Bruce D. Smith

Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods

Previously unknown sufficient conditions, a necessary condition, and reconstruction methods for image reconstruction from cone-beam projections are developed. A sufficient condition developed is contained in the following statement. Statement 5: If one every plane that intersects the object, there exists at least one cone-beam source point, then the object can be reconstructed. Reconstruction methods for an arbitrary configuration of source points that satisfy Statement 5 are derived. By requiring additional conditions on the configuration of source points, a more efficient reconstruction method is developed. It is shown that when the configuration of source points is a circle, Statement 5 is not satisfied. In spite of this, several suggestions are made for reconstruction from a circle of source points.

Reconstruction methods and completeness conditions for two Compton data models.

Two different models are proposed for the data produced in a Compton imaging device. A sequence of equations, which relate the model to the distribution of radioactivity that is being imaged, is developed for each of the two models. No series expansions are used in these developments. On the basis of these sequences of equations, a completeness condition is developed for each of the two models. These completeness conditions may prove useful in the future in determining appropriate shapes, configurations, and motions of the devices detectors. A computer simulation is performed to verify one of these sequences of equations. A computer simulation is also performed to demonstrate that this sequence of equations can produce more accurate images than a backprojection reconstruction method. In addition, a procedure is proposed that could mitigate the effects of the Klein-Nishina distribution, the Doppler broadening, and the variability in the data due to the random generation of photons.

Cone beam convolution formula

This paper presents a three-dimensional convolution formula for a general set of cone beam data. This formula is illustrated by examining at length the formula for a particular set of cone beam data--the set of cone beam data from source points which are located along a straight line. How the three-dimensional formula could be implemented for a general set of cone data is discussed.

Implementation, investigation, and improvement of a novel cone-beam reconstruction method (SPECT)

A three-dimensional reconstruction method which uses cone-beam data was implemented. Issues concerning implementation are investigated and discussed in detail. Computer simulations were used to determine the amount of image degradation occurring in each of the three steps that comprise the reconstruction method. In addition, computer simulations were used to investigate issues concerning sampling. Improvements in the implementation were made. Additionally, suggestions for future efforts to improve the implementation are made.

Fan-beam reconstruction from a straight line of source points

A fan-beam tomographic reconstruction algorithm is developed for source points distributed along a straight line. It is shown that, in theory, a perfect reconstruction is possible from an infinitely long straight line. Using computer simulations it is verified that using a finite segment of a straight line, it is possible to reconstruct images with quality comparable to those obtained when the source points are distributed along a circle. It is shown that the two parameters that most affect the image quality are the length of source point line, and the distance from the object to the source point line. In addition, a postreconstruction technique is developed that substantially improves the quality of images reconstructed from the straight line.

Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography

A novel cone-beam reconstruction method was proposed in 1985. The first objective of the work reported here is to implement this reconstruction method. The second objective is to compare it with the method developed by Feldkamp et al. (1984). Although the resulting reconstruction was not perfect, the proposed method did eliminate the axial distortion associated with Feldkamps method. A second cone-beam reconstruction method was proposed in 1987. Two major challenges arise when this method is implemented. One is to minimize the error that results from violating an assumed condition on the distribution of the cones. The second is to minimize the error that results from the convolution of a discontinuous function which is introduced to compensate for the redundancy in the data set. The third objective of the work reported here was to investigate several heuristic techniques to minimize these errors. Techniques were found that did mitigate these errors and using these techniques resulted in images that are more accurate than those resulting from the 1985 method.

Influence of Solicitation Pattern, Type of Practice Example, and Student Response on Pupil Behavior, Commitment to Discussion and Concept Attainment

Abstract This experimental study examined whether solicitation pattern, type of example, and student response influence the classroom behavior, commitment to discussion and concept attainment of high school social studies students. Participating teachers taught two concept application lessons according to detailed plans provided by the researcher. The classroom behaviors, inspection of stimulus material and attention to discussion were recorded as they occurred by the researcher on a low inference observation” system. Commitment was measured by a Thurstone successive intervals scale and concept learning by a test constructed for the experiment. Solicitation pattern had a significant effect on inspection and attention behaviors. Solicitation pattern interacted with response mode to exert a significant influence on commitment to discussion. Response sequence interacted with type of example to affect concept learning.

Cameras shapes for medical imaging Compton cameras

The shape of the detectors that comprise a Compton camera largely determines the geometric distribution of the set of data that is measured. It is our hypothesis that the geometric distribution of a set of data determines the informational contents of the data. The objective of the research reported here is to produce evidence that supports this hypothesis. We use a novel approach to measure the informational content of a set of data; namely, we will use the rank of the system matrix, which stems from the data set, as a numerical measure of the informational content of the set. The data was simulated using the surface integral model for Compton camera data. When just the data that scatter parallel to the face of the camera was used, it was found that the resulting system matrix was full rank. When just the data that scatter perpendicular to the face of the camera was used, it was found that the resulting system matrix was substantially less than full rank. Additional work is needed to determine if full rank matrices can be obtained using the conventional camera design that consists of two detectors that are planar shaped and parallel to each other.

Null-space smoothing of tomographic images using TV norm minimization

Smoothing is desirable in tomographic imaging when it reduces the effects of noise in the data and is undesirable when it smooths a small feature such as a tumor or a lesion so much that they become undetectable. Linear algebra can be used to identify a significant problem associated with reconstruction from incomplete data set; namely, the rank of the resulting system matrix is less then full. To maximize its benefit and to minimize its harm, when smoothing is used in this case, it seems desirable to give more credence to the row-space component of the reconstruction than the null-space because the tomographic data contains only information about the row-space component of the object. The objective of the work presented here is to propose and demonstrate a method, which is called null-space smoothing, for achieving this. The Methodology used involved computer generated data. ART is used to reconstruct the row-space component of the Shepp and Logan phantom. By solving a convex optimization problem, an image in the null-space was added to the reconstruction so that the resulting image had a minimum TV norm; thus, leaving the row-space component unchanged. It is concluded that although null-space smoothing can produce smooth images with an unchanged row-space component, more work needs to be done in the future to demonstrate its usefulness with real data.

Comparison of TV norm minimization and MLEM for reduction of metal artifacts in transmission tomography

The streaks in Computerized Tomography (CT) images, which are produced by metal implants in the human body, can severely diminish the diagnostic value of the images. The principal objective of this paper is to compare the performance of the Maximum Likelihood Expectation Maximization (MLEM) and Total-Variation (TV) minimization in mitigating these artifacts. To provide a benchmark, these algorithms are compared with two conventional algorithms, the Filtered Back-Projection (FBP) and Algebraic Reconstruction Technique (ART) with a positivity constraint (ART-PC). Computer generated measurements were used in these comparisons. Relative to those of FBP and ART-PC, the MLEM algorithm produced substantially less streaking. The image resulting from the TV-minimization, however, could not be distinguished from the phantom. Although the results presented for both the ASD-POCS and the MLEM algorithms are promising, further effort is needed to establish their usefulness with real data.