A general framework for nonlinear multigrid inversion

Determining optical material parameters with motion in structured illumination

A set of power measurements as a function of controlled nanopositioner movement of a planar film arrangement in a standing wave field is presented as a means to obtain the thicknesses and the dielectric constants to a precision dictated by noise in an exciting laser beam and the positioning and detector process, all of which can be refined with averaging. From a mutual information perspective, knowing the set of positions at which measurements are performed adds information. While applicable to any arrangement of planar films, the implementation considered involves thin transmissive membranes, as are employed in applications such as optomechanics. We show that measured power data as a function of object position provides sensitivity to the film refractive index and far-subwavelength thickness. Use of a cost function allows iterative retrieval of the film parameters, and a multi-resolution framework is described as a computationally efficient procedure. The approach is complementary to ellipsometry and could play an important role in routine film characterization studies for fields involving solid state material processing, as is common in the semiconductor device field.

Combining dual-tree complex wavelets and multiresolution in iterative CT reconstruction with application to metal artifact reduction

Background

This paper investigates the benefits of data filtering via complex dual wavelet transform for metal artifact reduction (MAR). The advantage of using complex dual wavelet basis for MAR was studied on simulated dental computed tomography (CT) data for its efficiency in terms of noise suppression and removal of secondary artifacts. Dual-tree complex wavelet transform (DT-CWT) was selected due to its enhanced directional analysis of image details compared to the ordinary wavelet transform. DT-CWT was used for multiresolution decomposition within a modified total variation (TV) regularized inversion algorithm.

Methods

In this study, we have tested the multiresolution TV (MRTV) approach with DT-CWT on a 2D polychromatic jaw phantom model with Gaussian and Poisson noise. High noise and sparse measurement settings were used to assess the performance of DT-CWT. The results were compared to the outcome of the single-resolution reconstruction and filtered back-projection (FBP) techniques as well as reconstructions with Haar wavelet basis.

Results

The results indicate that filtering of wavelet coefficients with DT-CWT effectively removes the noise without introducing new artifacts after inpainting. Furthermore, adoption of multiple resolution levels yield to a more robust algorithm compared to varying the regularization strength.

Conclusions

The multiresolution reconstruction with DT-CWT is also more robust when reconstructing the data with sparse projections compared to the single-resolution approach and Haar wavelets.

Scatter Corrections in X-Ray Computed Tomography: A Physics-Based Analysis

Fundamental limits for the calculation of scattering corrections within X-ray computed tomography (CT) are found within the independent atom approximation from an analysis of the cross sections, CT geometry, and the Nyquist sampling theorem, suggesting large reductions in computational time compared to existing methods. By modifying the scatter by less than 1 %, it is possible to treat some of the elastic scattering in the forward direction as inelastic to achieve a smoother elastic scattering distribution. We present an analysis showing that the number of samples required for the smoother distribution can be greatly reduced. We show that fixed forced detection can be used with many fewer points for inelastic scattering, but that for pure elastic scattering, a standard Monte Carlo calculation is preferred. We use smoothing for both elastic and inelastic scattering because the intrinsic angular resolution is much poorer than can be achieved for projective tomography. Representative numerical examples are given.

A phantom-based forward projection approach in support of model-based iterative reconstructions for HAADF-STEM tomography

We introduce a forward model for the computation of high angle annular dark field (HAADF) images of nano-crystalline spherical particles and apply it to image simulations for assemblies of nano-spheres of Al, Cu, and Au with a range of sizes, as well as an artificial bi-sphere, consisting of solid hemispheres of Al and Cu or Al and Au. Comparison of computed intensity profiles with experimental observations on Al spheres at different microscope accelerating voltages provides confidence in the forward model. Simulated tomographic tilt series for both HAADF and bright field (BF) images are then used to illustrate that the model-based iterative reconstruction (MBIR) approach is capable of reconstructing sphere configurations of mixed atomic number, with the correct relative reconstructed intensity ratio proportional to the square of the atomic number ratio.

In vivo mouse fluorescence imaging for folate-targeted delivery and release kinetics

Many cancer cells over-express folate receptors, and this provides an opportunity for both folate-targeted fluorescence imaging and the development of targeted anti-cancer drugs. We present an optical imaging modality that allows for the monitoring and evaluation of drug delivery and release through disulfide bond reduction inside a tumor in vivo for the first time. A near-infrared folate-targeting fluorophore pair was synthesized and used to image a xenograft tumor grown from KB cells in a live mouse. The in vivo results are shown to be in agreement with previous in vitro studies, confirming the validity and feasibility of our method as an effective tool for preclinical studies in drug development.

A model based iterative reconstruction algorithm for high angle annular dark field-scanning transmission electron microscope (HAADF-STEM) tomography

High angle annular dark field (HAADF)-scanning transmission electron microscope (STEM) data is increasingly being used in the physical sciences to research materials in 3D because it reduces the effects of Bragg diffraction seen in bright field TEM data. Typically, tomographic reconstructions are performed by directly applying either filtered back projection (FBP) or the simultaneous iterative reconstruction technique (SIRT) to the data. Since HAADF-STEM tomography is a limited angle tomography modality with low signal to noise ratio, these methods can result in significant artifacts in the reconstructed volume. In this paper, we develop a model based iterative reconstruction algorithm for HAADF-STEM tomography. We combine a model for image formation in HAADF-STEM tomography along with a prior model to formulate the tomographic reconstruction as a maximum a posteriori probability (MAP) estimation problem. Our formulation also accounts for certain missing measurements by treating them as nuisance parameters in the MAP estimation framework. We adapt the iterative coordinate descent algorithm to develop an efficient method to minimize the corresponding MAP cost function. Reconstructions of simulated as well as experimental data sets show results that are superior to FBP and SIRT reconstructions, significantly suppressing artifacts and enhancing contrast.

Small animal optical diffusion tomography with targeted fluorescence

Despite the broad impact in medicine that optics can bring, thus far practical approaches are limited to weak scatter or near-surface monitoring. We show a method that utilizes a laser topography scan and a diffusion equation model to describe the photon transport, together with a multiresolution unstructured grid solution to the nonlinear optimization measurement functional, that overcomes these limitations. We conclude that it is possible to achieve whole body optical imaging with a resolution suitable for finding cancer nodules within an organ during surgery, with the aid of a targeted imaging agent.

A filtering approach to edge preserving MAP estimation of images

The authors present a computationally efficient technique for maximum a posteriori (MAP) estimation of images in the presence of both blur and noise. The image is divided into statistically independent regions. Each region is modelled with a WSS Gaussian prior. Classical Wiener filter theory is used to generate a set of convex sets in the solution space, with the solution to the MAP estimation problem lying at the intersection of these sets. The proposed algorithm uses an underlying segmentation of the image, and a means of determining the segmentation and refining it are described. The algorithm is suitable for a range of image restoration problems, as it provides a computationally efficient means to deal with the shortcomings of Wiener filtering without sacrificing the computational simplicity of the filtering approach. The algorithm is also of interest from a theoretical viewpoint as it provides a continuum of solutions between Wiener filtering and Inverse filtering depending upon the segmentation used. We do not attempt to show here that the proposed method is the best general approach to the image reconstruction problem. However, related work referenced herein shows excellent performance in the specific problem of demosaicing.

Diffuse Optics for Tissue Monitoring and Tomography

This review describes the diffusion model for light transport in tissues and the medical applications of diffuse light. Diffuse optics is particularly useful for measurement of tissue hemodynamics, wherein quantitative assessment of oxy- and deoxy-hemoglobin concentrations and blood flow are desired. The theoretical basis for near-infrared or diffuse optical spectroscopy (NIRS or DOS, respectively) is developed, and the basic elements of diffuse optical tomography (DOT) are outlined. We also discuss diffuse correlation spectroscopy (DCS), a technique whereby temporal correlation functions of diffusing light are transported through tissue and are used to measure blood flow. Essential instrumentation is described, and representative brain and breast functional imaging and monitoring results illustrate the workings of these new tissue diagnostics.

SART-type image reconstruction from a limited number of projections with the sparsity constraint

Based on the recent mathematical findings on solving the linear inverse problems with sparsity constraints by Daubechiesx et al., here we adapt a simultaneous algebraic reconstruction technique (SART) for image reconstruction from a limited number of projections subject to a sparsity constraint in terms of an invertible compression transform. The algorithm is implemented with an exemplary Haar wavelet transform and tested with a modified Shepp-Logan phantom. Our preliminary results demonstrate that the sparsity constraint helps effectively improve the quality of reconstructed images and reduce the number of necessary projections.

A fast multilevel algorithm for wavelet-regularized image restoration

We present a multilevel extension of the popular "thresholded Landweber" algorithm for wavelet-regularized image restoration that yields an order of magnitude speed improvement over the standard fixed-scale implementation. The method is generic and targeted towards large-scale linear inverse problems, such as 3-D deconvolution microscopy. The algorithm is derived within the framework of bound optimization. The key idea is to successively update the coefficients in the various wavelet channels using fixed, subband-adapted iteration parameters (step sizes and threshold levels). The optimization problem is solved efficiently via a proper chaining of basic iteration modules. The higher level description of the algorithm is similar to that of a multigrid solver for PDEs, but there is one fundamental difference: the latter iterates though a sequence of multiresolution versions of the original problem, while, in our case, we cycle through the wavelet subspaces corresponding to the difference between successive approximations. This strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation. We establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer. We also provide experimental evidence that the improvement in convergence rate is essentially determined by the (unconstrained) linear part of the algorithm, irrespective of the type of wavelet. Finally, we illustrate the technique with some image deconvolution examples, including some real 3-D fluorescence microscopy data.

An Adaptive Multigrid method for modeling photon transport through biological tissues in bioluminescence tomography

In this paper, we develop an Adaptive Multigrid method (AMGM) to model photon transport through biological tissues in bioluminescence tomography. In our method, the smoothing operation on fine levels and residual correction on coarse levels in V-Cycle offer fast convergence rate for this forward problem. Using a heterogeneous phantom, the methodology is validated by Monte Carlo simulations, and the computation speed is much higher than conventional smoothing iteration methods on a single grid. In actual biomedical imaging applications, especially when there are many sources in small animal body, AMGM is potential to accurately simulate the forward problem at very low computation cost.

Multigrid tomographic inversion with variable resolution data and image spaces

A multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed. Since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements, the proposed algorithm further reduces the computation relative to conventional multigrid approaches, which change only the image space resolution at coarse scales. The advantage is particularly important for data-rich applications, where data resolutions may differ for different scales. Applications of the approach to Bayesian reconstruction algorithms in transmission and emission tomography with a generalized Gaussian Markov random field image prior are presented, both with a Poisson noise model and with a quadratic data term. Simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed-grid iterative coordinate descent method and a multigrid method with fixed-data resolution.

Estimation of kinetic model parameters in fluorescence optical diffusion tomography

We present a technique for reconstructing the spatially dependent dynamics of a fluorescent contrast agent in turbid media. The dynamic behavior is described by linear and nonlinear parameters of a compartmental model or some other model with a deterministic functional form. The method extends our previous work in fluorescence optical diffusion tomography by parametrically reconstructing the time-dependent fluorescent yield. The reconstruction uses a Bayesian framework and parametric iterative coordinate descent optimization, which is closely related to Gauss-Seidel methods. We demonstrate the method with a simulation study.