Mathematics and Computer Science in Medical Imaging

The attenuated Radon transform comes up in single particle emission computed tomography (SPECT). We describe some mathematical properties of the attenuated Radon transform and line out numerical procedures using the conjugate gradient algorithm. In the case of constant attenuation, a complete solution is possible, including the attenuation correction problem. We conclude with a suggestion for the attenuation correction problem in the general case.

A method for mapping the scattering interaction density of a certain class of objects from a measurement of the scattering amplitude is developed for the case of a longitudinal acoustic or scalar electromagnetic wave scattering from an inhomogeneous medium consisting of wave velocity fluctuations. The inversion procedure is exact, and explicitly takes into account all orders of multiple scattering. The interaction parameter map constitutes an image of the object. An interesting feature is that progressively higher resolution of the recovered image is obtained as higher order scattering is progressively incorporated into the inversion procedure.

Two criteria are presented, which provide a choice of the number of iterations for Landweber’s recursive algorithm applied to the inversion of an ill-posed linear equation. These stopping rules are shown to regularize the inversion algorithm, and their practical efficiency is discussed. Similar criteria are given to determine the number of singular components to be included in the development of the solution onto the singular system of the problem.

Commonly used iterative techniques for image reconstruction from projections include ART (Algebraic Reconstruction Technique) and SIRT (Simultaneous Iterative Reconstruction Technique). It has been shown that these are the two extremes of a general family of block-iterative image reconstruction techniques. Here we show that the initial performance of these commonly used extremes can be bested by other members of the family.

The expectation-maximization (EM) algorithm for maximum likelihood estimation can be applied to image reconstruction in positron emission tomography (PET) and time-of-flight assisted PET (TOFPET). To implement this algorithm, one can employ either the projection data acquired at various angles or the rotationally integrated image. In the case of TOFPET, the latter approach can reduce the required computation time substantially. Three approaches -- the conventional, direct and statistical methods -- that incorporate the effect of photon attenuation in the EM algorithm have been investigated. Preliminary results from computer simulation studies indicate that the direct method can provide a good compromise between image quality and computation time. Three acceleration approaches -- the geometric, additive and multiplicative methods -- that increase the rate of convergence of these EM reconstruction methods have been studied also.

Reconstructed images for single photon emission computed tomography (SPECT) with quantitative compensation for scatter and attenuation are provided using Inverse Monte Carlo (IMOC): Maximum likelihood estimation with Monte Carlo modeling of the photon interaction and detection probabilities. Quantitative compensation was evaluated by comparing region of interest values for compensated images of line sources scanned in water with line sources scanned in air. Compensation was demonstrated for both 360° 180° acquisition. Lesion contrast was investigated for cold spheres in an active background.

We consider the problem of reconstructing a 3-D object from its 2-D coded radiograph. A new approach to the solution of the problem is presented. The proposed method consists of computing a set of optimal decoding functions using the Kaczmarz algebraic iterative algorithm. To this end, an approximately space-invariant ‘3-D standard response’ is introduced which can be used to characterize any coded source imaging system. Each decoding function corresponds to a specific depth plane inside the object to be reconstructed. The result is a set of 2-D tomograms, each of which is obtained by correlating the coded radiograph with the corresponding decoding function. Two ways of computing the decoding functions are discussed: (i) considering only a single object slice; (ii) treating several immediately adjacent slices (possibly all of them) simultaneously. The proposed reconstruction method can be used for any planar arrangement of discrete sources and is thus capable of comparing the performance of various source point distributions. It is shown that a nine redundant source code provides for better reconstructions than a twelve circular array and a twelve nonredundant array (of same inertia). Finally, the reconstruction of a simulated five planes object using the nine redundant array code is presented.

The reconstruction problem for electrical impedance tomography can be formulated as a non-linear inverse problem. We suggest that the problem might be solved numerically using a regularized Newton’s method and study the ill-posedness of the linearized problem by numerically calculating the singular value decomposition. For the particular case of a two dimensional disc with pairs of electrodes driven the problem is found to be extremely ill-posed. At realistic signal-to-noise ratios the ill-posedness is reasonably independent of the angular separation between the drive electrodes.

Medical imaging has long needed a good method of shape description, both to quantitate shape and as a step toward object recognition. Despite this need none of the shape description methods to date have been sufficiently general, natural, and noise-insensitive to be useful. We have developed a method that is automatic and appears to have great hope in describing the shape of biological objects in both 0 and 3D.

The method produces a shape description in the form of a hierarchy by scale of simple symmetric axis segments. An axis segment that is a child of another has smaller scale and is seen as a branch of its parent. The scale value and parent-child relationship are induced by following the symmetric axis under successive reduction of resolution. The result is a figure-rather than boundary-oriented shape description that has natural segments and is insensitive to noise in the object description.

We extend this method to the description of grey-scale images. Thus, model-directed pattern recognition will not require pre-segmentation followed by shape matching but rather will allow shape properties to be included in the segmentation itself.

The approach on which this method is based is generally applicable to producing hierarchies by scale. It involves following a relevant feature to annihilation as resolution is reduced, defining the component that is annihilating as a basic subobject, and letting the component into which annihilation takes place become its parent in the hierarchy.

It is often desirable to display medical images of different parameters in correlation to each other. Multiparameter color images can be used to represent such information. We discuss the most common color models in computer graphics, presented in a new unifying framework. A new generalized color model, GIHS, is presented of which the existing color models are special cases realized by particular choices of parameter values. The use of the GIHS model for the representation of multiparameter images in general and some medical applications are discussed.

The evaluation of image processing algorithms implies the determination of the receiver operated characteristics (ROC’s) for typical details. This is in principle an experimental procedure. An alternative approach consists in calculating the ROC’s by using a model for the human visual system. A simple form of such a model is proposed. It is shown that realistic results can be obtained. The main conclusion to be made from this approach is perhaps that model calculations show more clearly the limitations of image processing.

The problem of detecting a signal in a noisy background is highlighted from the point of view of communication systems, sensory perception studies and assessment of the intrinsic quality of medical imaging equipment. The concepts of “psychometric curves” (PMC) and of “receiver operating characteristics” (ROC) are introduced.

The likelihood ratio approach is discussed to illustrate the techniques for constructing optimum detectors for communication systems, but at the same time to find the best figure of merit to perform the quality assessment. The concepts of “detectability index” and of the “area under ROC” are shown to be the most suitable figures of merit which can also profitably be employed in medical imaging experiments with human observers. Several medical imaging modalities are discussed and the methods to specify the detectability index from measurable characteristics of the imaging performance and of the intrinsic noise of the equipment are presented.

This paper demonstrates that fluctuations in the temporal phase of medical ultrasound pulse-echo signals can be used as accurate markers of interference-derived artefacts. It is shown that the incorporation of this insight into signal processing techniques can provide an efficient method for noise suppression, and hence maximize the extraction of information concerning the ultrasonic properties of the insonified object. This point is exemplified by applications in both ultrasound pulse-echo imaging and quantitative tissue parameter mapping, where it is shown that a priori knowledge of signal time-domain phase fluctuations leads to a novel speckle reduction technique, and a method for reducing the data necessary for effective pulse-echo attenuation estimation.

The imaging performance of echographic equipment is greatly depenlent on the characteristics of the ultrasound transducer. The paper is confined to single element focussed transducers which are still widely employed in modern equipment. Extrapolation of the results to array transducers maybe made to a certain extent. The performance is specified by the 2 dimensional point spread function (PSF) in the focal zone of the transmitted sound field. This PS? is fixed by the bandwidth of the transducer when the axial (depth) direction is considered, and by the central frequency and the relative aperture in the lateral direction. The PSF concept applies to the imaging of specular reflections and the PSF is estimated by scanning a single reflector. In case of scattering by small inhomogeneities within parenchymal tissues the concept of “speckle” formation has to be introduced. The speckle is due to interference phenomena at reception. It can be shown that the average speckle size in the focus is proportional to the PSF above defined for specular reflections. The dependencies of the speckle size on the distance of the tissue to the transducer (beam diffraction effects) and on the density of the scatterers were explored. It is concluded that with the necessary corrections tissue characterization by statistical analysis of the image texture can be meaningful.

The development of a computer model is described which will predict the concentration of contrast media (CM) in a blood vessel after the injection of CM and hence predict the angiographic appearances of the vessel. This model is used to calculate parametric images of CM concentration versus time and distance along the vessel for a wide range of experimental parameters. Blood velocity estimates are derived from these images at many points, both along the vessel and over the cardiac cycle. The model will be used to investigate the correct strategy for the injection of CM and to predict the precision of the resulting flow estimates.

An important goal of positron emission tomography (PET) is to measure, in vivo and in absolute units, the tissue concentration of a positron-emitting radiopharmaceutical. Over the past few years, such measurements have been made in a wide variety of PET studies using positron cameras constructed of rings of detectors of bismuth germanate or cesium fluoride crystals. Large area detectors, such as those based on the multiwire chamber, have yet to demonstrate such a capability in a clinical research environment. This paper highlights some of the problems of image quantitation that are encountered with the High Density Avalanche Chamber (HIDAC) positron camera.

Factor analysis of dynamic structures (FADS) extracts physiological information from a series of sequential images using the criterion that physiological data are non-negative. The technique consists of a factor analysis applied to the data followed by an oblique transformation of feature space. The result obtained is non-unique. Set theory, cluster analysis and constrained optimisation techniques are used here to define the transformation more precisely and to provide more realistic information.

In this paper a method is presented to delineate elliptically shaped objects in a scene. The detection algorithm is based on the Hough transformation. The transformation maps feature points onto the parameter space of ellipses. By applying cluster algorithms the best set of parameters of the ellipse can be estimated. The algorithm is able to detect elliptical contours even if they are only partly visualized, like the contour of the left ventricle in Thallium-201 scintigrams of patients with ischemic heart disease. Some results of the application of this transformation to synthetic and real scintigrams of elliptical objects are presented. Such an algorithm is also suitable for application to differently curved contours, as Long as the number of parameters describing the contour is relatively low.

The statistical properties of the pixels in an image, and the statistical properties of the noise can usually be estimated for a given application. From that knowledge, it is possible to determine an optimal non-linear filter, in the sense of maximal a posteriori likelihood via Bayes’ rule. This approach combines several types of non-linear filters (and the linear filter) under the same denominator. The results also reveal that median filtering is optimal for very noisy images with bi-exponential differential distributions.