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Acceleration of Iterative Image Restoration Algorithms

the extent to which the acceleration method can be applied in practice. The use **of** technique II should allow the use **of** iterative image restoration algorithms in applications where they were previously considered too slow. As mentioned previously, there is considerable scope for achieving higher levels **of** acceleration when more information is used in the acceleration process. The new acceleration method could also be used as a generic acceleration method when iterative algorithms are designed. This would allow the algorithm to be designed for producing the best result rather than having to compromise for speed considerations. 4 Conclusion Traditional acceleration methods using a gradient search approach have been presented but require careful use to prevent instability during restoration. A new method, called automatic acceleration, has been proposed and shown to achieve signi cantly increased acceleration over previous methods with little computational overhead. An example R{L restoration achieves an average speedup **of** 40 times after 250 iterations, and an ME method by 20 times after only 50 iterations. **Acceleration** is achieved without the need for a cost{function to be determined or minimized. The new method is stable, and an estimated acceleration factor has been derived and con rmed by experiment. The acceleration technique has been successfully applied to Richardson{Lucy, maximum entropy and Gerchberg{Saxton restoration algorithms and can be integrated with other iterative techniques. The technique shows considerable promise in accelerating a wide range **of** iterative image restoration algorithms, which should allow them to be used in applications where previously they would have been considered impractical. 5 Acknowledgments The authors thank Russell Smith **of** The University **of** Auckland and Richard Lane **of** Canterbury University for their contributions to this paper. The reviewers are also to be acknowledged for their useful comments and suggestions. D.S.C. Biggs also wishes to acknowledge the support **of** The University **of** Auckland and the Royal Society **of** New Zealand through the R.H.T. Bates Scholarship. References [1] T. J. Cornwell. Where have we been, where are we now, where are we going? In R. J. Hanisch and R. L. White, editors, The **Restoration** **of** HST **Image**s and Spectra II. Space Telescope Science Institute, Baltimore, Md. 1994. [2] William H. Richardson. Bayesian-based iterative method **of** image restoration. Journal **of** the Optical Society **of** America, Vol. 62(No. 1):55{59, January 1972. [3] L. B. Lucy. An iterative technique for the recti cation **of** observed images. The Astronomical Journal, Vol. 79(No. 6):745{754, June 1974. [4] V. Vardi, L. A. Shepp, and L. Kaufman. A statistical model for positron emission tomography. Journal **of** the American Statistical Association, Vol. 80(No. 389):8{37, March 1985. [5] Richard L. White. **Image** restoration using the damped Richardson{Lucy method. In R. J. Hanisch and R. L. White, editors, The **Restoration** **of** HST **Image**s and Spectra II. Space Telescope Science Institute, Baltimore, Md. 1994. [6] Edward S. Meinel. Origins **of** linear and non-linear recursive restoration algorithms. Journal **of** the Optical Society **of** America A,Vol. 3(No. 6):787{799, June 1986. [7] H.M. Adorf, R.N. Hook, L.B. Lucy, and F.D. Murtagh. Accelerating the Richardson{Lucy restoration algorithm. In 4th ESO/ST-ECF Data Analysis Workshop, pages 99{103. European Southern Observatory, Garching, Germany, 1992. [8] Timothy J. Holmes and Yi-Hwa Liu. **Acceleration** **of** maximum-likelihood image restoration for uorescence microscopy and other noncoherent imagery. Journal **of** the Optical Society **of** America A,Vol. 8(No. 6):893{907, June 1991. [9] Linda Kaufman. Implementing and accelerating the EM algorithm for positron emission tomography. IEEE Transactions on Medical Imaging, Vol. 6(No. 1):37{51, March 1987. [10] D. S. C. Biggs and M. Andrews. Conjugate gradient acceleration **of** maximum-likelihood image restoration. Electronics Letters, Vol. 31(No. 23):1985{1986, 9th Nov 1995. [11] R. G. Lane. Methods for maximum likelihood deconvolution. Journal **of** the Optical Society **of** America A,Vol. 13:1992{ 1998, 1996. [12] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipies in C. Cambridge University Press, 2nd edition, 1992. [13] J. R. Fienup. Phase retrieval algorithms: a comparison. Applied Optics, Vol. 21(No. 15):2758{2769, 1 August 1982. [14] Philip E. Gill, Walter Murray, and Margaret H. Wright. Practical Optimization. Academic, New York, 1981. [15] N.L. Bonavito, J.E. Dorband, and T. Busse. Maximum entropy restoration **of** blurred and oversaturated Hubble Space Telescope imagery. Applied Optics, Vol. 32, No. 29:5768{5774, 10 October 1993. Appendix A: **Acceleration** Parameter for Technique II We consider the iterative algorithm ( ), which generates a new estimate (xk+1) given the current vector (xk): xk+1 = (xk) = xk + gk; and xk slowly converges towards a solution at xs. The change gk introduced at iteration k can be considered a step in the gradient ascent direction toward the solution. The function to be maximized can be approximated locally by amultidimensional quadratic: m(x) = ,(x , xs) > A(x , xs); rm(x) = ,2A(x , xs); where A is a square, symmetric, and positive de nite matrix. Taking a step in the gradient direction rm(x) for each iteration produces a path through the multidimensional space towards the solution. This path can be described by the solution to the di erence equation in terms **of** the eigenvalues ( A) and eigenvectors (S) **of**A. gk = rm(xk) = ,2 A(xk , xs); A = S AS ,1 ; B = I , 2 A; B = I , 2 A; xk = S k B S,1 (x0 , xs) + xs; < 1 2 max( A) ; APPLIED OPTICS / Vol. 36, No. 8 / 10 March 1997 / pp. 1766{1775 / Page 8

xs is the vector xk converges. A and B are diagonal matricies, and is a scalar used to prevent B from becoming negative. We have no knowledge **of** the form **of** the quadratic or its eigenvalues or eigenvectors. The aim **of** acceleration is to move along the path (or close to it) faster than the iterative algorithm. If this is possible, then the accelerated method should result in the same solution given that the same path was taken. Using the current and the previous estimates and associated gradient information, the aim is to predict a new estimate further along the path. xk = S k B S,1 (x0 , xs) + xs; gk,1 = S k B(I , ,1 B )S,1 (x0 , xs); xk, = S k, B S,1 (x0 , xs) + xs; gk, ,1 = S k, B (I , ,1 B )S,1 (x0 , xs): The di erence between the current (xk) and previous estimate (xk, )is hk;, = xk , xk, = S k B (I , , B )S,1 (x0 , xs): Given estimates at xk and xk, the aim is to predict xk+ : xk+ = S k+ B S,1 (x0 , xs) + xs; gk+ ,1 = S k+ B hk; = xk+ , xk = S k+ B (I , ,1 B )S,1 (x0 , xs); (I , , B )S,1 (x0 , xs): Using a rst{order extrapolation we predict hk; by a least{squares projection **of** khk;, , where k is a scalar: ^hk; = khk;, ; 0 = (hk; , khk;, ) > hk;, ; k = hk; > hk;, hk;, > hk;, We do not have knowledge **of** hk; but we do have the gradient information: hk; = xk+ , xk = S k+ B I , , B (I , ,1 B ); : (I , , B )S,1 (x0 , xs); hk; S k B(I , ,1 B )S,1 (x0 , xs) = gk,1; hk;, k gk, ,1; gk,1 > gk, ,1 gk, ,1 > gk, ,1 From this we can predict a new point (yk) and apply the iterative algorithm to generate the next estimate (xk+1) and gradient (gk): yk = xk + k(xk , xk,1); xk+1 = (yk) = yk + gk; k+1 = gk > gk,1 gk,1 > gk,1 yk+1 = xk+1 + k+1(xk+1 , xk): ; : This formulation **of** the acceleration parameter requires no knowledge **of** the quadratic A or its eigenvalues and eigenvectors and does not require that they be constant over the entire error surface. Appendix B: Estimating the **Acceleration** Factor for Technique II Being able to estimate the acceleration at each step is useful in comparing the technique with unaccelerated restoration and determining how e ective the acceleration process is. Continuing from Appendix A, it can be seen that the acceleration achieved at iteration k is related to how effectively hk; is estimated. If k is the e ciency parameter, then the step actually achieved is hk; k k : ^hk; = khk;, k ; hk; k k = khk;, k ; and therefore the e ective number **of** unaccelerated iterations achieved by acceleration is k k. The value **of** k can be estimated from all previous values **of** k: k,1 X k = 1+ j=1 k,1 Y m=k,j The acceleration factor, k, is given by k k plus the single step **of** the iterative algorithm: k = 1+ kX j=1 kY m=k,j+1 The value **of** k is di cult to determine exactly, though it is proposed that it can be estimated by kgk,2k ^k = k kgk,1k gk,1 > gk,2 kgk,1k kgk,2k The estimates **of** acceleration achieved in this paper use M = 3 [Eq. 13] which correlate well with experimental results. APPLIED OPTICS / Vol. 36, No. 8 / 10 March 1997 / pp. 1766{1775 / Page 9 m: m: M :

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