## Abstract

An iterative technique is proposed for improving the quality of reconstructions from projections when the number of projections is small or the angular range of projections is limited. The technique consists of transforming repeatedly between image and transform spaces and applying *a priori* object information at each iteration. The approach is a generalization of the Gerchberg-Papoulis algorithm, a technique for extrapolating in the Fourier domain by imposing a space-limiting constraint on the object in the spatial domain. *A priori* object data that may be applied, in addition to truncating the image beyond the known boundaries of the object, include limiting the maximum range of variation of the physical parameter being imaged. The results of computer simulations show clearly how the process of forcing the image to conform to *a priori* object data reduces artifacts arising from limited data available in the Fourier domain.

© 1981 Optical Society of America

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### Equations (8)

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(1)
$${F}_{n}(u,v)={\iint}_{I}{f}_{n}(x,y)\hspace{0.17em}\text{exp}[i(ux+vy)]dxdy,$$
(2)
$${g}_{n+1}(x,y)=\frac{1}{{(2\pi )}^{2}}{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{G}_{n}(u,v)\hspace{0.17em}\text{exp}[-i(ux+vy)]dudv,$$
(3)
$${f}_{n+1}(x,y)={g}_{n+1}(x,y){P}_{I}(x,y),$$
(4)
$${P}_{I}(x,y)=\{\begin{array}{c}1\hspace{0.17em}\text{for}\hspace{0.17em}(x,y)\in I\\ 0\hspace{0.17em}\text{for}\hspace{0.17em}(x,y)\notin I.\end{array}$$
(5)
$${f}_{n+1}(x,y)=\{\begin{array}{l}{f}_{n+1}(x,y)\hspace{0.17em}\text{when}\hspace{0.17em}{t}_{l}(x,y)\le {f}_{n+1}(x,y)\le {t}_{u}(x,y),\hfill \\ {t}_{u}(x,y)\hspace{0.17em}\text{when}\hspace{0.17em}{t}_{u}(x,y)<{f}_{n+1}(x,y),\hfill \\ {t}_{l}(x,y)\hspace{0.17em}\text{when}\hspace{0.17em}{f}_{n+1}(x,y)<{t}_{l}(x,y),\hfill \end{array}$$
(6)
$$\begin{array}{l}{G}_{n}(u,v)={F}_{n}(u,v)+[F(u,v)-{F}_{n}(u,v)]{P}_{D}(u,v)\\ =\{\begin{array}{l}F(u,v)\hspace{0.17em}\text{for}\hspace{0.17em}(u,v)\in D\hfill \\ {F}_{n}(u,v)\hspace{0.17em}\text{for}\hspace{0.17em}(u,v)\notin D,\hfill \end{array}\end{array}$$
(7)
$${P}_{D}(u,v)=\{\begin{array}{l}1\hspace{0.17em}\text{for}\hspace{0.17em}(u,v)\in D\hfill \\ 0\hspace{0.17em}\text{for}\hspace{0.17em}(u,v)\notin D,\hfill \end{array}$$
(8)
$$\begin{array}{l}{E}_{n}={\iint}_{I}{\left|f(x,y)-{f}_{n}(x,y)\right|}^{2}dxdy\\ =\frac{1}{{(2\pi )}^{2}}{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{\left|F(u,v)-{F}_{n}(u,v)\right|}^{2}dudv>\frac{1}{{(2\pi )}^{2}}{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{\left|F(u,v)-{G}_{n}(u,v)\right|}^{2}dudv\\ ={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{\left|f(x,y)-{g}_{n+1}(x,y)\right|}^{2}dxdy>{\iint}_{I}{\left|f(x,y)-{f}_{n+1}(x,y)\right|}^{2}dxdy\\ ={E}_{n+1}.\end{array}$$