Basic Gray Level Transformations (2) Negative

Transcription

1 Gonzalez & Woods, 22 Basic Gay Level Tansfomations (2) Negative 23 Basic Gay Level Tansfomations (3) Log Tansfomation (Example fo Fouie Tansfom) Fouie spectum values ~1 6 bightest pixels dominant display values<1 Gonzalez & Woods,

2 Basic Gay Level Tansfomations (4) Powe-Law Tansfomation (1) Gonzalez & Woods, Basic Gay Level Tansfomations (5) Powe-Law Tansfomation (2) Gonzalez & Woods,

3 Gonzalez & Woods, 22 Basic Gay Level Tansfomations (6) Contast Stetching 1 = s1, 2 = s2 linea 1 = 2, s1 =, s2 = L 1 thesholding mean (d) (c) ( 1, s1) = ( min,) and ( 2, s2) = ( max, L 1) linea stetch to [, L 1] & max the minimum & maximum GL min 27 poo illumination, lack of dynamic ange in senso low contast image solution: incease dynamic ange by piecewise linea tansfomation that cab be abitaily complex, howeve elies on the use Image Enhancement in the Spatial Domain Histogam Equalization (HE) (1) HE yields output image with equally many pixels at evey gay level (flat histogam) which is useful befoe compaison/segmentation gay level in input image ( nomalized s.t [,1] ) s gay level tansfomed by s = T () (,s= is black &,s=1 is white) 28 3

4 Assume T() satisfies: HE (2) a. T() is single-valued & monotonically inceasing in 1 b. T The invese tansfom = T 1 (), s s 1 ( ) 1 fo 1 (monotonicity) exists (a) and the inceasing ode fom black to white (a) and ange (b) ae peseved 29 HE (3) Viewed as andom vaiables in [,1], the gay levels ae chaacteized by the (diffeent) pobability density functions (PDFs) p ( ) and p ( s) s dak image light image 3 4

5 31 HE (4) Following pobability theoy, if p ( ) and ps ( s) ae known and T 1 ( s) satisfies condition (a) then the PDF of the tansfomed gay levels is given by p d () s = p () ds (1) s The following enhancement techniques ae based on modifying the appeaance of an image by contolling the PDF of its gay levels via the tansfomation function T(). HE (5) Conside the (impotant in IP) tansfomation (2) p s = T ( ) = ( w) dw, 1 which is the cumulative distibution function (CDF) of. Since (1) the pdf is always positive, integal=aea and T is single valued condition (a); (2) integal of a pdf in [,1] is also in [,1] (b), that is, the two conditions above ae satisfied (check fo youself) 32 5

6 HE (6) Solving (1) fo the tansfomation in (2) yields d 1 () s = () = () ds ds d p p p s 1 1 = p ( ) = p ( ) dt () d d ( w) dw d p 1 = ( ) = 1 s 1 p p () That is, the CDF tansfomation yields a andom vaiable, s, having a unifom pobability density (thus inceasing the gay level dynamic ange). Notice, p () is always unifom independent of the fom of p s () s. 33 p HE (7) Example (Continuous Case) ( ) = othewise s = T ( ) = ( 2w + 2) dw 2 = + 2 = T 1 () s = 1± 1 s = 1 1 s [,1] 34 p d d () s = p () = (2 + 2) = s ds ds d 2(1 1 s) + 2 (1 1 s) = ds d 2( 1 s) (1 1 s) = 1 s 1 ds 6

7 35 HE (8) Fo digital images density pobability, The pobability of occuence of gay level k is n ( ) k p k =, 1, k =, L 1 fo L levels k n n is the total numbe of pixels in the image and n k is the numbe of pixels having gay level k The discete vesion of the CDF tansfomation k k n j s = T ( ) = p( ) =, k=, L 1 n k k j j= j= This tansfomation is called histogam equalization (show that T satisfies both conditions) HE (9) Unlike fo the continuous tansfomation, it cannot be poved that this discete tansfomation will poduce the discete equivalent of a unifom pdf, which would be a unifom histogam. Howeve, it does have the tendency of speading the histogam so that the levels of the histogam-equalized image will span a fulle ange of the gay scale 36 7

8 HE (1) Advantages: Gay level span the entie ange Automatic (based on image; no paamete selection) Simple to calculate 37 Gonzalez & Wintz, 1977 HE (11) Example (Discete Case) (1) 64x64, 8-level image having the following distibution (Fig. a) j1= s = T ( ) = p ( j ) = p ( ) =. 19 s = T( ) = p ( ) = 1 1 j= p( ) + p( 1) =.44 s2 =.65, s3 =.81, s4 =.89, s =.95, s =.98, s = j 38 8

9 HE (12) Example (Discete Case) (2) Only eight equally-spaced levels ae allowed, thus each of the tansfomed values must be assigned to its closest valid level s 1/ 7, s1 3/ 7, s2 5 / 7, s3 6 / 7, s 6 / 7, s 1, s 1, s (Fig. b) Thee ae only 5 distinct histogam-equalized gay levels, thus edefinition yields the levels s = / 7, s = 3/ 7, s = 5 / 7, s = 6 / 7, s = 79 s = 1/ 7, 123 s1 = 3/ 7, 85 s2 = 5 / 7, = 985 s3 = 6 / 7, = 448 s = 1 (Fig. c) 4 39 HE (13) Example (Discete Case) (3) 4 9

10 Castleman, 1996 HE (1) Illustation 1 output image with equally many pixels at evey gay level 41 Gonzalez & Wintz, 1977 HE (11) Illustation

11 Gonzalez & Woods, 22 HE (12) Illustation