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A storage-efficient algorithm for finding the regularized solution of a large, inconsistent system of equations

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) 1980年第4期25卷 361-366页

作者： Herman, G.T. Lent, A. Hurwitz, H. Medical Image Processing Group Department of Computer Science State University of New York at Buffalo Amherst NY 14226 4226 Ridge Lea Road United States Corporate Research and Development Center General Electric Company Schenectady NY 12301 United States

We propose an algorithm which for any real number r, any k × l matrix M and any k-vector y, finds the l-vector x which minimizes ∥x∥2 - r2∥Mx - y∥2, i.e. it finds the "regularized" solution to the e...

We propose an algorithm which for any real number r, any k × l matrix M and any k-vector y, finds the l-vector x which minimizes ∥x∥² - r²∥Mx - y∥², i.e. it finds the "regularized" solution to the equation Mx = y. (∥ ∥ denotes the 2-norm.) The algorithm is iterative with the following properties: (i) in a single step it needs access to only one row of the matrix M, (ii) it needs to store only the present estimate of the solution (size l) and a "residual vector" of size k, (iii) in a single step it updates only one component of the residual vector. Because of these properties the algorithm has been found useful in "solving" very large inconsistent systems of equations. Convergence of the algorithm to the desired solution is proved and the rate of convergence of the algorithm is illustrated. The algorithm is considered here, not because it is believed to be generally superior to more commonly used iterative methods, but because for certain very large problems it may be the only feasible method from the point of view of storage requirements. © 1980 Academic Press Inc. (London) Limited.

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On the Bayesian approach to image reconstruction

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Information and Control 1979年第1期42卷 60-71页

作者： Herman, Gabor T. Hurwitz, Henry Lent, Arnold Lung, Hsi-Ping Medical Image Processing Group Department of Computer Science The State University of New York at Buffalo Amherst NY 14226 United States General Electric Company Research and Development Center Schenectady NY United States

A new iterative method is proposed for finding the optimal Bayesian estimate of an unknown image from its projection data (experimentally obtained integrals of its grayness over thin strips). Convergence of the method is proved and its performance is illustrated. The method compares favorably with previously proposed procedures. © 1979 Academic Press, Inc.

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