The APEX Method in Image Sharpening and the Use of Low Exponent Lévy Stable Laws

Abstract

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 Lévy 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 of the same blurred image. Significantly, low exponent Lévy point spread functions were detected and used in all the above examples. Such low exponents are exceptional in physical applications where symmetric stable laws appear. In the present case, the physical meaning of these Lévy exponents is uncertain.

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