Mathematics of Hybrid Imaging: A Brief Review

The article provides a brief survey of the mathematics of newly being developed so-called “hybrid” (also called “multi-physics” or “multi-wave”) imaging techniques.

[1]  G. Bal,et al.  Inverse scattering and acousto-optic imaging. , 2009, Physical review letters.

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

[3]  Linh V. Nguyen,et al.  Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media , 2008 .

[4]  F. John Plane Waves and Spherical Means: Applied To Partial Differential Equations , 1981 .

[5]  Gunther Uhlmann,et al.  Inverse Problems: Theory and Applications , 2003 .

[6]  Linh V. Nguyen On singularities and instability of reconstruction in thermoacoustic tomography , 2009 .

[7]  Peter Kuchment,et al.  Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed , 2007, 0706.0598.

[8]  Joyce R. McLaughlin,et al.  Unique identifiability of elastic parameters from time-dependent interior displacement measurement , 2004 .

[9]  D. A. Popov,et al.  Image Restoration in Optical-Acoustic Tomography , 2004, Probl. Inf. Transm..

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

[11]  Linh V. Nguyen,et al.  Range conditions for a spherical mean transform and global extendibility of solutions of the darboux equation , 2010 .

[12]  Nedjeljko Frančula The National Academies Press , 2013 .

[13]  J Jossinet,et al.  Experimental measurement of the acousto-electric interaction signal in saline solution. , 2000, Ultrasonics.

[14]  Alexander Graham Bell,et al.  Upon the production and reproduction of sound by light , 1880 .

[15]  Azriel Z. Genack,et al.  Acousto-optic tomography with multiply scattered light , 1997 .

[16]  Tuan Vo-Dinh,et al.  Biomedical Photonics Handbook , 2003 .

[17]  Habib Ammari,et al.  An Introduction to Mathematics of Emerging Biomedical Imaging , 2008 .

[18]  R. Kruger,et al.  Photoacoustic ultrasound (PAUS)--reconstruction tomography. , 1995, Medical physics.

[19]  J. Jossinet,et al.  Quantitative assessment of ultrasound-induced resistance change in saline solution , 2000, Medical and Biological Engineering and Computing.

[20]  Jérôme Fehrenbach,et al.  Imaging by Modification: Numerical Reconstruction of Local Conductivities from Corresponding Power Density Measurements , 2009, SIAM J. Imaging Sci..

[21]  Editors , 1986, Brain Research Bulletin.

[22]  Rakesh,et al.  The range of the spherical mean value operator for functions supported in a ball , 2006 .

[23]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[24]  S. Arridge,et al.  Estimating chromophore distributions from multiwavelength photoacoustic images. , 2009, Journal of the Optical Society of America. A, Optics, image science, and vision.

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

[26]  Guillaume Bal,et al.  Quantitative thermo-acoustics and related problems , 2011 .

[27]  Alexander A. Oraevsky,et al.  Image reconstruction in 3D optoacoustic tomography system with hemispherical transducer array , 2002, SPIE BiOS.

[28]  Guillaume Bal,et al.  Inverse transport theory of photoacoustics , 2009, 0908.4012.

[29]  Peter Kuchment,et al.  Range conditions for a spherical mean transform , 2009, 0902.4272.

[30]  Angela W. Ma,et al.  Current Density Impedance Imaging , 2008, IEEE Transactions on Medical Imaging.

[31]  Gunther Uhlmann,et al.  Inverse boundary value problems and applications , 1992 .

[32]  G. Maret,et al.  Ultrasonic modulation of multiply scattered light , 1995 .

[33]  Guillaume Bal,et al.  On multi-spectral quantitative photoacoustic tomography in diffusive regime , 2012 .

[34]  G. Uhlmann,et al.  The Calderón problem with partial data , 2004, math/0405486.

[35]  Guillaume Bal,et al.  Inverse diffusion theory of photoacoustics , 2009, 0910.2503.

[36]  Alexandru Tamasan,et al.  Recovering the conductivity from a single measurement of interior data , 2009 .

[37]  Dustin Steinhauer A Reconstruction Procedure for Thermoacoustic Tomography in the Case of Limited Boundary Data , 2009 .

[38]  Plamen Stefanov,et al.  Thermoacoustic tomography arising in brain imaging , 2010, 1009.1687.

[39]  F. Natterer The Mathematics of Computerized Tomography , 1986 .

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

[41]  Frank K. Tittel,et al.  Laser-based optoacoustic imaging in biological tissues , 1994, SPIE LASE.

[42]  S. Helgason Integral Geometry and Radon Transforms , 2010 .

[43]  David Isaacson,et al.  Comment on Calderon's Paper: "On an Inverse Boundary Value Problem" , 1989 .

[44]  A. Nachman,et al.  Global uniqueness for a two-dimensional inverse boundary value problem , 1996 .

[45]  Huabei Jiang,et al.  Simultaneous reconstruction of acoustic and optical properties of heterogeneous media by quantitative photoacoustic tomography. , 2006, Optics express.

[46]  J. Craggs Applied Mathematical Sciences , 1973 .

[47]  Leonid Kunyansky A series solution and a fast algorithm for the inversion of the spherical mean Radon transform , 2007 .

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

[49]  Hongkai Zhao,et al.  A New Numerical Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed , 2011, 1101.3729.

[50]  Margaret Cheney,et al.  A Mathematical Tutorial on Synthetic Aperture Radar , 2001, SIAM Rev..

[51]  G. Uhlmann Inside out : inverse problems and applications , 2003 .

[52]  Victor Palamodov Remarks on the generalFunk transform and thermoacoustic tomography , 2010 .

[53]  Lihong V. Wang,et al.  Acousto-electric tomography , 2004, SPIE BiOS.

[54]  Dustin Steinhauer A Uniqueness Theorem for Thermoacoustic Tomography in the Case of Limited Boundary Data , 2009 .

[55]  Peter Kuchment,et al.  Range conditions for the exponential Radon transform , 1996 .

[56]  Peter Kuchment,et al.  A Range Description for the Planar Circular Radon Transform , 2006, SIAM J. Math. Anal..

[57]  P. Burgholzer,et al.  Photoacoustic tomography using a fiber based Fabry-Perot interferometer as an integrating line detector and image reconstruction by model-based time reversal method , 2007, European Conference on Biomedical Optics.

[58]  G. Bal Cauchy problem for Ultrasound Modulated EIT , 2012, 1201.0972.

[59]  B. Vainberg,et al.  ON THE SHORT WAVE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF STATIONARY PROBLEMS AND THE ASYMPTOTIC BEHAVIOUR AS t???? OF SOLUTIONS OF NON-STATIONARY PROBLEMS , 1975 .

[60]  G. Bal,et al.  Inverse diffusion from knowledge of power densities , 2011, 1110.4577.

[61]  Frank K. Tittel,et al.  Laser optoacoustic tomography for medical diagnostics: principles , 1996, Photonics West.

[62]  E. T. Quinto,et al.  Range descriptions for the spherical mean Radon transform. I. Functions supported in a ball , 2006, math/0606314.

[63]  A. Kirsch On the existence of transmission eigenvalues , 2009 .

[64]  Rakesh,et al.  The spherical mean value operator with centers on a sphere , 2007 .

[65]  John C. Schotland,et al.  Tomography and Inverse Transport Theory , 2011 .

[66]  Mark A. Anastasio,et al.  Reconstruction of speed-of-sound and electromagnetic absorption distributions in photoacoustic tomography , 2006, SPIE BiOS.

[67]  Peter Kuchment,et al.  Mathematics of thermoacoustic and photoacoustic tomography , 2007 .

[68]  D. V. Sushko,et al.  A parametrix for the problem of optical-acoustic tomography , 2002 .

[69]  Yao Sun,et al.  Simultaneous reconstruction of acoustic and optical properties of heterogeneous tissues by quantitative photoacoustic tomography , 2007, SPIE BiOS.

[70]  Lihong V. Wang,et al.  Reconstructions in limited-view thermoacoustic tomography. , 2004, Medical physics.

[71]  N. Holmer,et al.  Electrical Impedance Tomography , 1991 .

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

[73]  S. Jacques,et al.  Iterative reconstruction algorithm for optoacoustic imaging. , 2002, The Journal of the Acoustical Society of America.

[74]  Hongkai Zhao,et al.  An Efficient Neumann Series-Based Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed , 2011, SIAM J. Imaging Sci..

[75]  Eric Bonnetier,et al.  Electrical Impedance Tomography by Elastic Deformation , 2008, SIAM J. Appl. Math..

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

[77]  B. T. Cox,et al.  The challenges for quantitative photoacoustic imaging , 2009, BiOS.

[78]  Y. Egorov,et al.  Partial Differential Equations IV , 1992 .

[79]  Lihong V. Wang,et al.  Thermoacoustic tomography with correction for acoustic speed variations , 2006, Physics in medicine and biology.

[80]  R.W.B. Stephens,et al.  IEEE ultrasonics symposium , 1972 .

[81]  Leonid Kunyansky,et al.  Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra , 2010, 1009.0288.

[82]  G. Beylkin The inversion problem and applications of the generalized radon transform , 1984 .

[83]  R. Huesman,et al.  Emission computed tomography , 1979 .

[84]  Gabor T. Herman,et al.  Image Reconstruction From Projections , 1975, Real Time Imaging.

[85]  P. Kuchment Generalized Transforms of Radon Type and Their Applications , 2005 .

[86]  Yulia Hristova,et al.  Time reversal in thermoacoustic tomography—an error estimate , 2008, 0812.0606.

[87]  Peter Kuchment,et al.  2D and 3D reconstructions in acousto-electric tomography , 2010, 1011.3059.

[88]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[89]  P. Malliavin,et al.  Selected Papers of Alberto P. Calderon with Commentary , 2008 .

[90]  Eric Todd Quinto,et al.  The radon transform, inverse problems, and tomography : American Mathematical Society short course, January 3-4, 2005, Atlanta, Georgia , 2006 .

[91]  A. Bell On the production and reproduction of sound by light , 1880, American Journal of Science.

[92]  D. Colton,et al.  The interior transmission problem , 2007 .

[93]  Lihong V. Wang,et al.  Limited view thermoacoustic tomography , 2002, Proceedings of the Second Joint 24th Annual Conference and the Annual Fall Meeting of the Biomedical Engineering Society] [Engineering in Medicine and Biology.

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

[95]  Plamen Stefanov,et al.  Integral Geometry of Tensor Fields on a Class of Non-Simple Riemannian Manifolds , 2006 .

[96]  Eung Je Woo,et al.  Magnetic Resonance Electrical Impedance Tomography (MREIT) , 2011, SIAM Rev..

[97]  J. Ralston Solutions of the wave equation with localized energy , 1969 .

[98]  Fioralba Cakoni,et al.  The Existence of an Infinite Discrete Set of Transmission Eigenvalues , 2010, SIAM J. Math. Anal..

[99]  Victor Palamodov Remarks on the general Funk-Radon transform and thermoacoustic tomography , 2007 .

[100]  L. Ehrenpreis The Universality of the Radon Transform , 2003 .

[101]  John Sylvester,et al.  Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators , 2011, SIAM J. Math. Anal..

[102]  Jin Zhang,et al.  Half-time image reconstruction in thermoacoustic tomography , 2005, IEEE Transactions on Medical Imaging.

[103]  Peter Kuchment,et al.  Synthetic focusing in ultrasound modulated tomography , 2009, 0901.2552.

[104]  Eung Je Woo,et al.  Magnetic resonance electrical impedance tomography (MREIT) for high-resolution conductivity imaging , 2008, Physiological measurement.

[105]  Guillaume Bal,et al.  On multi-spectral quantitative photoacoustic tomography , 2011 .

[106]  Sun,et al.  Photoacoustic monopole radiation in one, two, and three dimensions. , 1991, Physical review letters.

[107]  N. Burq Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel , 1998 .

[108]  Eric Todd Quinto,et al.  The Radon Transform, Inverse Problems, and Tomography , 2006 .

[109]  S. Helgason The Radon Transform , 1980 .

[110]  V. Palamodov Reconstructive Integral Geometry , 2004 .

[111]  Peter Kuchment,et al.  Stabilizing inverse problems by internal data , 2011, 1110.1819.

[112]  Habib Ammari,et al.  Mathematical Modeling in Photoacoustic Imaging of Small Absorbers , 2010, SIAM Rev..

[113]  Yves Capdeboscq,et al.  A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , 2003 .

[114]  A. Manduca,et al.  Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. , 1995, Science.

[115]  Alexandru Tamasan,et al.  Conductivity imaging with a single measurement of boundary and interior data , 2007 .

[116]  Simon R. Arridge,et al.  Multiple Illumination Quantitative Photoacoustic Tomography using Transport and Diffusion Models , 2011 .

[117]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[118]  Lihong V. Wang,et al.  Ultrasound‐Modulated Optical Tomography , 2012 .

[119]  Frank Natterer,et al.  Mathematical methods in image reconstruction , 2001, SIAM monographs on mathematical modeling and computation.

[120]  Richard Courant,et al.  Methods of Mathematical Physics, Volume II: Partial Differential Equations , 1963 .

[121]  Wolfgang Bangerth,et al.  Reconstructions in ultrasound modulated optical tomography , 2009, 0910.2748.

[122]  S. Helgason Support of Radon Transforms , 1980 .

[123]  Kari Astala,et al.  Calderon's inverse conductivity problem in the plane , 2006 .

[124]  B. Vainberg,et al.  Asymptotic methods in equations of mathematical physics , 1989 .

[125]  Xu Xiao Photoacoustic imaging in biomedicine , 2008 .

[126]  B. Brown,et al.  Applied potential tomography. , 1989, Journal of the British Interplanetary Society.

[127]  Otmar Scherzer,et al.  Impedance-Acoustic Tomography , 2008, SIAM J. Appl. Math..

[128]  A. Pinkus,et al.  Fundamentality of Ridge Functions , 1993 .

[129]  John Sylvester,et al.  Transmission Eigenvalues , 2008, SIAM J. Math. Anal..

[130]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[131]  Leonid Kunyansky Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm , 2008 .

[132]  V. Tuchin Handbook of Optical Biomedical Diagnostics , 2002 .

[133]  Eric Todd Quinto,et al.  Singularities of the X-ray transform and limited data tomography , 1993 .

[134]  Eric Todd Quinto,et al.  Analysis, geometry, number theory : the mathematics of Leon Ehrenpreis , 2000 .

[135]  O. Scherzer Handbook of mathematical methods in imaging , 2011 .

[136]  T. Bowen Radiation-Induced Thermoacoustic Soft Tissue Imaging , 1981 .

[137]  Linh V. Nguyen A family of inversion formulas in thermoacoustic tomography , 2009, 0902.2579.

[138]  A. Pinkus,et al.  Approximation of Multivariate Functions , 2004 .

[139]  A. Tam Applications of photoacoustic sensing techniques , 1986 .

[140]  Kyle S. Hickmann Unique determination of acoustic properties from thermoacoustic data , 2010 .

[141]  Eric Todd Quinto,et al.  Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions , 1996 .

[142]  G. Uhlmann,et al.  The Calderón problem with partial data , 2004 .

[143]  Linh V. Nguyen,et al.  Range conditions for a spherical mean transform and global extension of solutions of Darboux equation , 2009 .

[144]  S. G. Gindikin,et al.  Selected Topics in Integral Geometry , 2003 .