A Geometry Driven Reconstruction Algorithm for the Mojette Transform

The Mojette transform is an entirely discrete form of the Radon transform developed in 1995. It is exactly invertible with both the forward and inverse transforms requiring only the addition operation. Over the last 10 years it has found many applications including image watermarking and encryption, tomographic reconstruction, robust data transmission and distributed data storage. This paper presents an elegant and efficient algorithm to directly apply the inverse Mojette transform. The method is derived from the inter-dependance of the “rational” projection vectors (pi, qi) which define the direction of projection over the parallel set of lines b = pil − qik. Projection values are acquired by summing the value of image pixels, f(k, l), centered on these lines. The new inversion is up to 5 times faster than previously proposed methods and solves the redundancy issues of these methods. @inproceedings{normand2006dgci, Author = {Normand, Nicolas and Kingston, Andrew and {\’E}venou, Pierre}, Booktitle = {Discrete Geometry for Computer Imagery}, Date = {2006-10}, Doi = {10.1007/11907350_11}, Editor = {Kuba, Attila and Ny{\’u}l, L{\’a}szl{\’o} G. and Pal{\’a}gyi, K{\’a}lm{\’a}n}, Month = oct, Pages = {122-133}, Publisher = {Springer Berlin / Heidelberg}, Series = {Lecture Notes in Computer Science}, Title = {A Geometry Driven Reconstruction Algorithm for the {M}ojette Transform}, Volume = {4245}, Year = {2006}} The original publication is available at www.springerlink.com A geometry driven reconstruction algorithm for the Mojette transform Nicolas Normand, Andrew Kingston and Pierre Évenou IRCCyN-IVC, École polytechnique de l’Université de Nantes, Rue Christian Pauc, La Chantrerie, 44306 Nantes Cedex 3, FRANCE. {nicolas.normand,andrew.kingston,pierre.evenou}@univ-nantes.fr Abstract. The Mojette transform is an entirely discrete form of the Radon transform developed in 1995. It is exactly invertible with both the forward and inverse transforms requiring only the addition operation. Over the last 10 years it has found many applications including image watermarking and encryption, tomographic reconstruction, robust data transmission and distributed data storage. This paper presents an elegant and efficient algorithm to directly apply the inverse Mojette transform. The method is derived from the inter-dependance of the “rational” projection vectors (pi, qi) which define the direction of projection (by summing the value of image pixels, f(k, l), centered) on the parallel set of lines b = pil − qik. It is up to 5 times faster than previously proposed methods and solves the redundancy issues of these methods. The Mojette transform is an entirely discrete form of the Radon transform developed in 1995. It is exactly invertible with both the forward and inverse transforms requiring only the addition operation. Over the last 10 years it has found many applications including image watermarking and encryption, tomographic reconstruction, robust data transmission and distributed data storage. This paper presents an elegant and efficient algorithm to directly apply the inverse Mojette transform. The method is derived from the inter-dependance of the “rational” projection vectors (pi, qi) which define the direction of projection (by summing the value of image pixels, f(k, l), centered) on the parallel set of lines b = pil − qik. It is up to 5 times faster than previously proposed methods and solves the redundancy issues of these methods.