iradon-sart

iradon_sart

skimage.transform.iradon_sart(radon_image, theta=None, image=None, projection_shifts=None, clip=None, relaxation=0.15) [source]

Inverse radon transform

Reconstruct an image from the radon transform, using a single iteration of the Simultaneous Algebraic Reconstruction Technique (SART) algorithm.

Parameters:

radon_image : 2D array, dtype=float Image containing radon transform (sinogram). Each column of the image corresponds to a projection along a different angle. The tomography rotation axis should lie at the pixel index radon_image.shape[0] // 2 along the 0th dimension of radon_image. theta : 1D array, dtype=float, optional Reconstruction angles (in degrees). Default: m angles evenly spaced between 0 and 180 (if the shape of radon_image is (N, M)). image : 2D array, dtype=float, optional Image containing an initial reconstruction estimate. Shape of this array should be (radon_image.shape[0], radon_image.shape[0]). The default is an array of zeros. projection_shifts : 1D array, dtype=float Shift the projections contained in radon_image (the sinogram) by this many pixels before reconstructing the image. The i’th value defines the shift of the i’th column of radon_image. clip : length-2 sequence of floats Force all values in the reconstructed tomogram to lie in the range [clip[0], clip[1]] relaxation : float Relaxation parameter for the update step. A higher value can improve the convergence rate, but one runs the risk of instabilities. Values close to or higher than 1 are not recommended.

Returns:

reconstructed : ndarray Reconstructed image. The rotation axis will be located in the pixel with indices (reconstructed.shape[0] // 2, reconstructed.shape[1] // 2).

Notes

Algebraic Reconstruction Techniques are based on formulating the tomography reconstruction problem as a set of linear equations. Along each ray, the projected value is the sum of all the values of the cross section along the ray. A typical feature of SART (and a few other variants of algebraic techniques) is that it samples the cross section at equidistant points along the ray, using linear interpolation between the pixel values of the cross section. The resulting set of linear equations are then solved using a slightly modified Kaczmarz method.

When using SART, a single iteration is usually sufficient to obtain a good reconstruction. Further iterations will tend to enhance high-frequency information, but will also often increase the noise.

References

[R370]

AC Kak, M Slaney, “Principles of Computerized Tomographic Imaging”, IEEE Press 1988.

[R371]

AH Andersen, AC Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm”, Ultrasonic Imaging 6 pp 81–94 (1984)

[R372]

S Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen”, Bulletin International de l’Academie Polonaise des Sciences et des Lettres 35 pp 355–357 (1937)

[R373]

Kohler, T. “A projection access scheme for iterative reconstruction based on the golden section.” Nuclear Science Symposium Conference Record, 2004 IEEE. Vol. 6. IEEE, 2004.

[R374]

Kaczmarz’ method, Wikipedia, http://en.wikipedia.org/wiki/Kaczmarz_method