A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.

/pdf/practical-cone-beam-algorithm-4tqh14lkkz.pdf

Practical cone-beam algorithm

Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density ? = ?(x, y, z) generates, and is to be reconstructed from, the number of counts n*(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate ? of ? which maximizes the probability p(n*|?) of observing the actual detector count data n* over all possible densities ?. Let independent Poisson variables n(b) with unknown means ?(b), b = 1, ···, B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ···, D with p(b, d) a one-step transition matrix, assumed known. We observe the total number n* = n*(d) of emissions in each detector unit d and want to estimate the unknown ? = ?(b), b = 1, ···, B. For each ?, the observed data n* has probability or likelihood p(n*|?). The EM algorithm of mathematical statistics starts with an initial estimate ?0 and gives the following simple iterative procedure for obtaining a new estimate ?new, from an old estimate ?old, to obtain ?k, k = 1, 2, ···, ?new(b)= ?old(b) ?Dd=1 n*(d)p(b,d)/??old(b?)p(b?,d),b=1,···B.

Maximum Likelihood Reconstruction for Emission Tomography

The authors define ordered subset processing for standard algorithms (such as expectation maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialize application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al. (1981) for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT, the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained. >

Accelerated image reconstruction using ordered subsets of projection data

We present a review of methods for the forward and inverse problems in optical tomography. We limit ourselves to the highly scattering case found in applications in medical imaging, and to the problem of absorption and scattering reconstruction. We discuss the derivation of the diffusion approximation and other simplifications of the full transport problem. We develop sensitivity relations in both the continuous and discrete case with special concentration on the use of the finite element method. A classification of algorithms is presented, and some suggestions for open problems to be addressed in future research are made.

https://iopscience.iop.org/article/10.1088/0266-5611/15/2/022/pdf

Optical tomography in medical imaging

Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.

/pdf/on-projection-algorithms-for-solving-convex-feasibility-2wyotvnuad.pdf

On Projection Algorithms for Solving Convex Feasibility Problems

Iterative reconstruction algorithms.

ART: Mathematics and applications a report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques

Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

The problem of recovering an image (a function of two variables) from experimentally available integrals of its grayness over thin strips is of great importance in a large number of scientific areas. An important version of the problem in medicine is that of obtaining the exact density distribution within the human body from X-ray projections. One approach that has been taken to solve this problem consists of translating the available information into a system of linear inequalities. The size and the sparsity of the resulting system (typically, 25,000 inequalities with fewer than 1 percent of the coefficients nonzero) makes methods using successive relaxations computationally attractive, as compared to other ways of solving systems of inequalities. In this paper, it is shown that, for a consistent system of linear inequalities, any sequence of relaxation parameters lying strictly between 0 and 2 generates a sequence of vectors which converges to a solution. Under the same assumptions, for a system of linear equations, the relaxation method converges to the minimum norm solution. Previously proposed techniques are shown to be special cases of our procedure with different choices of relaxation parameters. The practical consequences for image reconstruction of the choice of the relaxation parameters are discussed.

Relaxation methods for image reconstruction

Computerized tomography is used to calculate two independent images representing distributions of refractive index ( n ) and acoustic attenuation (α) within 1–3 mm thick cross-sections through excised organs, including canine hearts and human breasts. Values of n and α are calculated from profiles of propagation times and amplitudes, respectively, of digitized acoustic pulses obtained by rectilinear transmission scans of the tissue at multiple angles of view. Images of the local speed of ultrasound show high values in regions of muscle, breast parenchyma, medullary carcinoma and connective tissue and show low values in regions of fat. Images of acoustic attenuation show high values in regions of connective tissue and borders of scirrhus carcinoma and low values in regions of fat.

Arnold Lent

Papers

Iterative reconstruction algorithms.

ART: Mathematics and applications a report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques

Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

Relaxation methods for image reconstruction

Measurement of spatial distribution of refractive index in tissues by ultrasonic computer assisted tomography