Estimation of the Gaussian blur parameter by comparing histograms of gradients with a standard image

Objectives. The aim of this study is to develop a method for automatic quantitative estimation of the Gaussian blur parameter in digital images, which typically arises due to defocus of the optical system, various optical and camerainduced aberrations, as well as the influence of the propagation medium. This task is highly relevant for a wide range of applied fields, including remote sensing, forensic analysis, photogrammetry, medical imaging, automated inspection, and preprocessing of visual data prior to solving restoration, classification, or recognition problems.

Methods. The proposed method is based on comparing the two-dimensional histogram of gradients of the analyzed image with reference histograms precomputed for a high-sharpness image with similar texture and scale. The reference image is artificially blurred using convolution with a Gaussian kernel at various blur levels. For each level of blur, a two-dimensional gradient histogram is constructed, representing the distribution of directions and magnitudes of local intensity changes. The comparison with the corresponding histogram of the target image is performed after applying a logarithmic transformation and computing the Euclidean norm. This approach provides high sensitivity, interpretability, and numerical stability. The method does not require edge detection, neural network training, or labeled data, and can be implemented with minimal computational cost.

Results. Tests on synthetic data demonstrate that the proposed approach achieves high accuracy: the relative error in estimating the Gaussian blur parameter within the range of 0.7 to 2.0 pixels is less than 5%, and in most cases does not exceed 2–3%. The method is robust to noise, compression, local artifacts, and texture inhomogeneities.

Conclusions. The developed approach can be applied in automated image analysis systems as well as in blind deconvolution preprocessing tasks. It offers high accuracy, implementation simplicity, and reproducibility, providing reliable blur estimation under minimal data assumptions.

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