Computed Tomography Reconstruction Using Only One Projection Angle
Fawaz Hjouj, Mohamed Soufiane Jouini
Abstract
Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> represent a digitized version of an image <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f\left ({x,y }\right)$ </tex-math></inline-formula> . Assume that the image fits inside a rectangular region and this region is subdivided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\,\,\times \,\,N$ </tex-math></inline-formula> squares. We call these squares the shifted box functions. Thus <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f\left ({x,y }\right)$ </tex-math></inline-formula> is approximated by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M\,\,\times \,\,N$ </tex-math></inline-formula> matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> . This paper proofs that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> can be recovered exactly and uniquely from the Radon transform of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> using only one selected view angle with a well selected family of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$MN$ </tex-math></inline-formula> lines. The paper also proposes a precise method for computing the Radon transform of an image. The approach can be categorized as an algebraic reconstruction, but it is merely a theoretical contribution for the field of limited data tomography.