Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

The Apex Method in Image Sharpening and the Use of Low Exponent Levy Stable Laws



Alfred S. Carasso


The APEX method is an FFT-based direct blind deconvolution technique that can process complex high-resolution imagery in a few minutes of cpu time 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 L vy 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 very noticeable improvements in visual quality. The discussion includes a documented example of non-uniqueness where distinct point spread functions produce high-quality restorations of the same blurred image. Significantly, low exponent L vy 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 origin of these L vy exponents remains uncertain.
Siam Journal on Applied Mathematics
63 No. 2


APEX method, astronomical, blind deconvolution, direct methods, electronic imaging systems, image deblurring, Landsat, low exponent stable laws, SEM images


Carasso, A. (2003), The Apex Method in Image Sharpening and the Use of Low Exponent Levy Stable Laws, Siam Journal on Applied Mathematics, [online], (Accessed April 21, 2024)
Created January 1, 2003, Updated February 17, 2017