Five limited-data computed tomography algorithms are compared. The algorithms used are adapted versions of the algebraic reconstruction technique, the multiplicative algebraic reconstruction technique, the Gerchberg–Papoulis algorithm, a spectral extrapolation algorithm descended from that of Harris [J. Opt. Soc. Am. 54, 931–936 (1964)], and an algorithm based on the singular value decomposition technique. These algorithms were used to reconstruct phantom data with realistic levels of noise from a number of different imaging geometries. The phantoms, the imaging geometries, and the noise were chosen to simulate the conditions encountered in typical computed tomography applications in the physical sciences, and the implementations of the algorithms were optimized for these applications. The multiplicative algebraic reconstruction technique algorithm gave the best results overall; the algebraic reconstruction technique gave the best results for very smooth objects or very noisy (20-dB signal-to-noise ratio) data. My implementations of both of these algorithms incorporate a priori knowledge of the sign of the object, its extent, and its smoothness. The smoothness of the reconstruction is enforced through the use of an appropriate object model (by use of cubic B-spline basis functions and a number of object coefficients appropriate to the object being reconstructed). The average reconstruction error was 1.7% of the maximum phantom value with the multiplicative algebraic reconstruction technique of a phantom with moderate-to-steep gradients by use of data from five viewing angles with a 30-dB signal-to-noise ratio.

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

RPV is the number of ray sums per projection or view. L_{x} and L_{y} are the dimensions of the reconstruction space in arbitrary units; M and N represent the number of discretized object values in the x and y directions, respectively.

Table 2

Number of Iterations Used for Three Iterative Reconstruction Programs as a Function of Imaging Geometry

Computer Time and Memory Requirements for the Cosine Phantom with Geometry A^{a}

Algorithm

Iterations

Time (s)

Time (s) without Initialization

Seconds per Iteration

Memory (kbytes)

ART

300

870

522

1.74

270

MART

30

402

66

2.2

270

IH

300

312

312

1.04

1

HE

8

4

2

SVD

5580

2

751

The initialization time in this table is the time required to compute the matrix W of Eq. (5) or (17). Since this matrix depends only on the imaging and reconstruction geometries and not on the ray sum data, it need only be computed once for a given geometry. Time per iteration is based on the reconstruction timings without initialization. Timings are for a 25-MHz Intel 80386/387-based microcomputer. The SVD program requires more memory for the sparse matrix W than ART and MART because it is not stored optimally. The SVD reconstruction was performed on a VAX computer; the time was converted to microcomputer seconds by using a factor determined by running a similar program on both machines. Timings and memory requirements for ART, MART, and SVD (the series-expansion algorithms) are dependent on imaging and reconstruction geometry.

RPV is the number of ray sums per projection or view. L_{x} and L_{y} are the dimensions of the reconstruction space in arbitrary units; M and N represent the number of discretized object values in the x and y directions, respectively.

Table 2

Number of Iterations Used for Three Iterative Reconstruction Programs as a Function of Imaging Geometry

Computer Time and Memory Requirements for the Cosine Phantom with Geometry A^{a}

Algorithm

Iterations

Time (s)

Time (s) without Initialization

Seconds per Iteration

Memory (kbytes)

ART

300

870

522

1.74

270

MART

30

402

66

2.2

270

IH

300

312

312

1.04

1

HE

8

4

2

SVD

5580

2

751

The initialization time in this table is the time required to compute the matrix W of Eq. (5) or (17). Since this matrix depends only on the imaging and reconstruction geometries and not on the ray sum data, it need only be computed once for a given geometry. Time per iteration is based on the reconstruction timings without initialization. Timings are for a 25-MHz Intel 80386/387-based microcomputer. The SVD program requires more memory for the sparse matrix W than ART and MART because it is not stored optimally. The SVD reconstruction was performed on a VAX computer; the time was converted to microcomputer seconds by using a factor determined by running a similar program on both machines. Timings and memory requirements for ART, MART, and SVD (the series-expansion algorithms) are dependent on imaging and reconstruction geometry.