SIAM J. NUMER. ANAL. Vol. 15, No. 2, April 1978 Copyright (? 1978 Society forIndustrial and Applied Mathematics DIGITAL REMOVAL OF RANDOM MEDIA IMAGE DEGRADATIONS BY SOLVING THE DIFFUSION EQUATION BACKWARDS IN TIME* ALFRED S. CARASSO,t JAMES G. SANDERSONt AND JAMES M. HYMAN? Abstract.We considerthe image restoration problem withGaussian-likepointspread functions and reformulate it as an initialvalue problemfor the backwardsdiffusion equation. This approach leads to rigorousbounds on the reliability of the restoration, as a function of the noise variance,withoutany assumptions on thespectral characteristics of either signalor noise. In thelatter half ofthepaper, we then describe a powerful algorithm, based on theabove reformulation, and successfully use itto restore a turbulence degradedimage.Typically, a complete restoration and display requires10 seconds of CDC 7600 computing time, fora 128 x 128 image. We also describea restoration experiment whereGaussian blur was simulated, and multiplicative noiseadded according to Huang's model. The algorithm performs competently even at low signalto noise ratios. CONTENTS 1. Introduction. . . ..... . ......... . . .. . . . . . 344 2. Connection with parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . 346 3. The ill-posed nature ofthe restoration problem . . . . . . . . . . . . . . . . . . . . . 347 4. Logarithmic convexity inthe diffusion equation . . . . . . . . . . . . . . . . . . . . 350 5. Stabilization ofthe backwards problem for the diffusion equation . . . . . . . . . . . . . 351 6. An uncertainty principle for the restored image . . . . . . . . . . . . . . . . . . . . . 352 7. Pointwise reliability inthe restored image . . . . . . . . . . . . . . . . . . . . . . . 353 8. Numerical computation ofthe stabilized problem. The backward beamequation . . . . . . . 354 9. Computational considerations .357 10. Restoring a turbulence degraded image . . . . . . . . . . . . . . . . . . . . . . 358 11. Restoring a simulated noisy blurred image . . . . . . . . . . . . . . . . . . . . . . . 361 12. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . * 365 Acknowledgment ........... . ....... 366 References........... . ....... 367 1. Introduction. The problem ofimagerestoration is associated with thesolution of twodimensional integral equationsof theform (1.1) ~~~g(x, y) = -0 -0 h(( x, y)f((6,) de dql where g(x, y) is the recordedor "degraded" image,f(x,y) is the unknown "ideal" image, and h(4, -, x, y) is the point spread function, (p.s.f.),of the imaging process. Given h( , ) and g(x, y), theproblem offindingf(x, y) is seriously complicated bythe presence ofnose in therecorded image.In fact, (1.1) is ingeneral an ill-posed problem, and special precautions are necessary to resurrect f from g. In recent years,muchexcellent workhas been done in thisarea. We direct the readerto the reviewpapers of Sondhi [1] and Hunt [2] foran overview of the field, together with extensive references to the current literature. * Received by theeditors January 29 1977. t Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico. 87131. The work ofthis author was supported bytheArmy ResearchOffice under GrantDAAG29-76-G- 0153. t Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131. The workof this author was supported bytheU.S. Energy Researchand Development Administra- tionunderContract W7405-ENG-36. ? Courant Institute ofMathematical Sciences, New York University, New York,New York 10012. This author's work was supported bytheU.S. Energy Researchand Development Administration under Contract W7405-ENG-36. 344 This content downloaded on Thu, 27 Dec 2012 16:41:02 PM All use subject to JSTOR Terms and Conditions