A Stochastic Search Approach to Inverse Problems
Abstract
The focus of the thesis is on the development of a few stochastic search schemes for inverse problems and their applications in medical imaging. After the introduction in Chapter 1 that motivates and puts in perspective the work done in later chapters, the main body of the thesis may be viewed as composed of two parts: while the first part concerns the development of stochastic search algorithms for inverse problems (Chapters 2 and 3), the second part elucidates on the applicability of search schemes to inverse problems of interest in tomographic imaging (Chapters 4 and 5). The chapterwise contributions of the thesis are summarized below.
Chapter 2 proposes a Monte Carlo stochastic filtering algorithm for the recursive estimation of diffusive processes in linear/nonlinear dynamical systems that modulate the instantaneous rates of Poisson measurements. The same scheme is applicable when the set of partial and noisy measurements are of a diffusive nature. A key aspect of our development here is the filterupdate scheme, derived from an ensemble approximation of the timediscretized nonlinear Kushner Stratonovich equation, that is modified to account for Poissontype measurements. Specifically, the additive update through a gainlike correction term, empirically approximated from the innovation integral in the filtering equation, eliminates the problem of particle collapse encountered in many conventional particle filters that adopt weightbased updates. Through a few numerical demonstrations, the versatility of the proposed filter is brought forth, first with application to filtering problems with diffusive or Poissontype measurements and then to an automatic control problem wherein the exterminations of the associated cost functional is achieved simply by an appropriate redefinition of the innovation process.
The aim of one of the numerical examples in Chapter 2 is to minimize the structural response of a duffing oscillator under external forcing. We pose this problem of active control within a filtering framework wherein the goal is to estimate the control force that minimizes an appropriately chosen performance index. We employ the proposed filtering algorithm to estimate the control force and the oscillator displacements and velocities that are minimized as a result of the application of the control force. While Fig. 1 shows the time histories of the uncontrolled and controlled displacements and velocities of the oscillator, a plot of the estimated control force against the external force applied is given in Fig. 2.
(a) (b)
Fig. 1. A plot of the time histories of the uncontrolled and controlled (a) displacements and (b) velocities.
Fig. 2. A plot of the time histories of the external force and the estimated control force
Stochastic filtering, despite its numerous applications, amounts only to a directed search and is best suited for inverse problems and optimization problems with unimodal solutions. In view of general optimization problems involving multimodal objective functions with a priori unknown optima, filtering, similar to a regularized GaussNewton (GN) method, may only serve as a local (or quasilocal) search. In Chapter 3, therefore, we propose a stochastic search (SS) scheme that whilst maintaining the basic structure of a filtered martingale problem, also incorporates randomization techniques such as scrambling and blending, which are meant to aid in avoiding the socalled local traps. The key contribution of this chapter is the introduction of yet another technique, termed as the state space splitting (3S) which is a paradigm based on the principle of divideandconquer. The 3S technique, incorporated within the optimization scheme, offers a better assimilation of measurements and is found to outperform filtering in the context of quantitative photoacoustic tomography (PAT) to recover the optical absorption field from sparsely available PAT data using a bare minimum ensemble. Other than that, the proposed scheme is numerically shown to be better than or at least as good as CMAES (covariance matrix adaptation evolution strategies), one of the best performing optimization schemes in minimizing a set of benchmark functions.
Table 1 gives the comparative performance of the proposed scheme and CMAES in minimizing a set of 40dimensional functions (F1F20), all of which have their global minimum at 0, using an ensemble size of 20. Here, 10 5 is the tolerance limit to be attained for the objective function value and MAX is the maximum number of iterations permissible to the optimization scheme to arrive at the global minimum.
Table 1. Performance of the SS scheme and
Chapter 4 gathers numerical and experimental evidence to support our conjecture in the previous chapters that even a quasilocal search (afforded, for instance, by the filtered martingale problem) is generally superior to a regularized GN method in solving inverse problems. Specifically, in this chapter, we solve the inverse problems of ultrasound modulated optical tomography (UMOT) and diffraction tomography (DT). In UMOT, we perform a spatially resolved recovery of the meansquared displacements, p r of the scattering centres in a diffusive object by measuring the modulation depth in the decaying autocorrelation of the incident coherent light. This modulation is induced by the input ultrasound focussed to a specific region referred to as the region of interest (ROI) in the object. Since the ultrasoundinduced displacements are a measure of the material stiffness, in principle, UMOT can be applied for the early diagnosis of cancer in soft tissues. In DT, on the other hand, we recover the real refractive index distribution, n r of an optical fiber from experimentally acquired transmitted intensity of light traversing through it. In both cases, the filtering step encoded within the optimization scheme recovers superior reconstruction images visàvis the GN method in terms of quantitative accuracies.
Fig. 3 gives a comparative crosssectional plot through the centre of the reference and reconstructed p r images in UMOT when the ROI is at the centre of the object. Here, the anomaly is presented as an increase in the displacements and is at the centre of the ROI.
Fig. 4 shows the comparative crosssectional plot of the reference and reconstructed refractive index distributions, n r of the optical fiber in DT.
Fig. 3. Crosssectional plot through the center of the reference and reconstructed p r images.
Fig. 4. Crosssectional plot through the center of the reference and reconstructed n r distributions.
In Chapter 5, the SS scheme is applied to our main application, viz. photoacoustic tomography (PAT) for the recovery of the absorbed energy map, the optical absorption coefficient and the chromophore concentrations in soft tissues. Nevertheless, the main contribution of this chapter is to provide a singlestep method for the recovery of the optical absorption field from both simulated and experimental timedomain PAT data. A singlestep direct recovery is shown to yield better reconstruction than the generally adopted twostep method for quantitative PAT. Such a quantitative reconstruction maybe converted to a functional image through a linear map. Alternatively, one could also perform a onestep recovery of the chromophore concentrations from the boundary pressure, as shown using simulated data in this chapter. Being a Monte Carlo scheme, the SS scheme is highly parallelizable and the availability of such a machineready inversion scheme should finally enable PAT to emerge as a clinical tool in medical diagnostics.
Given below in Fig. 5 is a comparison of the optical absorption map of the SheppLogan phantom with the reconstruction obtained as a result of a direct (1step) recovery.
Fig. 5. The (a) exact and (b) reconstructed optical absorption maps of the SheppLogan phantom. The x and yaxes are in m and the colormap is in mm1.
Chapter 6 concludes the work with a brief summary of the results obtained and suggestions for future exploration of some of the schemes and applications described in this thesis.
Collections
Related items
Showing items related by title, author, creator and subject.

Analysis of MultiConductor Coupled Microstrip Lines with an Aperture in the Ground Plane for Compact Broadband Microwave Components
Packiaraj, D (20180411)In recent years, the wireless industry has witnessed tremendous development for the defense and commercial segments. The explosive growth in the modern radio frequency and microwave systems leads to an increased interest ... 
Integrated Common And Differential Mode Filters With Active Damping For Active Front End Motor Drives
Acharya, Anirudh B (20140730)IGBT based power converters acts as front end in the present day Adjustable Speed Drive (ASD). This offers many advantages and makes regenerative action possible. PWM rectifier operation produces electrically noisy DC bus ... 
Integrated Approach To Filter Design For Grid Connected Power Converters
Parikshith, B C (20101230)Design of filters used in gridconnected inverter applications involves multiple constraints. The filter requirements are driven by tight filtering tolerances of standards such as IEEE 5191992–IEEE Recommended Practices ...