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Deep Learning in Brain Segmentation
Published in Saravanan Krishnan, Ramesh Kesavan, B. Surendiran, G. S. Mahalakshmi, Handbook of Artificial Intelligence in Biomedical Engineering, 2021
In addition to the dice score, the Hausdorff distance is another metric for quantitative evaluation of segmentation performance. The Hausdorff distance of two sets X and Y measures the maximal distance between one point in a set to the nearest point in the other set. It can be calculated with the following equation: dH(X,Y)=maxx∈Xminy∈Yd(x,y)
H
Published in Phillip A. Laplante, Dictionary of Computer Science, Engineering, and Technology, 2017
Hausdorff distance an important distance measure, used in fractal geometry, among other places. Given a distance function d defined on a Euclidean space E, one derives from it the Hausdorff distance Hd on the family of all compact (i.e., bounded and topologically closed) subsets of E; for any two compact subsets K,L of E, Hd(K,L) is the least r > 0 such that each one of K, L is contained in the other’s dilation by a closed ball of radius r, that is: K⊆∪p∈LBr(p)andL⊆∪p∈KBr(p),
Tumor segmentation
Published in Ruijiang Li, Lei Xing, Sandy Napel, Daniel L. Rubin, Radiomics and Radiogenomics, 2019
Spyridon Bakas, Rhea Chitalia, Despina Kontos, Yong Fan, Christos Davatzikos
Hausdorff Distance (H) is mathematically defined as the maximum distance of a set to the nearest point in the other set [79]. In other words, it measures how close the boundaries of ROIPred are to those of ROIGT. H is used to assess the alignment between the contours of the segmentations and is defined as:
Multiscale topology optimization of pelvic bone for combined walking and running gait cycles
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
For 3D shapes represented by meshes, Hausdorff distance is a popular technique to measure the shape similarity between two objects. The Hausdorff distance quantifies the extent to which two objects resemble each other when they are aligned and superimposed on each other (Huttenlocher et al. 1993). Let A and B represent two data sets comprising nodal coordinates of meshes. Let be the distance between a given point and any point and hence is the minimum distance between a and all points in B. The one-sided Hausdorff distance is the maximum of all such minimum distances from points in A to points in B, i.e.
Statistical Modeling and Monitoring of Geometrical Deviations in Complex Shapes With Application to Additive Manufacturing
Published in Technometrics, 2022
Riccardo Scimone, Tommaso Taormina, Bianca Maria Colosimo, Marco Grasso, Alessandra Menafoglio, Piercesare Secchi
The Hausdorff distance is a metric applied by several authors in a wide range of applications (Kwan-Ho Lin et al. 2001; Aspert, Santa-cruz, and Ebrahimi 2002; Mémoli and Sapiro 2004; Zhao et al. 2005; Zhou and Wang 2009). It is a very general notion of metric given for any pair of subsets of a given metric space, thus including pairs of shapes, surfaces, meshes or point clouds in the usual 3D Euclidean space, regardless of their complexity or mutual relationship. As stated in Alt et al. (2003), the Hausdorff distance represents the most natural distance measure for pairs of geometrical objects where no point-to-point correspondence is available, a characteristic that makes it particularly suitable to determine differences between complex shapes, such as the ones we are focusing on. It is also the best-known metric between subsets of a metric space (Eiter and Mannila 1999; Conci and Kubrusly 2018). As far as the computation of distances between sets, other notions of distances have been proposed (for an extended overview, the reader is referred to Eiter and Mannila 1999; Gardner et al. 2014; Conci and Kubrusly 2018; and the therein). However, the Hausdorff distance has a much more general field of validity being a very general notion of metric applicable to any pair of shapes, surfaces, meshes or point clouds in the usual 2D or 3D Euclidean space, regardless of their complexity thanks to its computationally efficient tractability. This is also the reason why it is the default distance notion adopted in various software tools, like CloudCompare and MeshLab, in addition to being implemented in a large number of libraries (like the OpenCV library for image processing in C and Python), and other packages for Matlab and R.
An assessment of the efficiency of spatial distances in linear object matching on multi-scale, multi-source maps
Published in International Journal of Image and Data Fusion, 2018
Alireza Chehreghan, Rahim Ali Abbaspour
Considering Equations (9) and (10), Hausdorff distance is also known as ‘max–min’ distance. It satisfies the four properties of non-negativity, identity, symmetry and triangular inequality. Hausdorff distance and Euclidian distance are similar; however, contrary to Euclidian distance, Hausdorff distance is able to calculate the distance between all types of objects (point, line and polygon). As shown in Figure 2, the Hausdorff distance is highly sensitive to the objects’ shape. Thus, the change in the coordinates of vertices of one object would greatly influence the Hausdorff distance (Min et al. 2007).