Blind deconvolution of medical ultrasound images: parametric inverse filtering approach

©2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TIP.2007.910179The problem of reconstruction of ultrasound images by means of blind deconvolution has long been recognized as one of the central problems in medical ultrasound imaging. In this paper, this problem is addressed via proposing a blind deconvolution method which is innovative in several ways. In particular, the method is based on parametric inverse filtering, whose parameters are optimized using two-stage processing. At the first stage, some partial information on the point spread function is recovered. Subsequently, this information is used to explicitly constrain the spectral shape of the inverse filter. From this perspective, the proposed methodology can be viewed as a ldquohybridizationrdquo of two standard strategies in blind deconvolution, which are based on either concurrent or successive estimation of the point spread function and the image of interest. Moreover, evidence is provided that the ldquohybridrdquo approach can outperform the standard ones in a number of important practical cases. Additionally, the present study introduces a different approach to parameterizing the inverse filter. Specifically, we propose to model the inverse transfer function as a member of a principal shift-invariant subspace. It is shown that such a parameterization results in considerably more stable reconstructions as compared to standard parameterization methods. Finally, it is shown how the inverse filters designed in this way can be used to deconvolve the images in a nonblind manner so as to further improve their quality. The usefulness and practicability of all the introduced innovations are proven in a series of both in silico and in vivo experiments. Finally, it is shown that the proposed deconvolution algorithms are capable of improving the resolution of ultrasound images by factors of 2.24 or 6.52 (as judged by the autocorrelation criterion) depending on the type of regularization method used

Second Order Differences of Cyclic Data and Applications in Variational Denoising

In many image and signal processing applications, as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis or color image restoration in HSV or LCh spaces the data has its range on the one-dimensional sphere $\mathbb S^1$. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for $\mathbb S^1$-valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances the absolute cyclic second order differences have only values in [0,{\pi}]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm.Comment: 32 pages, 16 figures, shortened version of submitted manuscrip

Vision Sensors and Edge Detection

Vision Sensors and Edge Detection book reflects a selection of recent developments within the area of vision sensors and edge detection. There are two sections in this book. The first section presents vision sensors with applications to panoramic vision sensors, wireless vision sensors, and automated vision sensor inspection, and the second one shows image processing techniques, such as, image measurements, image transformations, filtering, and parallel computing

Provably robust estimation of modulo 1 samples of a smooth function with applications to phase unwrapping

Consider an unknown smooth function $f: [0,1]^d \rightarrow \mathbb{R}$, and say we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) + \eta_i)\mod 1$, for $x_i \in [0,1]^d$, where $\eta_i$ denotes the noise. Given the samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robust estimates of the clean samples $f(x_i) \bmod 1$. We formulate a natural approach for solving this problem, which works with angular embeddings of the noisy mod 1 samples over the unit circle, inspired by the angular synchronization framework. This amounts to solving a smoothness regularized least-squares problem -- a quadratically constrained quadratic program (QCQP) -- where the variables are constrained to lie on the unit circle. Our approach is based on solving its relaxation, which is a trust-region sub-problem and hence solvable efficiently. We provide theoretical guarantees demonstrating its robustness to noise for adversarial, and random Gaussian and Bernoulli noise models. To the best of our knowledge, these are the first such theoretical results for this problem. We demonstrate the robustness and efficiency of our approach via extensive numerical simulations on synthetic data, along with a simple least-squares solution for the unwrapping stage, that recovers the original samples of $f$ (up to a global shift). It is shown to perform well at high levels of noise, when taking as input the denoised modulo $1$ samples. Finally, we also consider two other approaches for denoising the modulo 1 samples that leverage tools from Riemannian optimization on manifolds, including a Burer-Monteiro approach for a semidefinite programming relaxation of our formulation. For the two-dimensional version of the problem, which has applications in radar interferometry, we are able to solve instances of real-world data with a million sample points in under 10 seconds, on a personal laptop.Comment: 68 pages, 32 figures. arXiv admin note: text overlap with arXiv:1710.1021

Smooth functions and their use in optical modeling and design

Analytical description of unknown smooth optical functions such as optical surface and wavefront phases will have profound importance in optical modeling and design. Polynomial models have been extensively used to describe smooth function. Various forms of polynomials for describing the smooth functions may be considered both in optical modeling and design. In optics, the Zernike polynomials are potential candidates to describe optical surface and wavefront phases. However, they are restrained to specific geometry and suffer from numerical instability, especially for describing complex functions. More recently, spline model functions were also investigated for describing the optical surface shape and wavefront phase

Image Deblurring Using Derivative Compressed Sensing for Optical Imaging Application

Reconstruction of multidimensional signals from the samples of their partial derivatives is known to be a standard problem in inverse theory. Such and similar problems routinely arise in numerous areas of applied sciences, including optical imaging, laser interferometry, computer vision, remote sensing and control. Though being ill-posed in nature, the above problem can be solved in a unique and stable manner, provided proper regularization and relevant boundary conditions. In this paper, however, a more challenging setup is addressed, in which one has to recover an image of interest from its noisy and blurry version, while the only information available about the imaging system at hand is the amplitude of the generalized pupil function (GPF) along with partial observations of the gradient of GPF's phase. In this case, the phase-related information is collected using a simplified version of the Shack-Hartmann interferometer, followed by recovering the entire phase by means of derivative compressed sensing. Subsequently, the estimated phase can be combined with the amplitude of the GPF to produce an estimate of the point spread function (PSF), whose knowledge is essential for subsequent image deconvolution. In summary, the principal contribution of this work is twofold. First, we demonstrate how to simplify the construction of the Shack-Hartmann interferometer so as to make it less expensive and hence more accessible. Second, it is shown by means of numerical experiments that the above simplification and its associated solution scheme produce image reconstructions of the quality comparable to those obtained using dense sampling of the GPF phase

Mathematics and Algorithms in Tomography

This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis