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THE APEX METHOD IN IMAGE SHARPENING AND THE USE OF LOW EXPONENT LEVY STABLE LAWS

IP.com Disclosure Number: IPCOM000125646D
Original Publication Date: 2002-Dec-11
Included in the Prior Art Database: 2005-Jun-09

Publishing Venue

National Institute of Standards and Technology

Related People

ALFRED S. CARASSO: INVENTOR

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 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 of the same blurred image. Significantly, low exponent Levy 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 Levy exponents is uncertain.

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Vol. 63, No. 2, pp. 593-618

   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.

   Key words. image deblurring; blind deconvolution; direct methods; electronic imaging systems; heavy-tailed distributions; low exponent stable laws; APEX method; SECB method; nonuniqueness; astronomical, Landsat, and SEM images; MRI and PET brain scans

AMS subject classifications. 35R25, 35B60, 60E07, 68U10

PII. S0036139901389318

   1. Introduction. The APEX method is an FFT-based direct blind deconvolu- tion technique introduced by the author in [9]. The significance of the present paper lies in the successful use of that method in sharpening a wide variety of real blurred images, as opposed to the synthetically blurred images discussed in [9]. The rea- sons behind these successful applications are not fully understood. Not all images can be usefully enhanced with the APEX method. The present paper is essentially self-contained and may be read independently of [9].

   Blind deconvolution seeks to deblur an image without knowing the point spread function (psf) describing the blur. Most approaches to that problem are iterative in nature. Because nonuniqueness is compounded with discontinuous dependence on data, such iterative procedures are not always well-behaved. When the iterative process is stable, several thousand iterations may be necessary to achieve useful recon- structions. However, as shown in [9], by limiting the class of blurs, noniterative direct procedures can be devised that accomplish blind deconvolution of 512 × 512 images in seconds on current desktop platforms.

   The APEX method assumes the image g(x, y) to have been blurred by a restricted type of shift-invariant psf h(x, y), one...