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Microlocal analysis of Radon transforms over quadric surfaces

Gaik Ambartsoumian, Raluca Felea, Venkateswaran P. Krishnan, Clifford J. Nolan, Eric Todd Quinto

Abstract

We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied.

Microlocal analysis of Radon transforms over quadric surfaces

Abstract

We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface . The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix , with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix and the hypersurface . If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied.
Paper Structure (16 sections, 7 theorems, 126 equations)

This paper contains 16 sections, 7 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The canonical relation of $\mathcal{R}$
  4. Case when $A$ is positive definite and $S$ is a smooth strictly convex hypersurface
  5. Case when $A$ is indefinite and $S$ is a smooth strictly convex hypersurface
  6. Case $k >1$
  7. Special case 1: $A$ diagonal with entries $\pm 1$ and $S$ a sphere:
  8. Special case 2: $A$ diagonal with entries $\pm 1$ and $S$ a smooth strictly convex hypersurface
  9. General $A$, $k>1$, $S$ strictly convex hypersurface
  10. Case $k=1$, $A$ diagonal, entries $\pm 1$
  11. Case when $A$ is diagonal, indefinite and $S$ is a cylinder
  12. Case $k >1$
  13. Case $k=1$, $S$ a cylinder
  14. Paraboloids and more general surfaces of integration
  15. The normal operator $\mathcal{R}^*\mathcal{R}$
  16. ...and 1 more sections

Key Result

Theorem 4.1

Let $\mathcal{R}: \mathcal{E}'(\mathbb{R}^n\setminus S) \rightarrow \mathcal{D}'(\Omega \times(0,\infty))$ be the Radon transform defined by def:psi and def:R. Assume $A$ is an invertible positive definite matrix, and Hess($q$) is definite (either positive or negative definite), then $\mathcal{R}$ i

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 4.1
  • proof
  • Remark 4.2
  • ...and 20 more