Gray-level histogram. Intensity (grey-level) transformation, or mapping. Use of intensity transformations:

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1 Faculty of Informatic Eötvö Loránd Univerity Budapet, Hungary Lecture : Intenity Tranformation Image enhancement by point proceing Spatial domain and frequency domain method Baic Algorithm for Digital Image Analyi: a coure Dmitrij Cetverikov with help of Attila Lerch, Judit Veretóy, Zoltán Megyei, Zolt Jankó and Levente Hajder Neighborhood operation and intenity tranformation Gray-level hitogram Some ueful intenity tranformation: Contrat tretching and intenity normaliation Image negation Nonlinear compreion of dynamic range Intenity licing Hitogram equaliation Spatial domain and frequency domain method Neighborhood operation Goal of image enhancement: Image i compoed of informative pattern modified by non-informative variation. Enhance informative pattern baed on image data. Example: noie filtering, geometric correction. Two type of image enhancement method Spatial domain method: Procedure that operate directly on image pixel. Frequency domain method : Procedure that operate in a tranformed domain. Example: power pectrum (Fourier), wavelet, Gabor tranformation Thi coure mainly deal with patial domain method. 3 Output value in image point (x, y) i determined by pixel belonging to a neighborhood of (x, y): g(x, y) = T [f(x, y)] f(x, y): input image; g(x, y): proceed (output) image; T : operator on f, defined over ome neighborhood of (x, y). (x,y) Image y A 3 3 neighbourhood (window) about a point (x, y) in an image. 4 x

2 Intenity (grey-level) tranformation, or mapping Neighborhood i of 1 1 ize: reduce to point (x, y) itelf (point proceing). Output value depend only on intenity in (x, y). Ue of intenity tranformation: Contrat tretching: Increae contrat. For implicity of notation: r. = f(x, y),. = g(x, y). Intenity tranformation T map r onto : Intenity range normaliation: (uually, [0, 55]). Make intenitie fall into pecified range = T (r) Baic propertie of intenity tranformation: Same intenitie tranform in the ame way: poition independent. Even local context neglected: no tructure taken into account. Can only reduce noie when noie intenity i ditinct; otherwie, can even amplify noie. 5 Background removal: Remove irrelevant intenitie. Pattern enhancement: Enhance relevant intenitie. Normalie image to compare image to compare image decription 6 Gray-level hitogram Gray-level hitogram p(k) i occurrence probability (frequency) of grey level k in an image: where p(k) = n k n k = 0, 1,..., L 1: grey level (L i number of grey level); n k : number of pixel with grey level k; n: total number of pixel. Intenity hitogram i computed in one can of image f(x, y) Allocate array p(k), et all p(k) = 0. Scan pixel (x, y). When f(x, y) = k, increment p new (k) = p old (k) + 1. Hitogram provide global decription of appearance: No tructural information Take tructured image (hape or periodic pattern). Re-arrange pixel in random way. Obtain random image with ame hitogram. 7 8

3 Baic type of grey-level hitogram: Too dark: grey level concentrate at the dark end. Too light: grey level concentrate at the light end. Low contrat (narrow dynamic range): grey level concentrate in the middle. Good contrat: ignificant pread of hitogram. too dark too light low contrat good contrat Image and their hitogram Some ueful intenity tranformation Contrat tretching Contrat tretching Intenity normaliation Image negation Nonlinear compreion of dynamic range Intenity licing Hitogram equaliation Purpoe: Increae dynamic range of intenitie in low-contrat image. Why low contrat? poor illumination low dynamic of enor wrong etting of len aperture Baic propertie of contrat tretching: Contrat tretching tranformation function mut be ingle-valued and monotonically increaing Preerve order of grey level: no artefact The greater the lope the higher the contrat (pread) at that range. 11 1

4 = = Piecewie-linear contrat tretching Dark Light Dark m Light CONTRAST STRETCHING r Dark Light Dark m BINARISATION Light Gray-level tranformation function for contrat tretching and binariation. Binariation (threholding) i a pecial cae of contrat tretching. Effect: Obtain binary image by etting to zero all intenitie below a threhold and to maximum value all other intenitie. Meaning: Separate object from background auming ditinct intenity clae. 13 r Point (r 1, 1 ) and (r, ) control hape of tranformation function. Binariation i pecial cae of piecewie-linear contrat tretching: r 1 = r, 1 = 0, = L 1. 0 r (r, ) 1 1 (r, ) Piecewie-linear approximation of contrat tretching. 14 Intenity normaliation (recaling) Purpoe: Normalie range of intenitie to make their value fall into a tandard range. low contrat tretched Example of contrat enhancement. Reaon: Image proceing operation may produce value that are beyond the initial range of intenitie. It may be deirable to preerve the range and torage type (e.g., byte/pixel). Solution: Let r [r 1, r ] be the original intenity and = T (r) a tranformation that modifie the range [r 1, r ] to [ 1, ], where cell image binaried cell Example of binariation (threholding). 1 = min r [r1,r ]T (r) = max r [r1,r ]T (r) 15 16

5 Image negation The output intenity hould be normalied o a to fall in the original range: = T () [r 1, r ] Thi i achieved by the following linear tranformation: Action: Revere the order of intenitie. Application: Negate image of poitive film and ue the reulting negative a normal lide. T () = r r 1 1 [T () 1 ] + r 1 Before applying the tranformation, check that 1. Prove that T () [r 1, r ]! 0 tranformation image negative Obtaining the negative of an image Compreion of dynamic range Action: Compre dynamic range of image. Application: Dynamic of proceed image may exceed the capabilitie of diplay or film. 0 r R grey-level tranformation function Only brightet part viible, dark part uppreed. Typical for ome medical imagery (e.g., x-ray) and Fourier pectra. Solution: image tranformation where c i a caling contant. = c log (1 + r ) 19 Compreing the dynamic range: Stretching the dark intenitie of an image with dark intenitie uppreed. 0

6 Gray-level licing Action: Highlight a pecific range of intenitie and/or uppre other intenitie. Application: Background removal and egmentation. Example: Highlight edge pixel when their intenitie fall into a narrow range. 0 A r B 0 A r B Tranformation function for intenity licing. Left: Highlight range [A, B] while uppreing other level. Right: Same, but preerving other level. Two baic verion of licing: Highlight a range, diminih other level. Threholding i a particular cae of thi. Highlight a range, preerve other level. image uppre other preerve other Example of intenity licing. 1 Hitogram pecification and equaliation Hitogram equaliation: continuou cae Hitogram pecification: Tranform intenitie o a to obtain a pecified hape of hitogram of output image. Example: Human perceive bet the image that have hyperbolic hitogram. The mot important cae of hitogram pecification: Hitogram equaliation, or hitogram flattening. Action: The output image hitogram become a uniform a poible. Purpoe: Increae dynamic range of image. Normalie image hitogram prior to comparion of image image decription 3 Gray-level r are normalied continuou quantitie: r [0, 1]. Conider the tranformation function = T (r) = r 0 p r (w) dw 0 r 1 p r (w): probability denity function (PDF, ditribution) of original intenity r. 0 T (r) 1: cumulative ditribution function (CDF) of r. T (r): ingle-valued and monotonically increaing. For thi tranformation function, the output intenity ha uniform ditribution. (Prove!) 4

7 Hitogram equaliation: dicrete cae 1 p (r) r = 0 1 r r p () Illutration of hitogram equaliation (flattening). Left: Original probability denity function. Center: Tranformation function. Right: Reulting uniform denity. The tranformation i obtained by umming up the bin of the grey-level hitogram: p r (r k ) = n k n k = T (r k ) = 0 r k 1 and k = 0, 1,..., L 1 k j=0 n j n = k p r (r j ) j=0 r k = T 1 ( k ) 0 k image equliation tretching dark adaptive equali. Dynamic range increaed, detail better viible. Viual grainine and patchine increaed becaue of too few grey level. Noie amplified. Adaptive verion i better. 7