{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function applies the Wiener filter to a noisy (degraded) image by an impulse response (or PSF). If the data model is" } ] }, { "__type": "Math", "__tag": 4058, "value": "y = Hx + n" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " is noise, " }, { "__type": "InlineMath", "__tag": 4057, "value": "H" }, { "__type": "Text", "__tag": 4046, "value": " the PSF, and " }, { "__type": "InlineMath", "__tag": 4057, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " the unknown original image, the Wiener filter is" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\hat x = F^\\dagger \\left( |\\Lambda_H|^2 + \\lambda |\\Lambda_D|^2 \\right)^{-1}\n\\Lambda_H^\\dagger F y" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "F" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "F^\\dagger" }, { "__type": "Text", "__tag": 4046, "value": " are the Fourier and inverse Fourier transforms respectively, " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Lambda_H" }, { "__type": "Text", "__tag": 4046, "value": " the transfer function (or the Fourier transform of the PSF, see " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": "), and " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Lambda_D" }, { "__type": "Text", "__tag": 4046, "value": " the regularization operator, which is a filter penalizing the restored image frequencies (Laplacian by default, that is, penalization of high frequencies). The parameter " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\lambda" }, { "__type": "Text", "__tag": 4046, "value": " tunes the balance between data (which tends to increase high frequencies, even those coming from noise) and regularization/prior (which tends to avoid noise artifacts)." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "These methods are then specific to a prior model. Consequently, the application or the true image nature must correspond to the prior model. By default, the prior model (Laplacian) introduces image smoothness or pixel correlation. 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[1] François Orieux, Jean-François Giovannelli, and Thomas", " Rodet, \"Bayesian estimation of regularization and point", " spread function parameters for Wiener–Hunt deconvolution\",", " J. Opt. Soc. Am. A 27, 1593–1607 (2010)", " https://www.osapublishing.org/josaa/abstract.cfm?URI=josaa-27-7-1593", " https://hal.archives-ouvertes.fr/hal-00674508", "", ".. [2] B. R. Hunt \"A matrix theory proof of the discrete", " convolution theorem\", IEEE Trans. on Audio and", " Electroacoustics, vol. au-19, no. 4, pp. 285–288, dec. 1971" ], "qa": "skimage.restoration.deconvolution:wiener", "arbitrary": [], "local_refs": [ "balance", "clip", "im_deconv", "image", "is_real", "psf", "reg" ] }