The APEX Method in Image Sharpening and the Use of Low Exponent L[e-acute]vy Stable Laws

The APEX method is an FFT-based direct blind deconvolution technique that can process complex high resolution imagery in seconds or minutes on current desktop platforms. The method is predicated on a restricted class of shift-invariant blurs that can be expressed as finite convolution products of two-dimensional radially symmetric Levy stable probability density functions. This class generalizes Gaussian and Lorentzian densities but excludes defocus and motion blurs. Not all images can be enhanced with the APEX method. However, it is shown that the method can be usefully applied to a wide variety of real blurred images, including astronomical, Landsat, and aerial images, MRI and PET brain scans, and scanning electron microscope images. APEX processing of these images enhances contrast and sharpens structural detail, leading to noticeable improvements in visual quality. The discussion includes a documented example of nonuniqueness, in which distinct point spread functions produce high-quality restorations ...

[1]  A. Carasso.  Linear and Nonlinear Image Deblurring: A Documented Study , 1999 .

[2]  Alfred S. Carasso,et al.  Error bounds in nonsmooth image deblurring , 1997 .

[3]  Fadil Santosa,et al.  Recovery of Blocky Images from Noisy and Blurred Data , 1996, SIAM J. Appl. Math..

[4]  Stanley Osher,et al.  Explicit Algorithms for a New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal , 2000, SIAM J. Sci. Comput..

[5]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[6]  Robert Weber.  The Ground-Based Electro-Optical Detection Of Deep-Space Satellites , 1978, Other Conferences.

[7]  C. B. Johnson.  Classification of Electron-optical Device Modulation Transfer Functions , 1972 .

[8]  C. Tsallis,et al.  Statistical-mechanical foundation of the ubiquity of Lévy distributions in Nature. , 1995, Physical review letters.

[9]  C B Johnson.  MTF parameters for all photographic films listed in Kodak pamphlet P-49. , 1976, Applied optics.

[10]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[11]  Mila Nikolova,et al.  Local Strong Homogeneity of a Regularized Estimator , 2000, SIAM J. Appl. Math..

[12]  Alfred S. Carasso,et al.  Overcoming Ho¨lder continuity in ill-posed continuation problems , 1994 .

[13]  S. Nudelman.  Photoelectronic imaging devices , 1971 .

[14]  C. Bruce Johnson,et al.  High-resolution microchannel plate image tube development , 1991, Electronic Imaging.

[15]  Charles S. Williams,et al.  Introduction To The Optical Transfer Function , 1989 .

[16]  竹中 茂夫.  G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance , 1996 .

[17]  Yann Gousseau,et al.  Are Natural Images of Bounded Variation? , 2001, SIAM J. Math. Anal..

[18]  M. Nikolova.  Image restoration by minimizing objective functions with nonsmooth data-fidelity terms , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[19]  R. Hufnagel,et al.  Modulation Transfer Function Associated with Image Transmission through Turbulent Media , 1964 .

[20]  Alfred S. Carasso,et al.  Logarithmic convexity and the “slow evolution” constraint in ill-posed initial value problems , 1999 .

[21]  David S. Bright,et al.  APEX method and real-time blind deconvolution of scanning electron microscope imagery , 2001 .

[22]  C. E. Catchpole,et al.  Microchannel plate inverter image intensifiers , 1971 .

[23]  C B Johnson.  Circular Aperture Diffraction Limited MTF: Approximate Expressions. , 1972, Applied optics.

[24]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[25]  Tony F. Chan,et al.  Total variation blind deconvolution , 1998, IEEE Trans. Image Process..

[26]  C. B. Johnson,et al.  Point-Spread Functions, Line-Spread Functions, and Edge-Response Functions Associated with MTFs of the Form exp [ - (omega /omega(c))(n)]. , 1973, Applied optics.

[27]  Stanley Osher,et al.  Total variation based image restoration with free local constraints , 1994, Proceedings of 1st International Conference on Image Processing.

[28]  Alfred S. Carasso,et al.  Direct Blind Deconvolution , 2001, SIAM J. Appl. Math..

[29]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[30]  Rachel Kuske,et al.  Rate of Convergence to a Stable Law , 2001, SIAM J. Appl. Math..