The imaging of shape perturbation and chromophore concentration using Diffuse Optical Tomography (DOT) data can be mathematically described as an ill-posed and non-linear inverse problem. The reconstruction algorithm for hyperspectral data using a linearized Born model is prohibitively expensive, both in terms of computation and memory. We model the shape of the perturbation using parametric level-set approach (PaLS). We discuss novel computational strategies for reducing the computational cost based on a Krylov subspace approach for parameteric linear systems and a compression strategy for the parameter-to-observation map. We will demonstrate the validity of our approach by comparison with experiments.
Analysis of the quasi-sinusoidal temporal signals measured by a Diffuse Optical Tomography (DOT) instrument can be used to determine both quantitative and qualitative characteristics of functional brain activities arising from visual and auditory simulations, motor activities, and cognitive tasks performances. Once the activated regions in the brain are resolved using DOT, the temporal resolution of this modality is such that one can track the spatial evolution (both the location and morphology) of these regions with time. In this paper, we explore a state-estimation approach using Extended Kalman Filters to track the dynamics of functionally activated brain regions. We develop a model to determine the size, shape, location and contrast of an area of activity as a function of time. Under the assumption that previously acquired MRI data has provided us with a segmentation of the brain, we restrict the location of the area of functional activity to the thin, cortical sheet. To describe the geometry of the region, we employ a mathematical model in which the projection of the area of activity onto the plane of the sensors is assumed to be describable by a low dimensional algebraic curve. In this study, we consider in detail the case where the perturbations in optical absorption parameters arising due to activation are confined to independent regions in the cortex layer. We estimate the geometric parameters (axis lengths, rotation angle, center positions) defining the best fit ellipse for the activation area's projection onto the source-detector plane. At a single point in time, an adjoint field-based nonlinear inversion routine is used to extract the activated area's information. Examples of the utility of the method will be shown using synthetic data.
KEYWORDS: Absorption, Diffusion, Brain, 3D image reconstruction, 3D modeling, Scattering, Reconstruction algorithms, 3D image processing, Data modeling, Brain imaging
We develop a parametric, shape-based image reconstruction algorithm
for the joint reconstruction of the optical absorption and diffusion coefficients in the brain using diffuse optical tomographic data. Specifically, we study the recovery of the geometry of an unknown number of 2D closed contours located on a 2D manifold (the cortex) in 3-space. We describe an approach for a brain model in which we assume the existence of a one-to-one map from the surface of the cortex to a subset of the plane. We use a new, parametric level set approach to map shapes on the plane to structures on the cortex. Our optimization-based reconstruction algorithm evolves shapes on the plane while finding absorption and reduced scattering values inside each shape. Preliminary numerical simulation results show the promise of our approach.
The finite difference frequency domain is used to study the scattering of buried non-metallic mine-like targets to determine the feasibility of identifying mines form shape features. It is shown that for constant cross-sectional target area - approximately 100 cm2 - the scattered fields of targets with roughly the same height-to-width aspect ratio at 500 MHz are virtually indistinguishable regardless of burial depth. A comparison of the field obtained for mine-like targets of different aspect ratios, but with constant area, buried at a depth of 5 cm, shows marked differences, as does scattered field for GPR frequencies above 700 MHz. The conclusion of this study is that while low GPR sensing frequencies may help to detect shallow anomalies, they do not supply any useful information about the shape details - particularly the edges - of the buried non-metallic mine-like targets.
In this paper we consider new methods for localizing and characterizing the structure of an anomalous areas embedded in an overall region of interest given sparse observations of diffuse photon density wavefields. Unlike traditional techniques which use the scattered field measurements first to form an image of the full region and then post-process the resulting reconstruction to localize areas of interest, our approach finds anomalies directly from the data. To accomplish this, we model the unknowns as a superposition of a slowly varying perturbation on a background of unknown structure. We assume that the perturbation is delineated from the background by a smooth perimeter which is modeled as a spline curve of unknown knot sequence. A greedy-type approach is employed to deform the curve in a manner which optimizes a cost function enforcing both consistency with the data along with a regularization term designed to reflect prior information we have concerning the likely structure of anomalies. As the algorithm progresses, we adaptively determine the optimal weighting coefficients describing both the texture of the anomaly as well as that of the background. Examples of this approach are provided for a diffuse photon density wave problem arising in a bio-imaging application.
KEYWORDS: Signal to noise ratio, Absorption, Reconstruction algorithms, Sensors, 3D image processing, Scattering, Algorithm development, Stereoscopy, Data modeling, Interference (communication)
In this paper we examine the performance of a number of linear techniques for reconstructing the 3-D distribution of absorption coefficient within a highly scattering medium using the diffuse photon density wave (DPDW) approximation. The simulation consists of a coplanar array of sources and detectors at the boundary of an infinite slab medium. The primary difficulty in the linear reconstruction of the 3D volume from a 2D array of measurement is that the forward matrix is both underdetermined and ill-conditioned.
Infrared techniques can be used to detect buried objects such as landmines or shallow-buried waste, by taking advantage of the diurnal heating cycle. Improvements in performance may be expected with the addition of a microwave source to heat the ground and the buried objects. We describe an experiment to demonstrate the heating effect and a two-dimensional model which describes the phenomena involved. It is suggested that variations in heating with angle of incidence and microwave wavelength may be useful discriminants.
In certain applications, the discretization of 2D integral equations can lead to system involving matrices with block Toeplitz-Toeplitz block structure. Iterative Krylov subspace methods are sometimes employed to find regularized solutions to the related 2D discrete ill-posed problems; however, preconditioned which filter noise are needed to speed convergence to regularized solutions. We describe a preconditioning techniques based on the Toeplitz-type structure of the matrix which generalizes the approaches in (1) and (2) to take advantage of symmetry and real arithmetic operations. We use fast sine transforms to transform the original system to a system whose matrix has partially reconstructible Cauchy-like blocks. The preconditioner is a block diagonal, rank m approximation to this matrix, with Cauchy-like blocks each augmented by an identity of appropriate dimension. We note that the initialization cost is in general less than that for the similar 2D preconditioner in (2) which does not take advantage of symmetry. Several examples are given which illustrate the success of our preconditioned methods.
In this paper, we introduce the L-hypersurface method for use in linear inverse problems. The new methods is intended to select multiple regularization parameters simultaneously. It is a multidimensional extension of classical L-curve method and hence does not require any specific knowledge about the noise level or signal semi-norm. We give examples of the L-hypersurface method applied the linear inverse problems posed in the wavelet domain and evaluate the performance of the new method on a signal restoration experiment.
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