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# Reply to “Comments on ‘The Discrete Periodic Radon Transform’”

Dept. of Electron. & Inf. Eng., Hong Kong Polytech. Univ., Hong Kong, China

IEEE Transactions on Signal Processing (Impact Factor: 2.81). 12/2010; DOI: 10.1109/TSP.2010.2059021 Source: IEEE Xplore

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**ABSTRACT:**This paper discusses the decomposition of the image by direction images, which is based on the concept of the tensor representation and its advanced form, the paired representation. The 2-D image is considered of the size N × N , where N is prime, a power of two, and a power of odd primes. The tensor and paired representations in the frequency-and-time domain define the image as a set of 1-D signals, which we call splitting-signals. Each of such splitting-signals is calculated as the sum of the image along the parallel lines, and it defines the direction image as a component of the original image. The unique decomposition of the image by direction images is described, and formulas for the inverse tensor and paired transforms are given. These formulas can be used for image reconstruction from projections, when splitting-signals or their direction images are calculated directly from the projection data. The number of required projections is uniquely defined by the tensor representation of the image.IEEE Transactions on Image Processing 10/2011; · 3.20 Impact Factor

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