scholarly journals ON OPTIMAL 2-D DOMAIN SEGMENTATION PROBLEM VIA PIECEWISE SMOOTH APPROXIMATION OF A SELECTIVE TARGET MAPPING

2019 ◽  
Vol 27 (2) ◽  
pp. 60 ◽  
Author(s):  
Volodymyr V. Hnatushenko ◽  
Peter I. Kogut ◽  
Mykola V. Uvarov
Keyword(s):  
Piecewise Smooth ◽  
Smooth Approximation ◽  
Segmentation Problem ◽  
Selective Target

AN IMAGE SEGMENTATION VARIATIONAL MODEL WITH FREE DISCONTINUITIES AND CONTOUR CURVATURE

2004 ◽  
Vol 14 (01) ◽  
pp. 1-45 ◽  
Author(s):  
GIOVANNI BELLETTINI ◽  
RICCARDO MARCH
Keyword(s):  
Image Segmentation ◽  
Input Image ◽  
Piecewise Smooth ◽  
Variational Model ◽  
Smooth Approximation ◽  
Different Depths ◽  
Free Discontinuities ◽  
Γ Convergence ◽  
Contour Curvature

We introduce a functional for image segmentation which takes into account the transparencies (or shadowing) and the occlusions between objects located at different depths in space. By minimizing the functional, we try to reconstruct a piecewise smooth approximation of the input image, the contours due to transparencies, and the contours of the objects together with their hidden portions. The functional includes a Mumford–Shah type energy and a term involving the curvature of the contours. The variational properties of the functional are studied, as well as its approximation by Γ-convergence. The comparison with the Nitzberg–Mumford variational model for segmentation with depth is also discussed.

A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves

2005 ◽  
Vol 38 (8) ◽  
pp. 1139-1151 ◽  
Author(s):  
Guillaume Lavoue ◽  
Florent Dupont ◽  
Atilla Baskurt
Keyword(s):  
Piecewise Smooth ◽  
Smooth Approximation ◽  
Polygonal Curves

scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

2021 ◽  
Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger
Keyword(s):  
Limit Cycles ◽  
Space Dimension ◽  
Blow Up ◽  
Exponential Convergence ◽  
Model System ◽  
Piecewise Smooth ◽  
Singularly Perturbed ◽  
Model Systems ◽  
Smooth System ◽  
Unique Limit

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

scholarly journals Computer-assisted contralateral side comparison of the ankle joint using flat panel technology

2021 ◽  
Author(s):  
Sarina Thomas ◽  
Lisa Kausch ◽  
Holger Kunze ◽  
Maxim Privalov ◽  
André Klein ◽  
...  
Keyword(s):  
Information Gain ◽  
Contralateral Side ◽  
Computer Assisted ◽  
Ankle Joints ◽  
Additional Radiation ◽  
Median Position ◽  
Operative Evaluation ◽  
Overall Performance ◽  
The Individual ◽  
Segmentation Problem

Abstract Purpose Reduction and osteosynthesis of ankle fractures is a challenging surgical procedure when it comes to the verification of the reduction result. Evaluation is conducted using intra-operative imaging of the injured ankle and depends on the expertise of the surgeon. Studies suggest that intra-individual variance of the ankle bone shape and pose is considerably lower than the inter-individual variance. It stands to reason that the information gain from the healthy contralateral side can help to improve the evaluation. Method In this paper, an assistance system is proposed that provides a side-to-side view of the two ankle joints for visual comparison and instant evaluation using only one 3D C-arm image. Two convolutional neural networks (CNN) are employed to extract the relevant image regions and pose information of each ankle so that they can be aligned with each other. A first U-Net uses a sliding window to predict the location of each ankle. The standard plane estimation is formulated as segmentation problem so that a second U-Net predicts the three viewing planes for alignment. Results Experiments were conducted to assess the accuracy of the individual steps on 218 unilateral ankle datasets as well as the overall performance on 7 bilateral ankle datasets. The experiments on unilateral ankles yield a median position-to-plane error of $$0.73\pm 1.36$$ 0.73 ± 1.36 mm and a median angular error between 2.98$$^\circ $$ ∘ and 3.71$$^\circ $$ ∘ for the plane normals. Conclusion Standard plane estimation via segmentation outperforms direct pose regression. Furthermore, the complete pipeline was evaluated including ankle detection and subsequent plane estimation on bilateral datasets. The proposed pipeline enables a direct contralateral side comparison without additional radiation. This has the potential to ease and improve the intra-operative evaluation for the surgeons in the future and reduce the need for revision surgery.

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