|
For many situations in clinical medicine and other areas, knowledge of the interior structure and properties of materials is of great practical value. Imaging schemes which employed high energy sources, such as x-rays, have been the method of choice, in part, because of the high quality images that can be produced. While the undesirable biological effects of ionizing radiation have been known for several decades, it has only been within the past 15-20 years that alternative strategies that evaluate endogenous sources or exogeneous, but non-ionizing, sources have been made available. Endogenous sources include the electrical and magnetic fields produced as a result of synaptic propagation of the chemical signals generated in neuro-muscular tissues. Electroencephalographic (EEG) [1] and magnetoencephalographic (MEG) [2] imaging methods are based on measurement of these signals. Knowledge that body tissues contain ionic species that will conduct electrical current has led to the development of elecirical impedance tomography (EIT) [3]. Within the electromagnetic spectrum, radio and microwave sources are also being explored for their suitability. When performed in the presence of a large external magnetic field, the former, has proven extremely successful in the form of magnetic resonance imaging [4,5]. The observation that mechanical energy will differentially propagate in tissue has led to acoustic imaging methods in the form of ultrasound imaging and more recently as acoustic tomography [6]. More recently, the search for identifying alternative sources for imaging studies has been extended into the near infrared range [7-9]. At these frequencies it is known that whereas light is intensely scattered by tissue, NIR photons will penetrate deeply, allowing measurements through the head of a neonate or an adult female breast [10]. The great sensitivity of optical measurements and the known relationship between tissue function and the oxygen-dependent spectral properties of hemoglobin and other heme proteins has underscored interest in this area. With the exception of MRI, the quality of images produced using these alternative sources has been significantly poorer than that achieved using high energy sources. Frequently, though, the informational value of the resultant images is not strictly dependent on an accurate mapping of anatomical structures. For example, the acquisition of signals in real-time has proven especially valuable for sonography and for BEG, MEG, and BIT imaging and the closely associated electrocardiographic imaging method. Further, the ability to directly relate the measured response to well defined physiological events has also served to extend the usefulness of these methods. 081944 1 140/931$6.OO SPIE Vol. 1887/77 While the precise reasons accounting for the image quality obtained vary with the different methods, a common aspect of all is the considerable uncertainty which can exists as to the spatial origin of, in the case of endogenous sources, or path taken by, in the case of penetrating energy, the detected signal. When electromagnetic or acoustic sources are employed, this uncertainty is a result scattering of the penetrating energy due to localized differences in permittivity that is produced as a consequence of the physiochemical and structural heterogeneity of tissues. The theoretical framework for evaluating such measurements is described by the wave equation and is the basis for Diffraction Tomography (DT) [1 1]. As general rule, problems whose detector responses are dominated by scattering are typically ill-posed as multiple descriptions of the underlying medium may be consistent with the measured data. Invariably, another attribute of these types of problems is the considerable computing effort required to produce the resultant image, especially if a map in 3-dimensions rather than 2-D is required. In practice, the cost and time constraints imposed by the computing effort dominate and is the key element in defining the application range that a particular measurement-algorithm scheme may have. Theoretically sound strategies that rely on brute force computations frequently have little practical value. The need to effectively balance the measurement-algorithm scheme with acceptable computing times has led to the development of a variety of image recovery strategies. The Born and Rytov approximations, for example, introduce the assumption that the intensity of scattered signal is small relative to the incident field thereby linearizing the problem [1 1]. Recent studies, however, have demonstrated that this approximation cannot provide sufficient accuracy in the resultant images [12,13}. Other schemes provide a nonlinear treatment of the data and obtain convergence by minimizing a specified cost functional. These methods differ in their starting points. Newton-type methods usually consider a homogeneous background and update Bom-Rytov approximations in an iterative process [12, 13]. Solutions of non-Newton methods are not dependent on employing a "good' initial first guess as to the properties of the medium [14-16]. In general, efforts to solve inverse scattering problems (ISP), especially in the 3-D case, are difficult because of the following factors: i) Ill-conditionedness; small random fluctuations in the data cause large variations in the solution; ii) inherent nonlinearity of measured data with respect to unknown coefficients; iii) very large scale computations because unkiiown coefficients essentially depend on three, rather than two spatial variables. Attempts to address these concerns invariably involve tradeoffs. For example, all of the approaches described above require minimization of some type of functional. In doing this one faces at least one of the following procedures which are rather lime consuming, even in the 2D case. i) Solution of a boundary value problem for Helmholtz-like equation; ii) computation of a large number of 2-D, or 3-D integrals on each iteration step; iii) implementation of regularization techniques that require computing multiple solutions of (i) and (ii) in order to choose a proper value of the regularization parameter; (iv) properties of minimizing cost functionals are usually not investigated theoretically, which, in principle, can lead to the presence of multiple local minimums that may further add to the computational burden for search of the global minimum, especially in the case of a large number of unknowns [17]. Clearly, the correct solution requires determining the global minimum. Overall, whereas various techniques have been successfully developed to address the issue of il-posedness, the practical applicability of these becomes increasingly problematic as the size of the computing problem grows (i.e. 3-D case) and alternative numerical strategies must be sought. Recently, a new approach for solving ISPs has been described by some of the co-authors [18-22]. We call this the Regularized Quasi-Reversibility approach (RQR). The coreof the RQR scheme consists of working directly with partial differential equations, rather than with associated integral equations (e.g. as for instance, Lippman-Schwinger integral equation). Computational efficiency is accomplished by using explicit precalculation of all needed integrals, a priori choice of the number of Fourier coefficients of the unknown function as a regularization parameter, and construction of quasisolutions ofneeded boundary value problems rather than construction of actual solutions. Alternatively, similar efficiencies are obtained through construction of a globally convex minimizing cost functional that has the attractive feature of a single minimum. Examination of the method indicates that it is computationally efficient and results presented here, and elsewhere [18,19,22] for the Helmholtz and time-dependent wave equations confirm this. For example accurate solutions in the 2-D case have been obtained for inverse problems containing 441 unknowns (solved iteratively) in less than one minute of CPU time using a YIMP-CRAY. In fact recent preliminary results (not shown) have indicated that 3-D problems involving 3,000-4,000 unknowns can be computed in less than 30 minutes. In this report we describe two versions of the RQR approach and present results demonstrating the accuracy and stability of the solutions, in the 2-D case, based on evaluation of time-independent as well as time-dependent data. We also provide a discussion which considers the potential applicability of these methods to evaluate time-resolved optical measurements from dense scattering media obtained using ultrafast laser sources operating at NIR frequencies. The compatibility of this approach with solution of the ISP based on the transport equation is also described
|
