# selected publications

## Developments in X-Ray Tomography V. Edited by Bonse, Ulrich. Proceedings of the SPIE, Volume 6318, pp. 63180Q (2006).

## Phase contrast CT: Fundamental theorem and fast image reconstruction algorithms

## A V Bronnikov

Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented at third-generation synchrotron radiation sources or by using a microfocus x-ray source. Promising results have recently been obtained in materials science and medicine. At the same time, the lack of a mathematical theory comparable with that of conventional CT limits the progress in this field. We suggest such a theory and prove a fundamental theorem that plays the same role in phase-contrast CT as the Fourier slice theorem does in absorption-based CT. The fundamental theorem allows us to derive fast image reconstruction algorithms in the form of filtered backprojection (FBP). The stability of the algorithms is discussed; experimental results are shown. More...

## J Opt Soc Am A 2002 Mar;19(3):472-80.

## Theory of quantitative phase-contrast computed tomography

## A V Bronnikov

Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented at third-generation synchrotron radiation sources or by using a microfocus x-ray source. Promising results have recently been obtained in materials science and medicine. At the same time, the lack of a mathematical theory comparable with that of conventional CT limits the progress in this field. Such a theory is now suggested, establishing a fundamental relation between the three-dimensional Radon transform of the object function and the two-dimensional Radon transform of the phase-contrast projection. A reconstruction algorithm is derived in the form of a filtered backprojection. The filter function is given in the space and spatial-frequency domains. The theory suggested enables one to quantitatively determine the refractive index of a weakly absorbing medium from x-ray intensity data measured in the near-field region. The results of computer simulations are discussed.

## IEEE Trans Med Imaging. 2000 May;19(5):451-62.

## Reconstruction of attenuation map using discrete consistency conditions

## A V Bronnikov

Methods of quantitative emission computed tomography require compensation for linear photon attenuation. A current trend in single-photon emission computed tomography (SPECT) and positron emission tomography (PET) is to employ transmission scanning to reconstruct the attenuation map. Such an approach, however, considerably complicates both the scanner design and the data acquisition protocol. A dramatic simplification could be made if the attenuation map could be obtained directly from the emission projections, without the use of a transmission scan. This can be done by applying the consistency conditions that enable us to identify the operator of the problem and, thus, to reconstruct the attenuation map. In this paper, we propose a new approach based on the discrete consistency conditions. One of the main advantages of the suggested method over previously used continuous conditions is that it can easily be applied in various scanning configurations, including fully three-dimensional (3-D) data acquisition protocols. Also, it provides a stable numerical implementation, allowing us to avoid the crosstalk between the attenuation map and the source function. A computationally efficient algorithm is implemented by using the QR and Cholesky decompositions. Application of the algorithm to computer-generated and experimentally measured SPECT data is considered. More...

## Phys Med Biol. 2000 Sep;45(9):2639-51.

## A filtering approach to image reconstruction in 3D SPECT

## A V Bronnikov

We present a new approach to three-dimensional (3D) image reconstruction using analytical inversion of the exponential divergent beam transform, which can serve as a mathematical model for cone-beam 3D SPECT imaging. We apply a circular cone-beam scan and assume constant attenuation inside a convex area with a known boundary, which is satisfactory in brain imaging. The reconstruction problem is reduced to an image restoration problem characterized by a shift-variant point spread function which is given analytically. The method requires two computation steps: backprojection and filtering. The modulation transfer function (MTF) of the filter is derived by means of an original methodology using the 2D Laplace transform. The filter is implemented in the frequency domain and requires 2D Fourier transform of transverse slices. In order to obtain a shift-invariant cone-beam projection-backprojection operator we resort to an approximation, assuming that the collimator has a relatively large focal length. Nevertheless, numerical experiments demonstrate surprisingly good results for detectors with relatively short focal lengths. The use of a wavelet-based filtering algorithm greatly improves the stability to Poisson noise. More...

## J Opt Soc Am A 2000 Nov;17(11):1993-2000.

## Cone-beam reconstruction by backprojection and filtering

## A V Bronnikov

A new analytical method for tomographic image reconstruction from cone-beam projections acquired on the source orbits lying on a cylinder is presented. By application of a weighted cone-beam backprojection, the reconstruction problem is reduced to an image-restoration problem characterized by a shift-variant point-spread function that is given analytically. Assuming that the source is relatively far from the imaged object, a formula for an approximate shift-invariant inverse filter is derived; the filter is presented in the Fourier domain. Results of numerical experiments with circular and helical orbits are considered. More...

## Optical Engineering 1999 38(2): 381-386.

## Virtual alignment of x-ray cone-beam tomography system using two calibration aperture measurements

## A V Bronnikov

In cone-beam tomography, relatively small misalignment of the imaging system is geometrically magnified and may cause severe distortion of the reconstructed image. We describe a method for alignment of a cone-beam tomography system built on an x-ray microfocus tube, an image intensifier, and a high-resolution CCD camera. To obtain geometrical parameters of system misalignment, we suggest measuring two 180-deg-opposed cone-beam radiographs of a specially manufactured calibration aperture. An advantage of the aperture over other calibration objects is that we can easily restore its idealized picture by applying a certain threshold to the measured data. The method permits the lateral displacement vector and lateral tilt angle to be accurately found. Unlike other alignment methods, our approach enables virtual system alignment by using mathematical processing of the measured data, rather than moving the parts of the system. The virtually aligned system data are used for 3-D image reconstruction by a standard filtered backprojection algorithm. Experimental results demonstrate considerable improvement of the image quality after applying the alignment method suggested.

## Nuclear Instruments and Methods in Physics Research A 1999 422: 909-13.

## Cone-beam tomography system used for non-destructive evaluation of critical components in power generation

## A V Bronnikov and D Killian

We present a three-dimensional (3D) X-ray tomography system used for non-destructive evaluation (NDE) of critical components in power generation. Thesystem consists of a microfocus X-ray tube, a precise manipulation table and a detection system comprising a high-resolution image intensifier coupled to a CCD camera, which enables circular-scan cone-beam data acquisition geometry. We use a Cu filter to reduce polychromaticity of the beam and apply a geometrical calibration technique to correct for misalignments and tiltof the system. Dedicated mathematical methods are applied to provide fast 3D image reconstruction and volume rendering. The whole process of measurement and visualization is fully automated. An example of NDE of turbine blades is considered.

## Inverse Problems 1999 15 1315-1324

## Numerical solution of the identification problem for the attenuated Radon transform

## A V Bronnikov

The attenuated Radon transform serves as a mathematical tool for single-photon emission computerized tomography (SPECT). The identification problem for the attenuated Radon transform is to find the attenuation coefficient, which is a parameter of the transform, from the values of the transform alone. Previous attempts to solve this problem used range theorems for the continuous attenuated/exponential Radon transform. We consider a matrix representation of the transform and formulate the corresponding discrete consistency conditions in the form of the orthogonal projection of the data vector onto the orthogonal complement of the column space of the matrix. The singular value decomposition is applied to compute the orthogonal projector and its Fr?et derivative. The numerical algorithm suggested is based on the Newton method with the Tikhonov regularization. Results of numerical experiments and inversion of the measured SPECT data are considered.

## Applied Optics 1998 37(20) pp.4437-4448

## Wavelet-based image enhancement in x-ray imaging and tomography

## A V Bronnikov and G Duifhuis

We consider an application of the wavelet transform to image processing in x-ray imaging and three-dimensional (3-D) tomography aimed at industrial inspection. Our experimental setup works in two operational modes digital radiography and 3-D cone- beam tomographic data acquisition. Although the x-ray images measured have a large dynamic range and good spatial resolution, their noise properties and contrast are often not optimal. To enhance the images, we suggest applying digital image processing by using wavelet-based algorithms and consider the wavelet-based multiscale edge representation in the framework of the Mallat and Zhong approach IEEE Trans. PAMI 14, 710 (1992) . A contrast-enhancement method by use of equalization of the multiscale edges is suggested. Several denoising algorithms based on modifying the modulus and the phase of the multiscale gradients and several contrast-enhancement techniques applying linear and nonlinear multiscale edge stretching are described and compared by use of experimental data. We propose the use of a filter bank of wavelet-based reconstruction filters for the filtered-backprojection reconstruction algorithm. Experimental results show a considerable increase in the performance of the whole x-ray imaging system for both radiographic and tomographic modes in the case of the application of the wavelet- based image-processing algorithms.

# new publications

## IEEE Transactions on Nuclear Sciences 2012 59(4) 1458-1464

## SPECT imaging with resolution recovery

## A V Bronnikov

Single-photon emission computed tomography (SPECT) is a method of choice for imaging spatial distributions of radioisotopes. Applications of this method are found in medicine, biomedical research and nuclear industry. This paper deals with improving spatial resolution in SPECT by applying correction for the point-spread function (PSF) in the reconstruction algorithm and optimizing the collimator. Several approaches are considered: the use of a depth-dependent PSF model for a parallel-beam collimator derived from experimental data, the extension of this model to a fan-beam collimator, a triangular approximation of the PSF for reconstruction acceleration, and a method for optimal fan-beam collimator design. An unmatched projector/backprojector ordered subsets expectation maximization (OSEM) algorithm is used for image reconstruction. Experimental results with simulated and physical phantom data of a micro-SPECT system show a significant improvement of spatial resolution with the proposed methods.More...

## Optics Express 2010 18(25) 25771-25785

## Nonlinear phase retrieval from single-distance radiograph

## J Moosmann, R Hofmann, A V Bronnikov, and T Baumbach

Phase contrast in the object plane of a phase object is retrieved from intensity contrast at a single object-detector distance. Expanding intensity contrast and phase shift in the detector plane in powers of object-detector distance, phase retrieval is performed beyond the solution to the linearized transport-of-intensity equation. The expansion coefficients are determined by the entire paraxial wave equation. The Laplacian of the phase shift in the object plane thus is written as a local expression linear in the intensity contrast and nonlinear in the phase shift in the object plane. A perturbative approach to this expression is proposed and tested with simulated phantom data.