Local inversion of the sonar transform regularized by the approximate inverse

A new reconstruction method is given for the spherical mean transform with centers on a plane in R 3 which is also called the sonar transform. Standard inversion formulas require data over all spheres, but typically, the data are limited in the sense that the centers and radii are in a compact set. Our reconstruction operator is local because, to reconstruct at x, one needs only spheres that pass near x, and the operator reconstructs singularities, such as object boundaries. The microlocal properties of the reconstruction operator, including its symbol as a pseudodifferential operator, are given. The method is implemented using the approximate inverse, and reconstructions are given. They are evaluated in light of the microlocal properties of the reconstruction operator. (Some figures in this article are in colour only in the electronic version)

[1]  G. Uhlmann,et al.  Thermoacoustic tomography with variable sound speed , 2009, 0902.1973.

[2]  Thomas Schuster,et al.  On a Regularization Scheme for Linear Operators in Distribution Spaces with an Application to the Spherical Radon Transform , 2005, SIAM J. Appl. Math..

[3]  Jerrold E. Marsden,et al.  Review: Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, and Nolan R. Wallach, Symplectic geometry and Fourier analysis , 1979 .

[5]  François Treves,et al.  Introduction to Pseudodifferential and Fourier Integral Operators , 1980 .

[6]  Frank Natterer,et al.  Photo-acoustic inversion in convex domains , 2012 .

[7]  G. Uhlmann,et al.  Nonlocal inversion formulas for the X-ray transform , 1989 .

[8]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , 1983 .

[9]  J. Klein.  Mathematical Problems in Synthetic Aperture Radar , 2010, 1010.4859.

[10]  L. Hörmander.  Fourier integral operators. I , 1995 .

[11]  V. Palamodov,et al.  Reconstruction from limited data of arc means , 2000 .

[12]  Alfred K. Louis.  Approximate inverse for linear and some nonlinear problems , 1995 .

[13]  Markus Haltmeier,et al.  Inversion of Spherical Means and the Wave Equation in Even Dimensions , 2007, SIAM J. Appl. Math..

[14]  L. Kunyansky,et al.  Explicit inversion formulae for the spherical mean Radon transform , 2006, math/0609341.

[15]  Rakesh,et al.  Determining a Function from Its Mean Values Over a Family of Spheres , 2004, SIAM J. Math. Anal..

[16]  John A. Fawcett,et al.  Inversion of N-dimensional spherical averages , 1985 .

[17]  Lihong V. Wang,et al.  Dark-Field Confocal Photoacoustic Microscopy , 2009 .

[18]  V. Guillemin.  The radon transform on zoll surfaces , 1976 .

[19]  Bent E. Petersen.  Introduction to the Fourier transform & pseudo-differential operators , 1983 .

[20]  L. Andersson.  On the determination of a function from spherical averages , 1988 .

[21]  Thomas Schuster,et al.  The approximate inverse in action III: 3D-Doppler tomography , 2004, Numerische Mathematik.

[22]  Peter Kuchment,et al.  Mathematics of thermoacoustic tomography , 2007, European Journal of Applied Mathematics.

[23]  Yehonatan Salman,et al.  An inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions , 2012, 1208.5739.

[24]  Gunther Uhlmann,et al.  Estimates for singular radon transforms and pseudodifferential operators with singular symbols , 1990 .

[25]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[26]  Alfred K. Louis,et al.  Approximate inverse for linear and some nonlinear problems , 1995 .

[27]  Stephen J. Norton,et al.  Ultrasonic Reflectivity Imaging in Three Dimensions: Exact Inverse Scattering Solutions for Plane, Cylindrical, and Spherical Apertures , 1981, IEEE Transactions on Biomedical Engineering.

[28]  Alexander Katsevich,et al.  Local Tomography for the Limited-Angle Problem , 1997 .

[29]  Eric Todd Quinto,et al.  The dependence of the generalized Radon transform on defining measures , 1980 .

[30]  Thomas Schuster,et al.  The Method of Approximate Inverse: Theory and Applications , 2007 .

[31]  Terry M. Peters.  Introduction to the Fourier Transform , 1998 .

[32]  Zhengyu Wang,et al.  Error Estimation for Nonlinear Complementarity Problems via Linear Systems with Interval Data , 2008 .

[33]  V. Palamodov.  A uniform reconstruction formula in integral geometry , 2011, 1111.6514.

[34]  William Rundell,et al.  Surveys on solution methods for inverse problems , 2000 .

[35]  Shlomo Sternberg,et al.  Geometric Asymptotics, Revised edition , 1977 .

[36]  Suresh Eswarathasan.  Microlocal analysis of scattering data for nested conormal potentials , 2011, 1103.6015.

[37]  Y. Egorov,et al.  Fourier Integral Operators , 1994 .

[38]  Thomas Schuster,et al.  The approximate inverse in action II: convergence and stability , 2003, Math. Comput..

[39]  Lihong V. Wang.  Photoacoustic imaging and spectroscopy , 2009 .

[40]  E. T. Quinto.  Support Theorems for the Spherical Radon Transform on Manifolds , 2006 .

[41]  P. Kuchment,et al.  On local tomography , 1995 .

[42]  S. Gindikin,et al.  Holomorphic horospherical transform on noncompactly causal spaces , 2006, math/0601728.

[43]  Otmar Scherzer,et al.  Filtered backprojection for thermoacoustic computed tomography in spherical geometry , 2005, Mathematical Methods in the Applied Sciences.

[44]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[45]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[46]  Jack K. Cohen,et al.  Velocity inversion procedure for acoustic waves , 1979 .

[47]  Plamen Stefanov,et al.  Is a Curved Flight Path in SAR Better than a Straight One? , 2012, SIAM J. Appl. Math..

[48]  David V. Finch,et al.  Microlocal Analysis of the X-Ray Transform with Sources on a Curve , 2003 .