Histogram Processing

Transcription

1 Hitogam Poceing Lectue 4 (Chapte 3) Hitogam Poceing The hitogam of a digital image with gay level fom to L- i a dicete function h( )=n, whee: i the th gay level n i the numbe of pixel in the image with that gay level =,, 2,, L- Nomalized hitogam: P( )=n / (M N) M N i the total numbe of pixel in the image um of all component = In hitogam opeation we will be woing motly with nomalized hitogam.

2 Hitogam Poceing The hape of the hitogam of an image doe povide ueful info about the poibility fo contat enhancement. Type of poceing: Hitogam equalization Hitogam matching (pecification) JERS- Why ae hitogam impotant? 2

3 Some baic tool: Cumulative ditibution function (CDF): t= i= ( ) ( ) Fx = P x = P t dt P i t= i= P Fx Pobability denity function (PDF): d fx Fx P d = P d An example Coin toing: H (-), T () F x < =.5 < Fx fx

4 Pixel a andom vaiable Conide each pixel a a andom vaiable The pobability of picing a pixel with a gay level i: n ( ) NM P The nomalized hitogam povide the ditibution of the pobabilitie of picing a given gay level What would be the pobability of picing a pixel with a gay level that i within the ange? i= = i= ( ) P ni MN The poblem domain Input image ( β ) Pf Tanfomation T Output image Pg h? h h? h 4

5 The diect poblem Given: P f T ( β ) - the hitogam of the input image - a given pixel tanfomation whee: T i ingle valued and monotonically inceaing ( ) T fo What would be the hitogam P g () of the output image? The diect poblem: Input image Output image Pf ( β ) T Pg h? h 5

6 Finding P g () g dt P P T ( T ) ( ) f = f ( ) = d dt x P dx x= T Finding P g (): an example Given T(x)=x+, find P g (): P g f ( ) ( ) x= T = ( ) P T Pf = = = Pf dt x dx ( ) h h 6

7 The invee poblem Given P f (β) and P g (), what hould be T o that P f (β) will be tanfomed to P g ()? Pf ( β ) T Pg h? h hitogam matching (pecification) Hitogam matching A both ide ae a function of T - () and ince thi function i ingle valued, we could apply T to both ide: ( ) T T ( T ) Pf ( T ) P ( ) ( ) f T ( T ) T Pf Pg = Pg ( T ) = = dt ( x) dt ( x) dt dx dx d x= T x= T ( T ) dt Pg T = d ( ) Pf 7

8 Hitogam matching Integating both ide: ( ) T dt x P T x dx = P T x dt x = P y dy = F T ( ( )) ( ( )) ( ) ( ) y= T ( x) g g g g dx f ( ) = P x dx F f ( ) Hitogam matching (cont.) We theefoe get: g ( ) = f ( ) = F T F ( ) ( ) = g ( f ) F F T F F g g g f ( ) T F F 8

9 An example: the continuou cae ( f ( )) F F x g 2 2x x x x Pg ( x) = Fg ( x) = othewie ( ) othewie 2 2 x x 2x x x Pf ( x) = Ff ( x) = othewie othewie 2 x x x x Fg ( x) = F ( x) = g othewie othewie 2 2x x x = othewie Hitogam matching (pecification): (The dicete cae) The pocedue fo hitogam-pecification baed enhancement i: Find the CDF of the oiginal image uing: = T ( ) = j= n j n n: total numbe of pixel, nj: numbe of pixel with gay level j, L: numbe of dicete gay level 9

10 Hitogam (matching) pecification (The dicete cae) Specify the deied denity function and obtain the tanfomation function G(z): z v = G( z) = P ( w) z i= z ni n p : the deiable PDF fo the output image Apply the invee tanfomation function z=g - () to the level obtained in tep. Hitogam (matching) pecification: (The dicete cae) The new, poceed veion of the oiginal image conit of gay level chaacteized by the pecified denity p z (z). In eence: z = G ( ) z = G [ T ( )]

11 Hitogam matching (pecification): (The dicete cae) T = Fg ( Ff ) Fg Pf ( t ) t= The oiginal CDF Ff Fg The equied CDF T P t dt P t ( ) = f ( ) f ( ) t= F g Finding F g - : the dicete cae Fo each input gay level in the CDF of f Find the element z in the CDF of g fo which (CDF(f)-CDF(g))=min Find the gay level that coepond to CDF(g z ) De-nomalize [ ] Add the pai [ ] to the LUT End

12 An example (uing non-nomalized hitogam) ( ) ( ) _ 4 h f h g gay level LUT in out An example: x f x g

13 An example (cont.) The invee CDF A woed example (cont.) out in

14 An example (cont.) The Taget image The tanfomed image An example (cont.) 2.5 x 4 x The Taget hitogam The tanfomed hitogam 4

15 Hitogam Equalization Given P f (β), what hould be T o that P g () would be unifom? Pf ( β ) T P g = L h? h Hitogam equalization (cont.) ( ) ( ) P T dt x P L P T L P x f g = = = f = f L dt x dx x= T dx x= T ( ) ( ) ( ) A both ide ae a function of T - () and ince thi function i ingle valued, we could ue intead of T - (): dt d integate ( ) ( ) = L P T = L P t dt L P t f f f t= 5

16 Hitogam Equalization (the dicete cae) n T( ) P ( ) j = = = j j= n j= Hitogam equalization(he) eult ae imila to contat tetching but offe the advantage of full automation, ince HE automatically detemine a tanfomation function to poduce a new image with a unifom hitogam. Hitogam equalization: an example of the dicete cae IN n_i Pi Sigma_Pi New_Gay OUT Image Hitogam Nommalized Hitogam Numbe of Pixel n_i Sigma_Pi Gay Level Numbe of Pixel Gay Level 6

17 Hitogam equalization A Compaion Between Hitogam Equalization and Hitogam Matching 7

18 A Compaion Between Hitogam Equalization and Hitogam Matching (cont.) A tanfomation function fo Hitogam Equalization 8

19 Hitogam matching (pecification): The pimay difficulty in applying the hitogam pecification method to image enhancement lie in being able to contuct a meaningful hitogam. So Eithe a paticula pobability denity function (uch a a Gauian denity) i pecified and then a hitogam i fomed by digitizing the given function, O a hitogam hape i pecified on a gaphic device and then i fed into the poceo executing the hitogam pecification algoithm. Hitogam matching (pecification) v. Hitogam equalization Hitogam equalization doe not allow inteactive image enhancement and geneate only one eult: an appoximation to a unifom hitogam. Sometime we need to be able to pecify paticula hitogam hape capable of highlighting cetain gay-level ange. 9