An alternative proof of Bukhgeim and Kazantsev's inversion formula for attenuated fan‐beam projections

A new multiresolution tomographic reconstruction method is investigated for a flat detector fan-beam geometry. To date, almost all connections between wavelets and computerized tomography have only been considered for parallel-beam geometry. Our approach relies on the use of radial and quincunx wavelets to generalize 2D parallel multiresolution reconstruction methods to 2D divergent geometries. In this paper, an approximate inversion formula for a quincunx multiresolution scheme is demonstrated. This efficient algorithm allows one to compute both the quincunx approximation and detail coefficients of an image from its fan-beam projections without extra computational cost compared to the standard fan-beam filtered backprojection algorithm. Obtaining the scale-space representation of the object during the reconstruction process allows one to incorporate directly wavelet-based techniques such as denoising, image analysis and also local tomography. Simulations on mathematical phantoms show that our wavelet tomographic scheme is acceptable for small beam angles but deteriorates at high angles due to non verified approximations.

A filtered backprojection (FBP) reconstruction algorithm for attenuated fan-beam projections has been derived based on Novikov's inversion formula. The derivation uses a common transformation between parallel-beam and fan-beam coordinates. The filtering is shift-invariant. Numerical evaluation of the FBP algorithm is presented. As a special application, we also present a shift-invariant FBP algorithm for fan-beam data reconstruction with uniform attenuation compensation. Several other fan-beam data reconstruction algorithms are discussed. In the attenuation-free case, our algorithm reduces to the conventional fan-beam FBP algorithm.

Filtered backprojection (FBP) reconstruction algorithm for attenuated fan-beam projections has been derived based on Novikov?s inversion formula. The filtering is shift-invariant. The derivation uses a relation of partial derivative between the Cartesian and polar coordinates. We implemented the FBP algorithm and presented the numerical results. As a special application, we give a shift-invariant FBP algorithm for fan-beam SPECT reconstruction with uniform attenuation compensation. Some additional equivalent fan-beam algorithms are also discussed. In the attenuation-free case, our algorithm reduces to the conventional fan-beam FBP algorithm.

§ 1. Introductory. The integralwhere b>0, was given by Hardy (1). It was proved by applying Mellin's inversion formula. An alternative proof, based on the differential equationsatisfied by Kn(x), has been given by the author (2).

§ 1. Introductory. In § 3 a generalisation of the formula [MacRobert, Phil. Mag., Ser. 7, XXXI, p. 258]where αp+1 = ½m + ½n, αp+2 = ½m - ½n, R(m ± n) > 0, and x is real and positive, will be established. In the course of the proof Hardy's formula [Mess, of Maths., LVI, (1927), p. 190],where R(b)>0, will be required. This was originally proved by an application of Mellin's Inversion Formula. An alternative proof is given in § 2, and some related formulae are deduced.

It is well known that in fan-beam X-ray computed tomography (CT) projection data acquired over 2pi contain redundant information and that accurate and stable reconstruction can be obtained from less than 2pi data. In single photon emission computed tomography (SPECT), since the photon absorption is distance-dependent, the conjugate projections are not identical even in the absence of noise. However, projection data over 2pi are still redundant for a noiseless image reconstruction except that the redundancy is not as apparent as in X-ray CT. We study the redundant information in the fan-beam SPECT projection data in this work and developed a reconstruction algorithm only requiring data measured over a short scan. Experiment with computer-simulated data is performed to establish the validity and efficiency of the solution. Besides the explicit inversion formula, the most significant property of the algorithm is that truncations are allowed in the projection data along the detector at each view angle, which makes the reconstruction of region of interest (ROI) feasible in some particular geometries.

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

The purpose of this paper is to introduce and study the linear canonical Fourier–Bessel wavelet transform. We prove an orthogonality relation, inversion formula and some inequality for linear canonical Fourier–Bessel wavelet transform. We first present a direct relationship between the linear canonical Bessel wavelet transform and ordinary Bessel wavelet transform. Based on this relation, we provide an alternative proof of the orthogonality relation for the linear canonical Bessel wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical Bessel wavelet transform.

Realizing symmetric set functions as hypergraph cut capacity

Over the years a number of tomographic applications have emerged in which only limited ranges of projection views are available, such as breast imaging PET scanners based on a pair of parallel planar or curved detectors. This paper addresses the topic of ridge functions and their possible usefulness for tomographic reconstruction from projections distributed in a limited range of views. In this work we derive the basis for speeding up the ridge functions procedure of reconstruction front equally spaced directions which constitute a part of the whole range of views. We describe the possibilities of fast reconstruction in terms of analytical inversion of a certain matrix, for which inversion was known only for the case of a complete range of views. A new inversion formula reducing the dimensionality of the reconstruction problem is derived. Two problems of reconstruction from truncated fan beam projections within the limited angle of view are considered with relevance to PET breast scanning geometries. Algebraic techniques for their solution are proposed. Numerical results of test reconstructions are presented and possible extensions to the 3D case are discussed.

In single photon emission computed tomography (SPECT), photon attenuation within the body is a major factor contributing to the quantitative inaccuracy in measuring the in vivo distribution of radioactivity. Usually the attenuation of the body is not uniform, but for brain imaging, it can be a good approximation to assume that the attenuation is uniformly distributed. For 2D parallel-beam geometry, an exact convolution backprojection algorithm to reconstruct image from attenuated Radon transform with constant attenuation had been developed by Tretiak and Metz (1980). The algorithm can be modified for attenuated fan-beam projections. Unlike the attenuated parallel-beam projections, the filter for attenuated fan-beam projections is no longer spatially invariant, instead, it is a space-variant filter. The algorithm with this spatially variant filter will take more computation time than the algorithm with convolution, but is an exact algorithm. This algorithm has been implemented and simulated using a mathematical phantom. Compared with parallel-beam reconstructions, fan-beam reconstructions have the same image quality.

A convolution backprojection algorithm was derived by Tretiak and Metz (198) to reconstruct two-dimensional (2-D) transaxial slices from uniformly attenuated parallel-beam projections. Using transformation of coordinates, this algorithm can be modified to obtain a formulation useful to reconstruct uniformly attenuated fan-beam projections. Unlike that for parallel-beam projections, this formulation does not produce a filtered backprojection reconstruction algorithm but instead has a formulation that is an inverse integral operator with a spatially varying kernel. This algorithm thus requires more computation time than does the filtered backprojection reconstruction algorithm for the uniformly attenuated parallel-beam case. However, the fan-beam reconstructions demonstrate the same image quality as that of parallel-beam reconstructions.

In this paper, Novikov's inversion formula of the attenuated two-dimensional (2D) Radon transform is applied to the reconstruction of attenuated fan-beam projections acquired with equal detector spacing and of attenuated cone-beam projections acquired with a flat planar detector and circular trajectory. The derivation of the fan-beam algorithm is obtained by transformation from parallel-beam coordinates to fan-beam coordinates. The cone-beam reconstruction algorithm is an extension of the fan-beam reconstruction algorithm using Feldkamp–Davis–Kress's (FDK) method. Computer simulations indicate that the algorithm is efficient and is accurate in reconstructing slices close to the central slice of the cone-beam orbit plane. When the attenuation map is set to zero the implementation is equivalent to the FDK method. Reconstructed images are also shown for noise corrupted projections.

physica status solidi (a)Volume 133, Issue 2 p. K57-K60 Short Note Charge Instability in MIS Structures on Silicon with PECVD Boron Nitride Thin Films A. N. Korshunov, A. N. Korshunov Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorM. L. Kosinova, M. L. Kosinova Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorE. G. Salman, E. G. Salman Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorYu. M. Rumyantsev, Yu. M. Rumyantsev Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorN. I. Fainer, N. I. Fainer Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorN. P. Sysoeva, N. P. Sysoeva Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorZ. L. Akkerman, Z. L. Akkerman Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this author A. N. Korshunov, A. N. Korshunov Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorM. L. Kosinova, M. L. Kosinova Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorE. G. Salman, E. G. Salman Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorYu. M. Rumyantsev, Yu. M. Rumyantsev Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorN. I. Fainer, N. I. Fainer Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorN. P. Sysoeva, N. P. Sysoeva Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this authorZ. L. Akkerman, Z. L. Akkerman Institute of Inorganic Chemistry, Russian Academy of Sciences, Siberian Branch, Novosibirsk Search for more papers by this author First published: 16 October 1992 https://doi.org/10.1002/pssa.2211330244Citations: 7 630090 Novosibirsk, Russia. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume133, Issue216 October 1992Pages K57-K60 RelatedInformation

Methods and algorithms for predicting the properties of chemical compounds by common fragments of their molecular graphs are described. The prediction algorithms are based on determination of a measure of structural proximity (distance) between molecular graphs, which depends on the size of their common fragment. The prediction procedure involves the following steps: partitioning the property classes of the training sample compounds into subclasses of structurally similar compounds; seeking structurally typical compounds and their fragments in each subclass; classifying control compounds according to their distances from the training sample compounds or fragments of classes; forming a set of essential fragments of samples potentially responsible for the properties exhibited by the compounds. The algorithms were successfully tested in the BACC system for analyzing and classifying biologically active compounds designed at the Institute of Mathematics, Siberian Branch, Russian Academy of Sciences.