Abstract—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. Index Terms—Bound optimization, confocal, convergence acceleration, deconvolution, fast, fluorescence, inverse problems,-regularization, majorize-minimize, microscopy, multigrid, multilevel, multiresolution, multiscale, nonlinear, optimization transfer, preconditioning, reconstruction, restoration, sparsity,

This paper introduces a novel method for reconstructing optical tomography images using pre-computed transforms. Our approach is to pre-compute and store the inverse matrix required for MAP reconstruction using lossy source coding techniques. We show how lossy source coding techniques can be used to store the large and non-sparse matrix by applying a wavelet transform in the image space and appropriate orthonormal transforms in the sensor space. Lossy coding dramatically reduces the the number of non-zero coefficients, thereby proportionately reducing both the required storage and computation time. However, if the number of sensor measurements is large, the storage and computation of the orthonormal transforms can become prohibitive. For this purpose, we introduce a general method for approximating any orthonormal transform by a series of sparse binary transforms. This sparse matrix transform technique is then used together with lossy coding to result in a fast reconstruction algorithm for optical tomography. Simulations indicate that the technique can dramatically reduce the storage and computation requirements in reconstruction by exploiting redundancy in the transformed matrices. Index Terms — Optical tomography, sparse matrices, image coding

contract CCR-0073357.

Abstract — A barrier to the use of optical tomography in practical applications is the high computational cost of iterative image reconstruction. This paper introduces a novel method for direct reconstruction of the image from a pre-computed and stored inverse matrix. Since the inverse matrix for optical tomography is generally quite large and not sparse, it is necessary to store the inverse matrix using lossy source coding techniques. A key innovation is the method used for matrix representation and the technique used for computing the required matrix-vector product. This representation is based on transforms of the image and sensor spaces which are designed to minimize reconstructed image distortion. Simulations indicate that the technique can dramatically reduce the storage and computation requirements by exploiting redundancy in the transformed matrix. I.

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Medical imaging typically requires the reconstruction of a limited region of interest (ROI) to obtain a high resolution image of the anatomy of interest. Although targeted reconstruction is straightforward for analytical reconstruction methods, it is more complicated for statistical iterative techniques, which must reconstruct all objects in the field of view (FOV) to account for all sources of attenuation along the ray paths from x-ray source to detector. A brute force approach would require the reconstruction of the full field of view in highresolution, but with prohibitive computational cost. In this paper, we propose a multi-resolution approach to accelerate targeted iterative reconstruction using the non-homogeneous ICD (NH-ICD) algorithm. NH-ICD aims at speeding up convergence of the coordinate descent algorithm by selecting preferentially those voxels most in need of updating. To further optimize ROI reconstruction, we use a multi-resolution approach which combines three separate improvements. First, we introduce the modified weighted NH-ICD algorithm, which weights the pixel selection criteria according to the position of the voxel relative to the ROI to speed up convergence within the ROI. Second, we propose a simple correction to the error sinogram to correct for inconsistencies between resolutions when the forward model is not scale invariant. Finally, we leverage the flexibility of the ICD algorithm to add selected edge pixels outside the ROI to the ROI reconstruction in order to minimize transition artifacts at the ROI boundary. Experiments on clinical data illustrate how each component of the method improves convergence speed and image quality.