James Damon
University of North Carolina at Chapel Hill, Mathematics, Faculty Member
Abstract. For compact regions Ω in R3 with generic smooth boundary B, we consider geometric properties of Ω which lie midway between their topology and geometry, and can be summarized by the term “geometric complexity”. The “geometric... more
Abstract. For compact regions Ω in R3 with generic smooth boundary B, we consider geometric properties of Ω which lie midway between their topology and geometry, and can be summarized by the term “geometric complexity”. The “geometric complexity”of Ω is captured by its Blum medial axis M, which is a Whitney stratified set whose local structure at each point is given by specific standard local types. We classify the geometric complexity by giving a structure theorem for the Blum medial axis M. We do so by first giving an algorithm for decomposing M using the local types into “irreducible components”, and then representing each medial component as obtained by attaching surfaces with boundaries to 4-valent graphs. The two stages are described by a two level extended graph structure. The top level describes a simplified form of the attaching of the irreducible medial components to each other, and the second level extended graph structure for each irreducible component specifies how to c...
Research Interests: Mathematics and Geometry
In Chap. 8 we gave the classification of stable view projections of type (FC). In this chapter we further give the classification of generic transitions of type (FC). We summarize the classification for the five classes of transitions in... more
In Chap. 8 we gave the classification of stable view projections of type (FC). In this chapter we further give the classification of generic transitions of type (FC). We summarize the classification for the five classes of transitions in Theorem 12.1 and in subsequent sections consider the individual cases.
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We have already completed the classifications of the realizations of the local transitions for both (SC) in Chap. 11 and (FC) Chap. 12 For (SF) we only consider stable view projections of stable (SF) stratifications which are regular or... more
We have already completed the classifications of the realizations of the local transitions for both (SC) in Chap. 11 and (FC) Chap. 12 For (SF) we only consider stable view projections of stable (SF) stratifications which are regular or strata regular, so there is no contribution from apparent contours in the images; and the classification of these view projections for both the local and multilocal cases was completed in Chap. 8 Thus, to complete the classification of stable views and transitions involves two remaining cases. One is for the transitions for the local interaction of all three geometric features, shade/shadow, and apparent contours (SFC); and the second is for the multilocal transitions. In this chapter, we complete the classification for the case (SFC) and in the next chapter we shall complete the classification for multilocal transitions.
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As already mentioned, a key part of our investigation involves the abstract classifications of mappings under \(_{\mathcal{V}}\mathcal{A}\)-equivalence for a special semianalytic stratification \(\mathcal{V}\). Initially the... more
As already mentioned, a key part of our investigation involves the abstract classifications of mappings under \(_{\mathcal{V}}\mathcal{A}\)-equivalence for a special semianalytic stratification \(\mathcal{V}\). Initially the stratification is simple, e.g. modeled by a distinguished smooth curve on a smooth surface, or a boundary curve of a smooth surface with boundary. In such cases, there is a finite classification in low codimension.
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We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces... more
We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces viewed as affine hyperplanes. For naturality, we want the geodesic flow to be invariant under rigid transformations and homotheties. Consequently, we do not use the dual projective space, as the geodesic flow in this space is not preserved under translations. Instead we give an alternate approach using a Lorentzian space, which is semi-Riemannian with a metric of index $1$. For this space for points corresponding to affine hyperplanes in $\R^n$, we give a formula for the geodesic between two such points. As a consequence, we show the geodesic flow is preserved by rigid transformations and homotheties of $\R^n$. Furthermore, we give a criterion that a vector field in a smoothly varying family of hyperplanes along a curve yields a Lorentzian parallel vec...
Research Interests: Mathematics and Geodesic
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Deformation of a generalized cylinder with a parameterized shape change of its centerline is a non-trivial task when the surface is represented as a high-resolution triangle mesh, particularly when self-intersection and local distortion... more
Deformation of a generalized cylinder with a parameterized shape change of its centerline is a non-trivial task when the surface is represented as a high-resolution triangle mesh, particularly when self-intersection and local distortion are to be avoided. We introduce a deformation approach that satisfies these properties based on the skeleton (densely sampled centerline and cross sections) of a generalized cylinder. Our approach uses the relative curvature condition to extract a reasonable centerline for a generalized cylinder whose orthogonal cross sections will not intersect. Given the desired centerline shape as a parametric curve, the displacements on the cross sections are determined while controlling for twisting effects, and under this constraint a vertex-wise displacement field is calculated by minimizing a quadratic surface bending energy. The method is tested on complicated generalized cylindrical objects. In particular, we discuss one application of the method for human ...
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This paper considers joint analysis of multiple functionally related structures in classification tasks. In particular, our method developed is driven by how functionally correlated brain structures vary together between autism and... more
This paper considers joint analysis of multiple functionally related structures in classification tasks. In particular, our method developed is driven by how functionally correlated brain structures vary together between autism and control groups. To do so, we devised a method based on a novel combination of (1) non-Euclidean statistics that can faithfully represent non-Euclidean data in Euclidean spaces and (2) a non-parametric integrative analysis method that can decompose multi-block Euclidean data into joint, individual, and residual structures. We find that the resulting joint structure is effective, robust, and interpretable in recognizing the underlying patterns of the joint variation of multiblock non-Euclidean data. We verified the method in classifying the structural shape data collected from cases that developed and did not develop into Autistic Spectrum Disorder (ASD). Zhiyuan Liu zhiy@cs.unc.edu Jörn Schulz jorn.schulz@uis.no Mohsen Taheri mohsen.taherishalmani@uis.no M...
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Before introducing the notion of equivalence we will use, we motivate our approach by briefly considering an earlier approach of Henry-Merle et al. [HM, DHM], and Donati-Stolfi [Dn, DS].
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In Chaps. 6 and 7 we have constructed abstract normal forms for view projections and in Chap. 8 we have identified the various physical situations to which these normal forms apply.
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For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The... more
For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The "geometric complexity" of Omega is captured by its Blum medial axis M, which is a Whitney stratified set whose local structure at each point is given by specific standard local types. We classify the geometric complexity by giving a structure theorem for the Blum medial axis M. We do so by first giving an algorithm for decomposing M using the local types into "irreducible components" and then representing each medial component as obtained by attaching surfaces with boundaries to 4--valent graphs. The two stages are described by a two level extended graph structure. The top level describes a simplified form of the attaching of the irreducible medial components to each other, and the second level extended graph structure for each irreduc...
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In previous work, we introduced a method for modeling a configuration of objects in 2D and 3D images using a mathematical "medial/skeletal linking structure." In this paper, we show how these structures allow us to capture... more
In previous work, we introduced a method for modeling a configuration of objects in 2D and 3D images using a mathematical "medial/skeletal linking structure." In this paper, we show how these structures allow us to capture positional properties of a multi-object configuration in addition to the shape properties of the individual objects. In particular, we introduce numerical invariants for positional properties which measure the closeness of neighboring objects, including identifying the parts of the objects which are close, and the "relative significance" of objects compared with the other objects in the configuration. Using these numerical measures, we introduce a hierarchical ordering and relations between the individual objects, and quantitative criteria for identifying subconfigurations. In addition, the invariants provide a "proximity matrix" which yields a unique set of weightings measuring overall proximity of objects in the configuration. Furth...
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We consider the Blum medial axis of a region in $$\mathbb R^n$$ R n with piecewise smooth boundary and examine its “rigidity properties,”by which we mean properties preserved under diffeomorphisms of the regions preserving the medial... more
We consider the Blum medial axis of a region in $$\mathbb R^n$$ R n with piecewise smooth boundary and examine its “rigidity properties,”by which we mean properties preserved under diffeomorphisms of the regions preserving the medial axis. There are several possible versions of rigidity depending on what features of the Blum medial axis we wish to retain. We use a form of the cross ratio from projective geometry to show that in the case of four smooth sheets of the medial axis meeting along a branching submanifold, the cross ratio defines a function on the branching sheet which must be preserved under any diffeomorphism of the medial axis with another. Second, we show in the generic case, along a Y -branching submanifold, that there are three cross ratios involving the three limiting tangent planes of the three smooth sheets and each of the hyperplanes defined by one of the radial lines and the tangent space to the Y -branching submanifold at the point, which again must be preserved...
For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of ‘type $\mathcal{V}$’ is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate... more
For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of ‘type $\mathcal{V}$’ is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{\,-1}(\mathcal{V})$. If $\mathcal{V}$ is a hypersurface germ, then so is $\mathcal{V}_0 $, and by transversality ${\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V}_0) = {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$ provided $n \gt {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$. So $\mathcal{V}_0, 0$ will exhibit singularities of $\mathcal{V}$ up to codimension n. For singularities $\mathcal{V}_0, 0$ of type $\mathcal{V}$, we introduce a method to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$. It is via the ‘characteristic cohomology’ of the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and l...
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... By contrast for the complex case. the local formula of Palamodov Page 149. A Global Weighted Version of Bezout's Theorem 129 [15] giving the local degree as the dimension of the local algebra yields the usual " local"... more
... By contrast for the complex case. the local formula of Palamodov Page 149. A Global Weighted Version of Bezout's Theorem 129 [15] giving the local degree as the dimension of the local algebra yields the usual " local" Bezout Theorem for the complex case. ...
... Page 8. 5 2 J. DAMON transversality condition and having it hold for an open dense set of functions. Also, it then still follows that there is a residual subset of functions in ' ( U ) ( a countable intersection of open dense... more
... Page 8. 5 2 J. DAMON transversality condition and having it hold for an open dense set of functions. Also, it then still follows that there is a residual subset of functions in ' ( U ) ( a countable intersection of open dense subsets ) which exhibit V at every point of ...
We study images of smooth or piecewise smooth objects illuminated by a single light source, with only background illumination from other sources. The objects may have geometric features (F), namely surface markings, boundary edges,... more
We study images of smooth or piecewise smooth objects illuminated by a single light source, with only background illumination from other sources. The objects may have geometric features (F), namely surface markings, boundary edges, creases and corners; and shade features (S), namely shade curves and cast shadow curves. We determine the local stable interactions between these and apparent contours (C) for