Geometric tomography. Radon transforms. Image Processing, Computer-Assisted -- methods. Tomography -- methods. Radon, Transformations de.

## Computerized Tomography '93 - Programme of International Symposium

Bildgebendes Verfahren. Mathematische Methode. Inverses Problem. Radon transforms -- Congresses.

Geometric tomography -- Congresses. The mathematical models, their theoretical aspects and the development of algorithms were treated. The proceedings contains surveys on reconstruction in inverse obstacle scat- tering, inversion in 3D, and constrained least squares pro- blems. Research papers include besides the mentioned imaging techniques presentations on image reconstruction in Hilbert spaces, singular value decompositions, 3D cone beam recon- struction, diffuse tomography, regularization of ill-posed problems, evaluation reconstruction algorithms and applica- tions in non-medical fields.

Contents: Theoretical Aspects: J. Boman: Helgason' s support theorem for Radon transforms-a newproof and a generalization -P.

Maass: Singular value de- compositions for Radon transforms- W. Madych: Image recon- struction in Hilbert space -R. Mukhometov: A problem of in- tegral geometry for a family of rays with multiple reflec- tions -V. Palamodov: Inversion formulas for the three-di- mensional ray transform - Medical Imaging Techniques: V. Friedrich: Backscattered Photons - are they useful for a surface - near tomography - P. Grangeat: Mathematical frame- work of cone beam 3D reconstruction via the first derivative of the Radon transform -P.

Grassin, B. Duchene, W. Tabbara: Dif- fraction tomography: some applications and extension to 3D ultrasound imaging -F.

### Introduction

Kress, A. Professor Quinto has successfully tested his limited data tomography algorithm on electron microscope data, and he will use these results as a basis for further refinement with the goal of incorporating scattering and other effects. Professor Quinto plans to prove uniqueness theorems for a model of synthetic aperture RADAR, research which is based on his pure mathematical work.

Within pure mathematics, the principal investigator plans to prove uniqueness and support theorems for spherical Radon transforms. He and Professor Agranovsky will use these results to characterize stationary sets of solutions of the wave equation. Stationary sets, where the solution is zero for all time, are very important and difficult to characterize. He plans to prove uniqueness theorems for Radon transforms on spheres on real-analytic manifolds and use these results to begin to characterize stationary sets on rank-one symmetric spaces.

## Geofizicheskiy Zhurnal

He is proving Morera theorems for the distinguished boundary of polydisks, and he plans to generalize them to complex manifolds. He hopes to prove inverse continuity for an important class of local limited data problems or clarify when they do not hold. This research encompasses both applied and pure mathematics: tomography and integral geometry. The pure research will be used to develop, understand, and justify the applied algorithms, and the applied problems will motivate much of the pure research.

A computed tomography algorithm will be developed for electron microscopy and tested jointly with colleagues at the Karolinska Institute in Sweden. Our goal is to produce accurate pictures of viruses and small molecules using an electron microscope.

Jointly with a colleague and an undergraduate student, he will develop an algorithm for emission tomography, a type of tomography that locates metabolic processes. He will develop pure mathematics that will show how well the algorithms work and where their limitations might occur. These pure mathematical underpinnings are required to ensure that his or any other algorithms are as effective as they can be.

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Panel II in Fig 6 demonstrates the correction effect by displaying the imaging results of the th row in the detector. Fig 6d shows the calibrated trajectory of the center of mass of the phantom. Compared with the one before correction in Fig 6a , it is much more continuous. Fig 6e and 6f are the CT slice images reconstructed by Eq 1 with the sinograms in Fig 6a and 6d. Observing the regions of interest marked the red arrows in Fig 6e and 6f , we can find that the result after correction matches the phantom much better than the one before correction.

### Table of contents

It validates the proposed correction method for the horizontal shift. Fig 6g compares the grey value profiles of the th row of the images with the phantom and supports quantitatively this conclusion. AB epoxy adhesive was used to fix the sample particle on the top of a pin under the help of an optical microscope. Finally the pin with the particle was mounted to the rotation sample stage. It is primarily composed of a condenser, sample stage, zone plate and CCD detector. The SR x-ray beam is focused onto the sample by a elliptically shaped capillary condenser.

Then the objective zone plate produces a magnified projection image of the sample on a scintillator crystal. When the size of the sample is smaller the depth of focus of the microscope, this imaging layout can be equivalently treated as the parallel-beam imaging geometry in Fig 1a. The x-ray energy was set to be 8keV. The first row is the original images recorded by CCD camera and the second row is after processing logarithm operation.

The first row is the original images recorded by CCD camera under five view angles. The second row is the ones after logarithm operation. Fig 8 presents the correction procedure and results for the vertical shift with the experimental data. Panel I in Fig 8 shows how to calculate the correction curve. In panel I, Fig 8a is one of the two dimensional projection images after logarithm operation.

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Fig 8b displays the plane integral curve at the first view angle marked by the solid blue line and the one at other view angle marked by the dashed red line. Fig 8c shows the cross correlation result of these two curves in Fig 8b. The peak of the cross correlation appears at the position indexed by So the correction value for this view angle is 7 since the center of the position index of the plane integral curve is The correction curve in Fig 8d is depicted after doing the operations in Fig 8c for all view angles.

Obviously, this curve shows that the stage is descending vertically with respect to view angle during the scanning. Although it exhibits some periodicity, this shift is generally random. Panel II in Fig 8 shows the correction results of one typical slice, the th row in the detector. Fig 8e and 8h are the sinogram and the CT image before correction. Fig 8f and 8i are after correction. Fig 8g and 8j show the differences of sinograms and CT images before and after correction. Some observations can be made for Panel II. Due to the vertical shift of the stage, the sinogram of the th row is incomplete and misaligned by the neighbor rows.

Before correction, some structures marked by the red arrow are distorted or disappear and the edge is blurred. In contrast, these problems are mitigated after correction. The results in Panel II demonstrate the validity of the proposed correction method for vertical shift of the stage.