Bunce-Deddens Algebras as Quantum Gromov-Hausdorff Distance Limits of Circle Algebras

2021 ◽  
Vol 94 (1) ◽  
Author(s):  
Konrad Aguilar ◽  
Frédéric Latrémolière ◽  
Timothy Rainone
Keyword(s):  
Hausdorff Distance ◽  
Deddens Algebras

Efficient Hausdorff Distance Calculation Algorithm Based on Z-Order and Octree

2018 ◽  
Vol 30 (10) ◽  
pp. 1794
Author(s):  
Dejun Zhang ◽  
Fazhi He ◽  
Long Tian ◽  
Zhuyang Xie ◽  
Lu Zou
Keyword(s):  
Hausdorff Distance ◽  
Calculation Algorithm ◽  
Distance Calculation

scholarly journals Fully automated preoperative segmentation of temporal bone structures from clinical CT scans

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
C. A. Neves ◽  
E. D. Tran ◽  
I. M. Kessler ◽  
N. H. Blevins
Keyword(s):  
Facial Nerve ◽  
Inner Ear ◽  
Temporal Bone ◽  
Hausdorff Distance ◽  
Sigmoid Sinus ◽  
Ct Scans ◽  
Automated Segmentation ◽  
Dice Coefficient ◽  
Lateral Skull Base ◽  
End To End

AbstractMiddle- and inner-ear surgery is a vital treatment option in hearing loss, infections, and tumors of the lateral skull base. Segmentation of otologic structures from computed tomography (CT) has many potential applications for improving surgical planning but can be an arduous and time-consuming task. We propose an end-to-end solution for the automated segmentation of temporal bone CT using convolutional neural networks (CNN). Using 150 manually segmented CT scans, a comparison of 3 CNN models (AH-Net, U-Net, ResNet) was conducted to compare Dice coefficient, Hausdorff distance, and speed of segmentation of the inner ear, ossicles, facial nerve and sigmoid sinus. Using AH-Net, the Dice coefficient was 0.91 for the inner ear; 0.85 for the ossicles; 0.75 for the facial nerve; and 0.86 for the sigmoid sinus. The average Hausdorff distance was 0.25, 0.21, 0.24 and 0.45 mm, respectively. Blinded experts assessed the accuracy of both techniques, and there was no statistical difference between the ratings for the two methods (p = 0.93). Objective and subjective assessment confirm good correlation between automated segmentation of otologic structures and manual segmentation performed by a specialist. This end-to-end automated segmentation pipeline can help to advance the systematic application of augmented reality, simulation, and automation in otologic procedures.

scholarly journals Retraining Convolutional Neural Networks for Specialized Cardiovascular Imaging Tasks: Lessons from Tetralogy of Fallot

2021 ◽  
Author(s):  
Animesh Tandon ◽  
Navina Mohan ◽  
Cory Jensen ◽  
Barbara E. U. Burkhardt ◽  
Vasu Gooty ◽  
...  
Keyword(s):  
Tetralogy Of Fallot ◽  
Hausdorff Distance ◽  
Volumetric Analysis ◽  
Dice Similarity Coefficient ◽  
Algorithm Performance ◽  
Comprehensive Method ◽  
Combined Use ◽  
Median Error ◽  
Structural Abnormalities ◽  
Testing Data

AbstractVentricular contouring of cardiac magnetic resonance imaging is the gold standard for volumetric analysis for repaired tetralogy of Fallot (rTOF), but can be time-consuming and subject to variability. A convolutional neural network (CNN) ventricular contouring algorithm was developed to generate contours for mostly structural normal hearts. We aimed to improve this algorithm for use in rTOF and propose a more comprehensive method of evaluating algorithm performance. We evaluated the performance of a ventricular contouring CNN, that was trained on mostly structurally normal hearts, on rTOF patients. We then created an updated CNN by adding rTOF training cases and evaluated the new algorithm’s performance generating contours for both the left and right ventricles (LV and RV) on new testing data. Algorithm performance was evaluated with spatial metrics (Dice Similarity Coefficient (DSC), Hausdorff distance, and average Hausdorff distance) and volumetric comparisons (e.g., differences in RV volumes). The original Mostly Structurally Normal (MSN) algorithm was better at contouring the LV than the RV in patients with rTOF. After retraining the algorithm, the new MSN + rTOF algorithm showed improvements for LV epicardial and RV endocardial contours on testing data to which it was naïve (N = 30; e.g., DSC 0.883 vs. 0.905 for LV epicardium at end diastole, p < 0.0001) and improvements in RV end-diastolic volumetrics (median %error 8.1 vs 11.4, p = 0.0022). Even with a small number of cases, CNN-based contouring for rTOF can be improved. This work should be extended to other forms of congenital heart disease with more extreme structural abnormalities. Aspects of this work have already been implemented in clinical practice, representing rapid clinical translation. The combined use of both spatial and volumetric comparisons yielded insights into algorithm errors.

scholarly journals Impact of inter-reader contouring variability on textural radiomics of colorectal liver metastases

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Francesco Rizzetto ◽  
Francesca Calderoni ◽  
Cristina De Mattia ◽  
Arianna Defeudis ◽  
Valentina Giannini ◽  
...  
Keyword(s):  
Liver Metastases ◽  
Hausdorff Distance ◽  
Intraclass Correlation ◽  
Three Dimensional ◽  
Liver Lesion ◽  
Dice Coefficient ◽  
Rf Values ◽  
Enhanced Computed Tomography ◽  
Relative Change ◽  
The Impact

Abstract Background Radiomics is expected to improve the management of metastatic colorectal cancer (CRC). We aimed at evaluating the impact of liver lesion contouring as a source of variability on radiomic features (RFs). Methods After Ethics Committee approval, 70 liver metastases in 17 CRC patients were segmented on contrast-enhanced computed tomography scans by two residents and checked by experienced radiologists. RFs from grey level co-occurrence and run length matrices were extracted from three-dimensional (3D) regions of interest (ROIs) and the largest two-dimensional (2D) ROIs. Inter-reader variability was evaluated with Dice coefficient and Hausdorff distance, whilst its impact on RFs was assessed using mean relative change (MRC) and intraclass correlation coefficient (ICC). For the main lesion of each patient, one reader also segmented a circular ROI on the same image used for the 2D ROI. Results The best inter-reader contouring agreement was observed for 2D ROIs according to both Dice coefficient (median 0.85, interquartile range 0.78–0.89) and Hausdorff distance (0.21 mm, 0.14–0.31 mm). Comparing RF values, MRC ranged 0–752% for 2D and 0–1567% for 3D. For 24/32 RFs (75%), MRC was lower for 2D than for 3D. An ICC > 0.90 was observed for more RFs for 2D (53%) than for 3D (34%). Only 2/32 RFs (6%) showed a variability between 2D and circular ROIs higher than inter-reader variability. Conclusions A 2D contouring approach may help mitigate overall inter-reader variability, albeit stable RFs can be extracted from both 3D and 2D segmentations of CRC liver metastases.

scholarly journals On the Nielsen root theory of n-valued maps

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert F. Brown
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Lie Group ◽  
Hausdorff Distance ◽  
Root Number ◽  
Root Numbers ◽  
Fixed Point Result

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

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