improving filtered backprojection reconstruction by data dependent filtering filtered backprojection one of the most widely used reconstruction methods in tomography requires a large number of low noise projections to yield accurate reconstructions in many applications of tomography complete projection data of high quality cannot be obtained because of practical considerations algebraic methods tend to handle such problems better but are computationally more expensive in this paper we introduce a new method that improves the filtered backprojection method by using a custom data dependent filter that minimizes the projection error of the resulting reconstruction we show that the computational cost of the new method is significantly lower than that of algebraic methods experiments on both simulation and experimental data show that the method is able to produce more accurate reconstructions than filtered backprojection based on popular static filters when presented with data with a limited number of projections or statistical noise present furthermore the results show that the method produces reconstructions with similar accuracy to algebraic methods but is faster at producing them finally we show that the method can be extended to exploit certain forms of prior knowledge improving reconstruction accuracy in specific cases

computational analysis and improvement of sirt iterative x ray computed tomography ct algorithms have the potential for producing high quality images but are computationally very demanding especially when applied to high resolution problems focusing on simultaneous iterative reconstruction technique sirt we provide an eigenvalue based scheme for automatically determining a near optimal value of the relaxation parameter this accelerates the convergence rate of sirt to the point where only half the number of iterations normally required is needed we also modify the way sirt uses preconditioning to solve a weighted least squares problem the resulting algorithm which we call psirt is associated with a smaller memory footprint and calls for less data to be communicated in a distributed memory implementation experimental residual norm and timing results are provided based on cone beam micro ct mouse data including for an ordered subsets study

an improved algorithm for reprojecting rays through pixel images it is often desired to calculate line integrals through a field of reconstructed ct density pixels for the purpose of improving ct image quality two algorithms widely published and discussed in the past are known to either degrade spatial resolution or generate errors in the results due to the discontinuous square pixel modeling of the reconstructed image an algorithm is described based on linear interpolation between pixels which provides superior accuracy without unnecessary loss of resolution it was tested on simulated data for a head section and on a narrow gaussian density distribution the experimental results demonstrated improved performance the method is expected to prove useful for many types of post reconstruction processing including beam hardening missing data and noise supression algorithms

superlinear indefinite systems beyond lotka volterra models abstract this paper analyzes the dynamics of a superlinear indefinite parabolic system as a byproduct a number of new results related to population dynamics and economy are obtained among them it is shown that the presence of refuge areas in competitive environments is an optimal mechanism to avoid extinction and that incorporating local symbiosis in competitive environments increases productivity and allows avoiding extinction of the weaker species undoubtedly a paradigm of global markets and possibly of earth biodiversity our analysis combines a series of well known results for systems with some very recent pioneering findings within the context of superlinear indefinite equations

a second order estimate for blow up solutions of elliptic equations abstract we investigate second term asymptotic behavior of boundary blow up solutions to the problems u b x f u x subject to the singular boundary condition u x in a bounded smooth domain r n b x is a non negative weight function the nonlinearly f is regularly varying at infinity with index 1 that is lim u f u f u for every 0 and the mapping f u u is increasing on 0 the main results show how the mean curvature of the boundary appears in the asymptotic expansion of the solution u x our analysis relies on suitable upper and lower solutions and the karamata regular variation theory

blow up rates of radially symmetric large solutions abstract this paper adapts a technical device going back to j lopez gomez optimal uniqueness theorems and exact blow up rates of large solutions j differential equations 224 2006 385 439 to ascertain the blow up rate of the unique radially symmetric large solution given through the main theorem of j lopez gomez uniqueness of radially symmetric large solutions discrete contin dyn syst supplement dedicated to the 6th aims conference poitiers france 2007 pp 677 686 the requested underlying estimates are based upon the main theorem of s cano casanova j lopez gomez existence uniqueness and blow up rate of large solutions for a canonical class of one dimensional problems on the half line j differential equations 244 2008 3180 3203 precisely we show that if is a ball or an annulus f c 0 is positive and non decreasing v c 0 c 2 0 satisfies v 0 0 v u 0 v u 0 for every u 0 and v u h u p 1 as u for some h 0 and p 1 then for each 0 u u f dist x v u u possesses a unique positive large solution in l which must be radially symmetric by uniqueness and we can estimate the exact blow up rate of l x at in terms of p h and f see theorem 1 1

uniqueness and blow up rate of large solutions for elliptic equation δu λu b x h u abstract in this paper we establish the blow up rate of the large positive solution of the singular boundary value problem u u b x h u in u on where is a smooth bounded domain in r n the weight function b x is a non negative continuous function in the domain h u is locally lipschitz continuous and h u u is increasing on 0 and h u h u p for sufficiently large u with h 0 and p 1 naturally the blow up rate of the problem equals its blow up rate for the very special but important case when h u h u p we distinguish two cases i is a ball domain and b is a radially symmetric function on the domain in theorem 1 1 ii is a smooth bounded domain and b satisfies some local condition on each boundary normal section assumed in theorem 1 2 the blow up rate is explicitly determined by functions b and h in case i the singular boundary value problem has a unique solution u satisfying lim d x 0 u x k h b x x 0 1 where d x dist x b r and k are defined by b r r r s r b t d t d s k 1 c 0 1 1 p 1 1 p 1 in case ii the blow up rates of the solutions to the boundary value problem are established and the uniqueness is proved

boundary blow up elliptic problems with nonlinear gradient terms by karamata regular variation theory and perturbation method we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in rn q 0 g c1 0 g 0 0 g is regularly varying at infinity with index with 0 and b is nonnegative nontrivial in which may be vanishing on the boundary

asymptotic estimates of boundary blow up solutions to the infinity laplace equations abstract in this paper we study the asymptotic behavior of boundary blow up solutions to the equation u b x f u in where is the laplacian the nonlinearity f is a positive increasing function in 0 and the weighted function b c is positive in and may vanish on the boundary we first establish the exact boundary blow up estimates with the first expansion when f is regularly varying at infinity with index p 3 and the weighted function b is controlled on the boundary in some manner furthermore for the case of f s s p 1 c g s with the function g normalized regularly varying with index q 0 we obtain the second expansion of solutions near the boundary it is interesting that the second term in the asymptotic expansion of boundary blow up solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the boundary blow up problems involving the classical laplacian

fast approximation of algebraic reconstruction methods for tomography most reconstruction algorithms for transmission tomography can be subdivided in two classes variants of filtered backprojection fbp and iterative algebraic methods fbp is very fast and yields accurate results when a large number of projections are available with high signal to noise ratio and a full angular range algebraic methods require much more computation time yet they are more flexible in dealing with limited data problems and noise in this paper we propose an algorithm that combines the best of these two approaches for a given linear algebraic method a filter is computed that can be used within the fbp algorithm the fbp reconstructions that result from using this filter strongly resemble the algebraic reconstructions and have many of their favorable properties while the required reconstruction time is similar to standard fbp based on a series of experiments for both simulation data and experimental data we demonstrate the merits of the proposed algorithm

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

the exact blow up rates of large solutions for semilinear elliptic equations abstract in this paper combining the method of lower and upper solutions with the localization method we establish the boundary blow up rate of the large positive solutions to the singular boundary value problem u b x f u x u x x where is a smooth bounded domain in r n the weight function b x is a non negative continuous function in the domain which vanishes on the boundary of the underlying domain at different rates according to the point of the boundary f u is locally lipschitz continuous satisfying the keller osserman condition and f u u is increasing on 0 it is worth emphasizing that we obtain the main results for a large class of nonlinear terms f which is regularly varying at infinity with index p r that is for all 0 lim u f u f u p instead of the restriction f u h u p for sufficiently large u and some positive constants h 0 p 1 as in the series of papers of j l gomez

the exact boundary blow up rate of large solutions for semilinear elliptic problems in this paper we establish the blow up rate of the large positive solution of the singular boundary value problem u u a x up u where is a bounded smooth domain in rn the weight function a x in front of the nonlinearity can vanish on the boundary of the domain at different rates according to the point x0 of the boundary the decay rate of the weight function a x may not be approximated by a power function of distance near the boundary we combine the localization method of j lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 with some previous radially symmetric results of t ouyang z xie the uniqueness of blow up solution for radially symmetric semilinear elliptic equation nonlinear anal 64 9 2006 2129 2142 to prove that any large solution u x must satisfy limx x0u x k bx0 dist x 1for each x0 where bx0 r 0r 0sbx0 t dtds k 1 c0 1 1p 1 1p 1 c0 limr 0 0rbx0 t dt 2bx0 r bx0 r and bx0 r is the boundary normal section of a x at x0 i e bx0 r a x0 rnx0 r 0 r 0 and nx0 stands for the outward unit normal vector at x0

multiple positive solutions to elliptic boundary blow up problems abstract we prove the existence of multiple positive radial solutions to the sign indefinite elliptic boundary blow up problem u a x a x g u 0 x 1 u x x 1 where g is a function superlinear at zero and at infinity a and a are the positive negative part respectively of a sign changing function a and 0 is a large parameter in particular we show how the number of solutions is affected by the nodal behavior of the weight function a the proof is based on a careful shooting type argument for the equivalent singular ode problem as a further application of this technique the existence of multiple positive radial homoclinic solutions to u a x a x g u 0 x r n is also considered

existence and the dynamical behaviors of the positive solutions for a ratio dependent predator prey system with the crowing term and the weak growth abstract this paper deals with a ratio dependent predator prey system with the crowing term and the weak growth in the prey equation under the condition that the coefficient is less than a critical value 1 d 0 we obtain existence of multiple positive steady state solutions of the predator prey system and the dynamical behaviors of its positive solutions our results show that the predator and the prey possess not only the common coexistence but also the very weak coexistence which both of the predator and the prey are very low meantime the persistence of the positive solutions for the corresponding parabolic type system sometime depends strictly on the ratio of its initial data therefore our results may be used to explain some special phenomena which under some bad environment the predator and the prey may still coexist

blow up rate and uniqueness of singular radial solutions for a class of quasi linear elliptic equations we establish the uniqueness and the blow up rate of the large positive solution of the quasi linear elliptic problem pu up 1 b x h u in br x0 with boundary condition u on br x0 where br x0 is a ball centered at x0 rn with radius r n 3 2 p 0 are constants and the weight function b is a positive radially symmetrical function we only require h u to be a locally lipschitz function with h u up 1 increasing on 0 and h u uq 1 for large u with q p 1 our results extend the previous work z xie uniqueness and blow up rate of large solutions for elliptic equation u u b x h u j differential equations 247 2009 344 363 from case p 2 to case 2 p

existence uniqueness and blow up rate of large solutions for a canonical class of one dimensional problems on the half line this paper shows the existence and the uniqueness of the positive solution l t of the singular boundary value problem u t f t h u t t 0 u 0 u 0 where f is a continuous non decreasing function such that f 0 0 and h is a non negative function satisfying the keller osserman condition moreover it also ascertains the exact blow up rate of l t at t 0 in the special case when there exist h 0 and p 1 such that h u hup for sufficiently large u naturally the blow up rate of the problem in such a case equals its blow up rate for the very special but important case when h u hup for all u 0 so our results are substantial improvements of some previous findings of j lopez gomez uniqueness of large solutions for a class of radially symmetric elliptic equations in s cano casanova j lopez gomez c mora corral eds spectral theory and nonlinear analysis with applications to spatial ecology world scientific 2005 pp 75 110 and j lopez gomez optimal uniqueness theorems and exact blow up rates of large solutions j differential equations 224 2006 385 439

a boundary blow up elliptic problem with an inhomogeneous term abstract by a perturbation method and constructing comparison functions we reveal how the inhomogeneous term h affects the exact asymptotic behaviour of solutions near the boundary to the problem u b x g u h x u 0 in u where is a bounded domain with smooth boundary in r n 0 g c 1 0 is increasing on 0 g 0 0 g is regularly varying at infinity with positive index the weight b which is non trivial and non negative in may be vanishing on the boundary and the inhomogeneous term h is non negative in and may be singular on the boundary

boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms abstract in this paper we study the boundary behavior of solutions to boundary blow up elliptic problems u u q b x f u x u where is a bounded domain with smooth boundary in r n q 0 b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary and f is rapidly varying or normalized regularly varying at infinity

the boundary blow up rate of large solutions abstract in this paper we ascertain the blow up rate of the large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain at different rates according to the point of the boundary x all previous results in the literature assumed the decay rate of the underlying weight function to be the same at any point of this hypothesis substantially simplified the mathematical analysis of the problem as it allowed constructing global sub and supersolutions in an open neighborhood of obtaining general results requires localizing at each particular point of the boundary making particularly involved the mathematical analysis of the problem

second order expansion for blowup solutions of semilinear elliptic problems abstract in this paper we mainly study the second order expansion of solutions to the elliptic problems u b x f u u 0 x u where is a bounded domain with a smooth boundary in r n n 2 b c which is positive in and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p 1 our analysis is based on the sub supersolution method and karamata regular variation theory

uniqueness of large positive solutions for a class of radially symmetric cooperative systems abstract this paper shows the uniqueness of the large solution of a class of radially symmetric sublinear elliptic systems of cooperative type this result is based on the strong maximum principle rather than on the blow up rates of the large solutions as it is common in most the available results for single equations

retracted blow up and global existence for a modified two component camassa holm equation the present work is mainly concerned with blow up phenomena and global existence for a modified two component camassa holm equation firstly a new blow up criterion and the precise blow up rate are presented then a new global existence result for strong solutions to the equation is given

an intuitive discussion on the ideal ramp filter in computed tomography i in x ray computed tomography ct the ideal ramp filter is a generalized function defined by the inverse fourier transform similar to dirac s discussion on the delta function we present an intuitive discussion on the ideal ramp filter with this concise discussion one obtains a better understanding of the filter backprojection algorithm fbp and can easily construct new practical filters