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Calibrating Image Roughness by Estimating Lipschitz Exponents, with Applications to Image Restoration



Alfred S. Carasso, Andras Vladar


Most commonly occurring images f(x,y) are not smoothly differentiable functions of the variables x and y. Rather, these images display edges, localized sharp features, and other significant fine scale details or texture. Correct characterization and calibration of the lack of smoothness in such images is important in various image processing tasks. So-called Lipschitz spaces appear to be the appropriate mathematical framework for accommodating non smooth images. The L1 Lipschitz exponent ?? for the given image, where 0 < ?? ?? 1, measures the fine scale roughness of that image, provided the image is relatively noise free. Finely textured imagery has low values for ??, while large values of ?? indicate that the image is relatively smooth. This paper describes a recently developed mathematical technique for estimating ??. The technique is based on successively blurring the image by convolution with increasingly narrower Gaussians, using commonly available Fast Fourier Transform algorithms. Instructive examples are used to illustrate the quantitative changes in ?? that occur when an image is either degraded or restored. Of particular interest are the documented changes in ?? that accompany APEX blind deconvolution of real images from Additional applications include the routine monitoring of image sharpness and imaging performance in diverse imaging systems, the evaluation of image reconstruction software quality, the detection of possibly abnormal fine scale features in some medical applications, and the monitoring of surface finish in industrial applications.
NIST Interagency/Internal Report (NISTIR) - 7438
Report Number


Andromeda galaxy, APEX method, Gauss singular integral, image metrology, image roughness, Lipschitz exponents


Carasso, A. and Vladar, A. (2007), Calibrating Image Roughness by Estimating Lipschitz Exponents, with Applications to Image Restoration, NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online],, (Accessed April 23, 2024)
Created July 31, 2007, Updated October 12, 2021