Backprojection Operator Research Articles

A new algorithm to compute the discrete inverse radon transform

The aim of this paper is to compute the discrete inverse Radon transform over ℝ n . The Radon transform is a function with domainS n−1×ℝ. It is shown that under different measure this function can be defined with domain ℝ n . In this case one can compute the discrete inverse Radon transform in the Cartesian coordinate system without interpolating from polar to Cartesian coordinates or using the backprojection operator.

Detection of edges from projections

In a number of applications of computerized tomography, the ultimate goal is to detect and characterize objects within a cross section. Detection of edges of different contrast regions yields the required information. The problem of detecting edges from projection data is addressed. It is shown that the class of linear edge detection operators used on images can be used for detection of edges directly from projection data. This not only reduces the computational burden but also avoids the difficulties of postprocessing a reconstructed image. This is accomplished by a convolution backprojection operation. For example, with the Marr-Hildreth edge detection operator, the filtering function that is to be used on the projection data is the Radon transform of the Laplacian of the 2-D Gaussian function which is combined with the reconstruction filter. Simulation results showing the efficacy of the proposed method and a comparison with edges detected from the reconstructed image are presented.

Simultaneous linearized inversion of velocity and density profiles for multidimensional acoustic media

In this paper, the multidimensional inverse scattering problem for an acoustic medium is formulated as a generalized tomographic problem. The medium is probed by wide-band plane-wave sources, and the scattered field is observed along straight-line receiver arrays. The objective is to reconstruct, simultaneously, the velocity and density profiles of the medium. Using the Born approximation and applying a proper filter to the observed scattered field, generalized projections of the velocity and density scattering potentials are obtained. The generalized projections are weighted integrals of the scattering potentials; in two dimensions, the weighting functions are concentrated along parabolas, in three dimensions, they are concentrated over circular paraboloids. The resulting problem of generalized tomography (or inverse generalized Radon transform, or integral geometry) is solved in exact analytical form. The solution is expressed as a backprojection operation followed by a two- or three-dimensional space-invariant filtering operation. In the Fourier domain, the resulting image is a linear combination of the velocity and density scattering potentials, where the coefficients depend on the angle of incidence of the probing wave. Therefore, in principle, two experiments with different angles of incidence are necessary to reconstruct the velocity and density scattering potentials separately. It is shown that this involves a numerically unstable procedure, especially for bandlimited data, necessitating the use of several angles of incidence to regularize the inversion.

A study of reconstruction artifacts in cone beam tomography using filtered backprojection and iterative EM algorithms

Reconstruction artifacts in cone beam tomography are studied for filtered backprojection (Feldkamp) and iterative EM algorithms. The filtered backprojection algorithm uses a voxel-driven, interpolated backprojection to reconstruct the cone beam data, whereas the iterative EM algorithm performs ray-driven projection and backprojection operations for each iteration. Two weighting schemes for the projection and backprojection operations in the EM algorithm are studied. One weights each voxel by the length of the ray through the voxel and the other equates the value of a voxel to the functional value of the midpoint of the line intersecting the voxel, which is obtained by interpolating between eight neighboring voxels. Cone beam reconstruction artifacts such as rings, bright vertical extremities, and slice-to-slice cross-talk are not found with parallel beam and fan beam geometries. When using filtered backprojection and iterative EM algorithms, the line-length weighting is susceptible to ring artifacts which are improved by using interpolated projector-backprojectors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An iterative reconstruction algorithm for single photon emission computed tomography with cone beam geometry

AbstractAn iterative EM reconstruction algorithm for single photon emission computed tomography is implemented for cone beam geometry that uses a ray‐driven projector‐backprojector. The cone beam projector‐backprojector models the attenuated Radon transform of a source distributed within an attenuator as line integrals of discrete voxels representing the source and attenuation distributions. The attenuation coefficient distribution for each voxel is assumed to be equal to the average value over a cubical region, and the integral of the source concentration through each voxel is obtained by interpolating between the source values at eight neighboring voxels. The calculation of the attenuation factors requires a specification of the attenuation distribution, estimated either from an assumed constant distribution with an estimated body outline or from transmission measurements. The distribution of attenuation coefficients is stored in memory, and the attenuation factors are calculated for each voxel during the projection and backprojection operations instead of using precalculated values. Computer simulations of half‐scan and full‐scan reconstructions show that the algorithm is able to compensate for attentuation and to suppress the propagation of artifacts that can result with limited angular sampling. The simulations verify that the algorithm has important application to cardiac SPECT imaging; because of variable attentuation through the thorax, it is necessary to use an iterative algorithm with an attenuation map in order to quantify radiopharmaceutical distributions in the heart.

Linograms in Image Reconstruction from Projections

The notion of a linogram is introduced. It corresponds to the notion of a sinogram in the conventional representation of projection data in image reconstruction. In the sinogram, points which correspond to rays which go through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve. In the linogram, however, these points fall on a straight line. Thus, backprojection corresponds to integration along straight lines in the linogram. A theorem is proved expressing the backprojection operator in terms of the Radon transform and simple changes of variables. Consequences of this theorem are discussed. A novel image reconstruction method based on the theorem is presented.

Microwave projection imaging for refractive objects: a new method

The distorted-wave Born and geometrical-optics approximations are used to establish a connection between coherent swept-frequency microwave scattering data and the projections of refractive objects. A reconstruction algorithm based on a generalized backprojection operator is then derived. The algorithm is numerically efficient, robust, and applicable to a wider class of scatterers than are traditional inversion techniques. Results obtained by using both simulated and experimental data sets suggest that the new inversion technique provides improved images over those obtained by using traditional algorithms for the refractive structures tested to date. The method complements the weighted generalized backprojection operator proposed by Beylkin [ Commun. Pure Appl. Math.37, 579 ( 1984); J. Math. Phys.26, 99 ( 1985)].

Novel microwave projection imaging for determination of internal structure

A novel microwave imaging technique based on the generalised weighted backprojection operator is introduced and tested using experimental data. Images of the internal structure of penetrable objects using this technique are compared and contrasted with images found from the classical back-projection method. It is demonstrated that the new method provides high-quality images of both discrete and continuous internal structure without the distortions characteristic of traditional methods.

A Numerically Stable Circular Harmonic Reconstruction Algorithm

A numerically stable algorithm based on the orthogonal function solution of the 2-D Radon transform is given. This form of image reconstruction is often called circular harmonic transform (CHT) reconstruction. Series of Zernike polynomials are evaluated by solving a system of nonhomogeneous difference equations. The solution is based on a recursion formula for Zernike polynomials. We show that the method is numerically stable for the paraxial case. This algorithm overcomes the loss of spatial and contrast resolution associated with CHT algorithms. When compared with ramp-filtered back-projection, it is more resistant to ringing and it is not subject to systematic errors that seem to be caused by the discretization of the back-projection operator. The execution time is proportional to ${{9N^3 } / {32}}$ for $N^2 $ points on a polar grid, so that it is comparable to back-projection methods in execution time.

Reconstructive tomography with diffracting wavefields

A unified theoretical framework based on the scalar wave equation is presented for transmission tomography of weakly inhomogeneous objects embedded in a known uniform background medium. The developed formulation differs from the usual treatments of diffraction tomography in that it is not limited to applications employing plane-wave illumination and planar measurement boundaries and can be extended to weakly inhomogeneous objects embedded in non-uniform backgrounds and to the class of strongly scattering objects. The new formulation is based on the mathematical operations of propagation and backpropagation which are generalisations of the conventional tomographic operations of projection and backprojection to the case of diffracting wavefields. The projection and backprojection operations are then obtained from the generalised operations in the zero wavelength limit. A class of reconstruction algorithms are developed within the new formulation that are generalisations, to the case of diffraction tomography, of the class of iterative ART (algebraic reconstruction technique) algorithms of conventional (X-ray) tomography. The reconstruction algorithms developed (called generalised ART algorithms) are shown to be applicable to situations where the tomographic data are sparse (limited-view problem) and to the incorporation of a priori information into the reconstruction process.

An attenuated projector-backprojector for iterative SPECT reconstruction

A new ray-driven projector-backprojector which can easily be adapted for hardware implementation is described and simulated in software. The projector-backprojector discretely models the attenuated Radon transform of a source distributed within an attenuating medium as line integrals of discrete pixels, obtained using the standard sampling technique of averaging the emission source of attenuation distribution over small square regions. Attenuation factors are calculated for each pixel during the projection and backprojection operations instead of using precalculated values. The calculation of the factors requires a specification of the attenuation distribution, estimated either from an assumed constant distribution and an approximate body outline or from transmission measurements. The distribution of attenuation coefficients is stored in memory for efficient access during the projection and backprojection operations. The reconstruction of the source distribution is obtained by using a conjugate gradient or SIRT type iterative algorithm which requires one projection and one backprojection operation for each iteration.

Approximate inversion of the 3 D radon transform

AbstractThe nuclear magnetic resonance (NMR) zeugmatography, a new technology in diagnostic medicine, leads to the problem of inverting the Radon transform in three dimensions. In contrast to even dimensions in odd dimensions the inversion formula is local. In three dimensions it consists of a backprojection operator, which brings no problems for the implementation, and taking the second derivative of the data. This is usually approximated by the central difference quotient. Here the effect of this approximate inversion operator is studied, error estimates for this ill‐posed problem are given and a stepsize is computed which minimizes the sum of discretization and data error.

A filtered backpropagation algorithm for diffraction tomography

A reconstruction algorithm is derived for parallel beam transmission computed tomography through two-dimensional structures in which diffraction of the insonifying beam must be taken into account. The algorithm is found to be completely analogous to the filtered backprojection algorithm of conventional transmission tomography with the exception that the backprojection operation has to be replaced by a back propagation process whereby the complex phase of a field measured over a line outside the object is made to propagate back through the object space. The algorithm is applicable to diffraction tomography within either the first Born or Rytov approximations. Application of the algorithm to three-dimensional structures is also discussed.

A filtered backpropagation algorithm for diffraction tomography.

A reconstruction algorithm is derived for parallel beam transmission computed tomography through two-dimensional structures in which diffraction of the insonifying beam must be taken into account. The algorithm is found to be completely analogous to the filtered backprojection algorithm of conventional transmission tomography with the exception that the backprojection operation has to be replaced by a back propagation process whereby the complex phase of a field measured over a line outside the object is made to propagate back through the object space. The algorithm is applicable to diffraction tomography within either the first Born or Rytov approximations. Application of the algorithm to three-dimensional structures is also discussed.

The reconstruction of fan-beam data by filtering the back-projection

Expressions for the fan-beam projection and back-projection operators are given along with an evaluation of the impulse response for the composite of the projection and back-projection operations. The fan-beam geometries for both curved and flat detectors have back-projection operators such that the impulse response for the composite of the projection and back-projection operations is equal to 1/| r − r 0| if the data are taken over 360°, and thus two-dimensional Fourier filter techniques can be used to reconstruct transverse sections from fan-beam projection data. It is shown that the impulse response for the composite of the projection and back-projection operators for fan-beam geometry is not spatially invariant if the data are taken over 180° as is true for parallel-beam projection data. The convolution algorithm is approximately four times faster and requires 40% less memory than the filter of the back-projection algorithm. However, the filter of the back-projection algorithm has the advantage of easily implementing different frequency space filters which lends itself to easily changing the noise propagation vs resolution properties of the convolution kernel.

A new approach to interpolation in computed tomography.

AbstractMost X-ray scanners presently in production use a method of image reconstruction called filtered back-projection. There are two limitations in data sampling which require that approximations be used during the back-projection operation, and hence result in error in the reconstructed image. I