Microlocal Analysis of an FBP Algorithm for Truncated Spiral Cone Beam Data

In this article we propose an FBP-type algorithm for inversion of spiral cone beam data, study its theoretical properties, and illustrate performance of the algorithm by numerical examples. In particular, it is shown that the algorithm does not reconstruct f exactly, but computes the result of applying a pseudo-differential operator (PDO) with singular symbol to f. Away from critical directions the amplitude of this PDO is homogeneous of order zero in the dual variable, bounded, and approaches one as the pitch of the spiral goes to zero. Numerical experiments presented in the article show that even when the pitch is relatively large, the accuracy of reconstruction is quite high. On the other hand, under certain circumstances, the algorithm produces artifacts typical of all FBP-type algorithms.     

Consistency of a rank test against general alternatives of change points (surfaces) and continuous trend

Considered are modifications of a rank test of randomness for the one- and multi-dimensional regular design cases as well as for the one- and multi-dimensional random design cases. The null hypothesis is that all observations are independent and identically distributed. The main result is the proof of consistency of the test in each of the above cases against two general alternatives.Alternative 1: there exists a pairwise disjoint partion U _i =1 ^m D _i =D, where D d1, is a bounded domain inside which one makes observations, such that (1) if an observation point falls insideD _i , then the corresponding observed value is the realization of a random variable _i i = l,...,m; (2) there exists an ordering % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-Tha7jabe67a4Hqbdiab% +LgaPnaaBaaaleaacaWGRbaabeaakiab-1ha9naaDaaaleaacaWGRb% Gaeyypa0JaaGymaaqaaiaad2gaaaaaaa!4C2D!\[\{ \xi i_k \} _{k = 1}^m \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaaaaa!454D!\[\xi i_k \] is stochastically smaller than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaakmaaBaaaleaacqGHRaWkcaaIXaaabeaakiaacY% cacaWGRbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGG% SaGaamyBaiabgkHiTiaaigdaaaa!509B!\[\xi i_k _{ + 1} ,k = 1,...,m - 1\], (3) the partition is independent of the number of observation points. Note thatm, this ordering, and the sets D _i are not known a priori: one tests only for the existence of such a partition. Note also that in the one-dimensional case the initial sequence need not be stochastically monotone under the alternative.Alternative 2: there exists an arbitrary asymptotically continuous trend in location. Asymptotically continuous means that the trend converges to some continuous, not identically constant function as the number of data points goes to infinity. This function need not be monotone.A numerical example illustrating the use of the obtained results for image analysis (edge detection) is presented.     

Vibrations of an elastic orthotropic layer with a cavity

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–97, January–February, 1991.     

Consistency of rank tests against some general alternatives

The proof of consistency of rank tests against some general alternatives is given. The alternatives considered include arbitrary, not necessarily monotone, trend in location or change points (change surfaces) under the infill asymptotics. One-dimensional and multi-dimensional cases are studied. The numerical experiment illustrating the usage of the above results for image analysis (edge detection) is presented.     

Possibility and consequences of T violation in nucleon decay

It is argued that superunified theories suggest that T violation in nucleon decay might be less well hidden than it is in ordinary weak interactions. Some experimental probes for this possibility are suggested.     

Stability Estimates for Helical Computer Tomography

In this article we analyze an inversion formula for helical computer tomography proposed earlier by the author. Our first result is a global stability estimate. The formula is continuous of order 1 in the Sobolev norms. Then the formula is extended to distributions. Originally it was derived only for C₀ functions. It turns out that there exist distributions, to which the formula does not apply. These exceptional distributions are characterized in terms of their wave fronts. Finally, we show that microlocally away from a critical set the continuity estimate can be mproved: The order goes down from 1 to 1/2.     

Lacunary eta-quotients modulo powers of primes

The Ramanujan Journal - An integral power series is called lacunary modulo M if almost all of its coefficients are divisible by M. Motivated by the parity problem for the partition function, Gordon...     

Local tomography with nonsmooth attenuation

Local tomography for the Radon transform with nonsmooth attenuation is proposed and justified. The main theoretical tool is analysis of singularities of pseudodifferential operators with nonsmooth symbols. Results of numerical testing of local tomography are presented.

     

An improved exact filtered backprojection algorithm for spiral computed tomography

Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography scan with a two-dimensional detector array. The detector array is supposed to be of limited extent in the axial direction. The main property of the formula is that it can be implemented in a truly filtered backprojection fashion. First, one performs shift-invariant filtering of a derivative of the cone beam projections, and, second, the result is back-projected in order to form an image. Compared with an earlier reconstruction algorithm proposed by the author, the new one is two times faster, requires a smaller detector array, and does not impose restrictions on how big the patient is inside the gantry. Results of numerical experiments are presented.     

An efficient,green, and easy‐to‐scale‐up strategy for target‐oriented isolating cadinene sesquiterpenoids from Eupatorium adenophorum Spreng

A green and efficient strategy was established and optimized for target‐oriented extraction, enrichment and separation of cadinene sesquiterpenoids from Eupatorium adenophorum Spreng., using the combination of supercritical fluid extraction, molecular distillation, and industrial preparative chromatography for the first time. The extraction conditions of supercritical fluid extraction were initially optimized by orthogonal experimental design. Under the optimum conditions, the contents of 9‐oxo‐10,11‐dehydroageraphorone and 10Hβ‐9‐oxo‐ageraphorone, which were 55.00% and 6.01%, respectively, were much higher than conventional extraction methods. Then, the molecular distillation enrichment method was established and investigated by response surface methodology technology, which showed strong specificity for enriching target compounds and removing impurities from crude extracts. Under the optimum conditions of molecular distillation, total contents of cadinene sesquiterpenoids were increased to 89.19%. Finally, a total of 146 mg of 9‐oxo‐10,11‐dehydroageraphorone and 29 mg of 10Hβ‐9‐oxo‐ageraphorone were easily obtained by industrial preparative chromatography, from 200 mg of distillation fraction, with purities over 99%. The contents of target components were analyzed by HPLC, and structures of them were identified by high‐resolution MS, ¹H‐NMR, and ¹³C‐NMR spectroscopy. These results indicate that it is a simple, effective, and eco‐friendly strategy, which is easily converted into industrial scale.