in SE(2)

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