Reconstruction in tomography with diffracting sources
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In this dissertation, we ﬁrst derive exact reconstuction algorithms for thermoacoustic tomography (TAT) and broadband diﬀraction tomography (a linearized inverse scattering problem) using derived time-reversal formulas. Then we focus on some important practical problems of TAT including the limited-view problem, the eﬀects of acoustic heterogeneity, and fast reconstruction algorithms. In Chapter II, we propose time-reversal methods and apply them to tomography with diﬀrating sources. We ﬁrst develop time-domain methods to time-reverse a transient scalar wave using only the ﬁeld measured on an arbitrary closed surface enclosing the initial sources. Under certain conditions, a time-reversed ﬁeld can be obtained with the delay-and-sum algorithm (backprojection to spheres) used in synthetic aperture imaging.Consequently, the physicalmeaningandthe validconditions of this widely used algorithm are revealed quantitatively for the ﬁrst time from basic physics. Then exact reconstruction for TAT and broadband diﬀraction tomography is proposed by time-reversing the measured ﬁeld back to the time when each source or secondary source is excited. The theoretical conclusions are supported by a numerical simulation ofthree-dimensional diﬀraction tomography.The extension ofour time-reversal methods to the case using Green function in a heterogeneous medium is also discussed. In Chapter III, the limited-view problem is studied for TAT. We deﬁne a "detection region," within which all points have suﬃcient detection views. It is explained analytically and shown numerically that the boundaries of any object inside this region can be recovered stably.Otherwise some sharp details become blurred.One can identify in advance the parts of the boundaries that will be aﬀected if the detection view is insuﬃcient. Computations are conducted for both numerically simulated and experimental data. The reconstructions conﬁrm our theoretical predictions. In Chapter IV, the eﬀects of wavefront distortions induced by acoustic heterogeneities in breast TAT are studied. Amplitude distortions are shown to be insigniﬁcant for diﬀerent scales of acoustic heterogeneities. After that we consider the eﬀects of phase distortions (errors in time-of-ﬂight) in our numerical studies. The numerical results on the spreads of point sources and boundaries caused by the phase distortions are in good agreement with the proposed formula. We also demonstrate that the blurring of images can be compensated for by using the distribution of acoustic velocityin the tissues in the reconstructions. In Chapter V, we discuss exact and fast Fourier-domain reconstruction algorithms for TAT in planar and circular conﬁgurations.
Xu, Yuan (2003). Reconstruction in tomography with diffracting sources. Doctoral dissertation, Texas A&M University. Texas A&M University. Available electronically from