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# Analog Reconstruction From X Ray Projections

IP.com Disclosure Number: IPCOM000079131D
Original Publication Date: 1973-May-01
Included in the Prior Art Database: 2005-Feb-26
Document File: 3 page(s) / 50K

IBM

## Related People

Chang, SK: AUTHOR [+2]

## Abstract

This description relates to a projection method used to evaluate the lateral, two-dimensional, density distribution of a lengthy object which cannot be sliced into cross sections. For example, X-ray projections which may be used to investigate the leg of a living patient.

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Analog Reconstruction From X Ray Projections

This description relates to a projection method used to evaluate the lateral, two-dimensional, density distribution of a lengthy object which cannot be sliced into cross sections. For example, X-ray projections which may be used to investigate the leg of a living patient.

In accordance with the method, the two-dimensional density distribution is represented by f(x,y) and the length of the object lies in the z direction. An X-ray projection in the y direction can be regarded as F(y)(x) = integral f(x,y) dy. The Fourier transform (FT) of F(y) in the x direction becomes, integral integral f(x,y)e/-iux/ dx dy, which is just F(u, O), where F(u,v) identical to integral integral f(x,y) e/-iux/ e/-ivy/ dx dy, that is, the two-dimen FT of f(x,y).

Similarly, by taking the FT of the projection at an incident angle Theta with respect to the y-axis, the value of F along the line u' making an angle Theta with respect to the u-axis is known, as illustrated in Fig. 1.

By taking a sufficient number of projections, F(u,v) is known at a reasonable number of points to enable reverse FT for the desired density distribution f(x,y).

Previously, the mathematical operations were performed with a digital computer. What is now described is an analog means for the same purpose.

The scheme of the setup is shown in Fig. 2. As usual, a number of X-ray projections are recorded on film. After development, the films are emerged in an index-matching liquid in a container which consists of transparent flat walls, to be Fourier transformed by a lens. This index-matching operation eliminates unwanted phase variation on the films. An opaque mask with a horizontal strip opening covers the film so that f(x,y,z) is Fourier transformed at a well-defined value of z. The FT is recorded with a hologram using an off-axis reference beam. On the hologram, another opaque mask is used.

As in Fig. 3, this mask has an opening that takes the shape of a diagonal sector and is rotated to a new position for each projection. The angle of rotation in radians is simply Pi divided by the number of projections used. When another z p...