Linograms in Image Reconstruction from Projections

Abstract

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.

Planogram rebinning with the frequency-distance relationship

“…The relations (3,4,6) were first shown by Defrise et al in an unpublished note [9]. Also note that from (4) with y = −V 0 /U 0 we have that (10) This relationship is known as the frequency-distance relationship.…”

“…We name our rebinning algorithm the planogram frequency-distance rebinning algorithm (PFDR). The PFDR algorithm is derived in the planogram coordinate system [2] which is the native data format for PET systems with panel detectors and is the 3-D extension of the 2-D linogram transform developed by Edholm [3].…”

Fourier Reconstruction of Functions from their Nonstandard Sampled Radon Transform

“…Fourier reconstruction algorithms for standard sampled Radon transforms have recently attained new attention in connection with fast Fourier transform algorithms for nonequispaced data [10,32,11,26]. These new algorithms improve known Fourier reconstruction algorithms as the UFR/gridding algorithm [17,18,23,27] or the linogram algorithm [7,28] with respect to the reconstruction speed and image quality. In contrast to the convenient filtered backprojection which ensures a good quality of the image at the expense of O(N 3 ) arithmetic operations, Fourier reconstruction methods require only O(N 2 log N) arithmetic operations.…”

Characterization of and correction for artifacts in linogram MRI

“…7 , compared to the 2DFT images, is a result of the number of views used in the reconstruction. To reconstruct a 256 x 256 image, as was done here, requires about 402 views for PR and about 512 views for LR to prevent any blurring (see (4)). The number of views used is substantially less than the ideally required number; this accounts for the blurring noticed.…”