Growth, motion and 1-parameter families of symmetry sets

Bruce, J. W.; Giblin, Peter

doi:10.1017/s030821050001917x

Cited by 63 publications

(93 citation statements)

References 14 publications

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“…Such descriptions have involved the use of a skeleton or medial axis to extract shape features (e.g. Blum, 1967Blum, , 1973Bruce and Giblin, 1986;Talbot and Vincent, 1992;Ogniewicz, 1993;Attali and Montanvert, 1994;Kimia et al, 1995;Näf et al, 1996Näf et al, , 1997August et al, 1999;Golland et al, 1999). Other approaches have included physically based shape representations such as thin-plate-splines and fiducials (e.g.…”

Section: Introductionmentioning

confidence: 99%

Amygdala–hippocampal shape differences in schizophrenia: the application of 3D shape models to volumetric MR data

Shenton

Gerig

McCarley

et al. 2002

Psychiatry Research: Neuroimaging

125

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Evidence suggests that some structural brain abnormalities in schizophrenia are neurodevelopmental in origin. There is also growing evidence to suggest that shape deformations in brain structure may reflect abnormalities in neurodevelopment. While many magnetic resonance (MR) imaging studies have investigated brain area and volume measures in schizophrenia, fewer have focused on shape deformations. In this MR study we used a 3D shape representation technique, based on spherical harmonic functions, to analyze left and right amygdala-hippocampus shapes in each of 15 patients with schizophrenia and 15 healthy controls matched for age, gender, handedness and parental socioeconomic status. Left/right asymmetry was also measured for both shape and volume differences. Additionally, shape and volume measurements were combined in a composite analysis. There were no differences between groups in overall volume or shape. Left/right amygdalahippocampal asymmetry, however, was significantly larger in patients than controls for both relative volume and shape. The local brain regions responsible for the left/right asymmetry differences in patients with schizophrenia were in the tail of the hippocampus (including both the inferior aspect adjacent to parahippocampal gyrus and the superior aspect adjacent to the lateral geniculate nucleus and more anteriorly to the cerebral peduncles) and in portions of the amygdala body (including the anterior-superior aspect adjacent to the basal nucleus). Also, in patients, increased volumetric asymmetry tended to be correlated with increased left/right shape asymmetry. Furthermore, a combined analysis of volume and shape asymmetry resulted in improved differentiation between groups. Classification function analyses correctly classified 70% of cases using volume, 73.3% using shape, and 87% using combined volume and shape measures. These findings suggest that shape provides important new information toward characterizing the pathophysiology of schizophrenia, and that combining volume and shape measures provides improved group discrimination in studies investigating brain abnormalities in schizophrenia. An evaluation of shape deformations also suggests local abnormalities in the amygdala-hippocampal complex in schizophrenia.

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Section: Introductionmentioning

confidence: 99%

Amygdala–hippocampal shape differences in schizophrenia: the application of 3D shape models to volumetric MR data

Shenton

Gerig

McCarley

et al. 2002

Psychiatry Research: Neuroimaging

125

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“…Mathematically, the AASS and the MPTL 'go together' in the same way that the classical symmetry set and evolute, or the ADSS and the affine evolute go together: in each case the pair makes up a single mathematical entity called a full bifurcation set. A good deal is known about the structure of such sets, including the structure of full bifurcation sets arising from families of curves (see Bruce and Giblin, 1986). For instance, the symmetry set has endpoints in the cusps of the evolute, and in the same way the AASS has endpoints in cusps of the MPTL.…”

Section: As Mentioned Above It Is Not Clear To Us How Far the Area Dmentioning

confidence: 99%

Area-Based Medial Axis of Planar Curves

Niethammer

Betelú

Sapiro

et al. 2004

International Journal of Computer Vision

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A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.

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“…The symmetry set together with the (euclidean) evolute constitute the full bifurcation set of the family of distance-squared functions on γ (see [6]). The analogous symmetry set in the affine case is the affine distance symmetry set (ADSS): the closure of the locus of points x ∈ R 2 on two affine normals and affine-equidistant from the corresponding points on the curve.…”

Section: The Affine Distance Symmetry Setmentioning

confidence: 99%

“…The local structure of the ADSS was classified in these articles, on the assumption that the curve contained no inflexions. The present article extends this to curves with inflexions and gives a complete list of the transitions on the ADSS of generic 1-parameter families of curves, following the analogous procedure given in [6] for the euclidean symmetry set. We find that ovals (strictly convex smooth closed curves) behave very much as do generic curves relative to the euclidean symmetry set.…”

Section: Introductionmentioning

confidence: 99%

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Affine-Distance symmetry sets

Giblin¹,

Holtom²

2003

MATH. SCAND.

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The affine distance symmetry set (ADSS) of a plane curve is an affinely invariant analogue of the euclidean symmetry set (SS) [7], [6]. We list all transitions on the ADSS for generic 1-parameter families of plane curves. We show that for generic convex curves the possible transitions coincide with those for the SS but for generic non-convex curves, further transitions occur which are generic in 1-parameter families of bifurcation sets, but are impossible in the euclidean case. For a nonconvex curve there are also additional local forms and transitions which do not fit into the generic structure of bifurcation sets at all. We give computational and experimental details of these.

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Growth, motion and 1-parameter families of symmetry sets

Cited by 63 publications

References 14 publications

Amygdala–hippocampal shape differences in schizophrenia: the application of 3D shape models to volumetric MR data

Amygdala–hippocampal shape differences in schizophrenia: the application of 3D shape models to volumetric MR data

Area-Based Medial Axis of Planar Curves

Affine-Distance symmetry sets

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