Hermann Schomberg

Ultrasonic diagnostic device

A device and method for the determination of the internal structure of a body by means of ultrasonic waves which penetrate the body. The invention enables quantitative determination to be made of the relative acoustic parameters (absorption coefficient, body density) as a function of the location.

The gridding method for image reconstruction by Fourier transformation

The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform f;. The method involves a window function w; and proceeds in three steps. First, the convolution g;=w;*f; is computed numerically on a Cartesian grid, using the available samples of f;. Then, g=wf is computed via the inverse discrete Fourier transform, and finally f is obtained as g/w. Due to the smoothing effect of the convolution, evaluating w;*f; is much less error prone than merely interpolating f;. The method was originally devised for image reconstruction in radio astronomy, but is actually applicable to a broad range of reconstructive imaging methods, including magnetic resonance imaging and computed tomography. In particular, it provides a fast and accurate alternative to the filtered backprojection. The basic method has several variants with other applications, such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the Radon (Hough) transform.

Off-resonance correction of MR images

In magnetic resonance imaging (MRI), the spatial inhomogeneity of the static magnetic field can cause degraded images if the reconstruction is based on inverse Fourier transformation. This paper presents and discusses a range of fast reconstruction algorithms that attempt to avoid such degradation by taking the field inhomogeneity into account. Some of these algorithms are new, others are modified versions of known algorithms. Speed and accuracy of all these algorithms are demonstrated using spiral MRI.

Gridding-based direct Fourier inversion of the three-dimensional ray transform

We describe a fast and accurate direct Fourier method for reconstructing a function f of three variables from a number of its parallel beam projections. The main application of our method is in single particle analysis, where the goal is to reconstruct the mass density of a biological macromolecule. Typically, the number of projections is extremely large, and each projection is extremely noisy. The projection directions are random and initially unknown. However, it is possible to determine both the directions and f by an iterative procedure; during each stage of the iteration, one has to solve a reconstruction problem of the type considered here. Our reconstruction algorithm is distinguished from other direct Fourier methods by the use of gridding techniques that provide an efficient means to compute a uniformly sampled version of a function g from a nonuniformly sampled version of Fg, the Fourier transform of g, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of f, the function to be reconstructed, in order to obtain Ff on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct f from this sampled version of Ff. This stage requires a proper weighting of the samples of Ff to compensate for their nonuniform distribution. We use a fast method for computing appropriate weights that exploits the special properties of the spherical sampling grid for Ff and involves the computation of a Voronoi diagram on the unit sphere. We demonstrate the excellent speed and accuracy of our method by using simulated data.

Improvements in spiral MR imaging.

The basic principles of spiral MR image acquisition and reconstruction are summarised with the aim to explain how high quality spiral images can be obtained. The sensitivity of spiral imaging to off-resonance effects, gradient system imperfections and concomitant fields are outlined and appropriate measures for corrections are discussed in detail. Phantom experiments demonstrate the validity of the correction approaches. Furthermore, in-vivo results are shown to demonstrate the applicability of the corrections under in-vivo conditions. The spiral image quality thus obtained was found to be comparable to that obtainable with robust spin warp sequences.

Four-dimensional cardiac reconstruction from rotational x-ray sequences: first results for 4D coronary angiography

The tomographic reconstruction of the beating heart requires dedicated methods. One possibility is gated reconstruction, where only data corresponding to a certain motion state are incorporated. Another one is motioncompensated reconstruction with a pre-computed motion vector field, which requires a preceding estimation of the motion. Here, results of a new approach are presented: simultaneous reconstruction of a three-dimensional object and its motion over time, yielding a fully four-dimensional representation. The object motion is modeled by a time-dependent elastic transformation. The reconstruction is carried out with an iterative gradient-descent algorithm which simultaneously optimizes the three-dimensional image and the motion parameters. The method was tested on a simulated rotational X-ray acquisition of a dynamic coronary artery phantom, acquired on a C-arm system with a slowly rotating C-arm. Accurate reconstruction of both absorption coefficient and motion could be achieved. First results from experiments on clinical rotational X-ray coronary angiography data are shown. The resulting reconstructions enable the analysis of both static properties, such as vessel geometry and cross-sectional areas, and dynamic properties, like magnitude, speed, and synchrony of motion during the cardiac cycle.

Performance of image intensifier-equipped X-ray systems for three-dimensional imaging

Abstract The performance of image intensifier (II)-equipped C-arm systems for three-dimensional (3D) imaging was investigated. The three-dimensional image quality was evaluated in terms of spatial resolution (modulation transfer function, MTF), contrast resolution, geometrical accuracy and homogeneity depending on the image intensifier format, focal spot size, number of projections and the angular span covered during the data acquisition. Experiments have been performed on a vascular C-arm system equipped with a 38-cm image intensifier. Several objects, including CT performance, MTF and pelvis phantoms, were scanned under various conditions. It was shown that for reasonable acquisition parameters, a contrast resolution below 100 HU could be obtained with standard acquisition strategies. Focusing on the spatial resolution, an almost isotropic three-dimensional resolution of up to 22 lp/cm at 10% modulation could be obtained when a 17-cm II was used. The homogeneity in the resulting images was limited by the remaining scatter and truncations. The resulting geometrical accuracy was in the order of the voxel size.

Magnetic particle imaging: Model and reconstruction

Magnetic Particle Imaging is an emerging reconstructive imaging method that can create images of the spatial distribution of magnetizable nanoparticles in an object. A magnetic particle image is reconstructed by solving a discrete approximation to a linear integral equation that models the data acquisition. So far, an explicit formula for the kernel of this integral equation has been missing, forcing one to determine the matrix of the linear equation to be solved by time consuming measurements. Also, this matrix is huge and dense so that the reconstruction times tend to be long. Here, we present an explicit formula for the kernel of the modeling integral operator, transform this operator into a spatial convolution operator, and point out fast reconstruction algorithms that make use of Nonuniform Fast Fourier Transforms.

Nonlinear Image Reconstruction from Projections of Ultrasonic Travel Times and Electric Current Densities

Image reconstruction from projections may be tried with ultrasound or electric current as projecting phenomena. For both these phenomena we outline an experiment supplying the projections and derive a mathematical model describing the experiment. These models are nonlinear variations of the Radon transform. We present practical algorithms for their numerical inversion and report on numerical tests carried out with simulated and real projection data. Our starting-point is the linear case.

Image reconstruction from truncated cone-beam projections

In cone-beam CT, an image of an object is reconstructed from a series of cone-beam projections acquired while the source moves along some trajectory around the object. An accurate reconstruction is possible, provided the source trajectory satisfies a certain completeness condition and none of the cone-beam projections is truncated. Complete source trajectories may be realized by a suitable measuring system, but in medical cone-beam CT, some truncation of the projections is inevitable. An image reconstructed from truncated projections by a standard cone-beam reconstruction algorithm may suffer from severe artifacts. In principle, these artifacts can be reduced by extrapolating the truncated projections prior to the reconstruction. In this paper, a favorable extrapolation method is described in more detail, and the effectiveness of the method is demonstrated using simulated data.