Point Processing of Images. Point Processing of Images. EECE/CS 253 Image Processing. Point Processing

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EECE/CS 253 me Processin Lecture Notes: Lecture Notes: The Point Processin of mes Richrd Aln Peters Deprtment of Electricl Enineerin nd

its vlue lone; it it does not depend on the vlues of the pixel s neihbors This wor is licensed under the Cretive Commons

Cretive Commons, 543 Howrd Street, 5th Floor, Sn Frncisco, Cliforni, 94105, USA 2 Point Processin of mes Point Processin Brihtness nd

1 EECE/CS 253 me Processin Lecture Notes: Lecture Notes: The Point Processin of mes Richrd Aln Peters Deprtment of Electricl Enineerin nd Computer Science Fll Semester 2007 Point Processin of mes n diitl ime, point = pixel Point processin trnsforms pixel s vlue s function of its vlue lone; it it does not depend on the vlues of the pixel s neihbors This wor is licensed under the Cretive Commons Attribution-Noncommercil 25 License To view copy of this license, visit or send letter to Cretive Commons, 543 Howrd Street, 5th Floor, Sn Frncisco, Cliforni, 94105, USA 2 Point Processin of mes Point Processin Brihtness nd contrst djustment Gmm correction Historm equliztion Historm mtchin Color correction - mm - brihtness oriinl + brihtness + mm historm mod - contrst oriinl + contrst historm EQ 3 4

The Historm of Gryscle me Let be 1-bnd (ryscle) ime (r, is is n 8-bit

for for = 1, 1, 2, 2, 3, 3,,, 256, is is n n inteer h () () = number of

(4-bit) ime lower RHC: number of pixels with intensity blc mrs pixels

intensity The Historm of Gryscle me Blc mrs pixels with intensity Plot

2 The Historm of Gryscle me Let be 1-bnd (ryscle) ime (r, is is n 8-bit inteer between 0 nd 255 Historm, h,, of of : : 256-element rry, h h (), for for = 1, 1, 2, 2, 3, 3,,, 256, is is n n inteer h () () = number of of pixels in in tht hve vlue -1 The Historm of Gryscle me 16-level (4-bit) ime lower RHC: number of pixels with intensity blc mrs pixels with intensity 5 6 The Historm of Gryscle me Blc mrs pixels with intensity The Historm of Gryscle me Blc mrs pixels with intensity Plot of historm: number of pixels with intensity Plot of historm: number of pixels with intensity 7 8

The Historm of Gryscle me ( +1) h = the number Luminosity of pixels in

24-bit) then then (r,c,b) is is n n inteer between 0 nd nd 255 255

(:,:,1) (:,:,1) with with intensity vlue vlue h G (+1) G (+1) = # of

B (+1) B (+1) = # of of pixels pixels in in (:,:,3) (:,:,3) with with

h(+1,1,1) = # of of pixels pixels in in with with red redintensity

reenintensity vlue vlue h(+1,1,3) = # of of pixels pixels in in with

is is one one historm per per color bnd R, R, G, G, & B B Luminosity

3 The Historm of Gryscle me ( +1) h = the number Luminosity of pixels in with rylevel The Historm of Color me f f is is 3-bnd ime (truecolor, 24-bit) then then (r,c,b) is is n n inteer between 0 nd nd Either hs hs 3 historms: h R (+1) R (+1) = # of of pixels pixels in in (:,:,1) (:,:,1) with with intensity vlue vlue h G (+1) G (+1) = # of of pixels pixels in in (:,:,2) (:,:,2) with with intensity vlue vlue h B (+1) B (+1) = # of of pixels pixels in in (:,:,3) (:,:,3) with with intensity vlue vlue or or 1 vector-vlued historm, h(,1,b) where h(+1,1,1) = # of of pixels pixels in in with with red redintensity vlue vlue h(+1,1,2) = # of of pixels pixels in in with with reen reenintensity vlue vlue h(+1,1,3) = # of of pixels pixels in in with with blue blueintensity vlue vlue 9 10 The Historm of Color me There is is one one historm per per color bnd R, R, G, G, & B B Luminosity historm is is from 1 bnd = (R+G+B)/3 Luminosity h L The Historm of Color me h R h G h B 11 12

Vlue or Luminnce Historms Vlue Historm vs Avere of R,G,&B Historms The vlue historm of 3-bnd (truecolor) ime,, is the historm of the vlue ime, 1 V, 3 = [ R + G + B( r ] Where R, G, nd B re the red,

Mtlb % Multi-bnd historm clcultor function h=historm() [R C B]=size(); % llocte the historm h=zeros(256,1,b); % rne throuh the intensity vlues for =0:255 h(+1,1,:) = sum(sum(==)); % ccumulte end

elements tht tht hve hve vlue vlue The The result result is is n n RxCxB RxCxBloicl loiclrry rry tht tht [R C B]=size(); hs hs 11 wherever wherever (r,c,b) (r,c,b) = nd nd 0 s 0 s everywhere

tht tht vlue vlue in in the the 256x1xB 256x1xB historm historm % rne throuh the intensity t t h(+1,1,b) h(+1,1,b) vlues for =0:255 h(+1,1,:) = sum(sum(==)); % ccumulte end sum(sum(==)) sum(sum(==))

4 Vlue or Luminnce Historms Vlue Historm vs Avere of R,G,&B Historms The vlue historm of 3-bnd (truecolor) ime,, is the historm of the vlue ime, 1 V, 3 = [ R + G + B( r ] Where R, G, nd B re the red, reen, nd blue bnds of The luminnce historm of is the historm of the luminnce ime, = 0299 R G B( r L, R,G,B,&V historms V & v v of of R,G,&B historms Multi-Bnd Historm Clcultor in Mtlb % Multi-bnd historm clcultor function h=historm() [R C B]=size(); % llocte the historm h=zeros(256,1,b); % rne throuh the intensity vlues for =0:255 h(+1,1,:) = sum(sum(==)); % ccumulte end return; Multi-Bnd Historm Clcultor in Mtlb % Multi-bnd historm clcultor function h=historm() Loop Loop throuh throuh ll ll intensity intensity levels levels (0-255) (0-255) T T the the elements elements tht tht hve hve vlue vlue The The result result is is n n RxCxB RxCxBloicl loiclrry rry tht tht [R C B]=size(); hs hs 11 wherever wherever (r,c,b) (r,c,b) = nd nd 0 s 0 s everywhere everywhere else else % llocte the historm Compute Compute the the number number of of ones ones in in ech ech bnd bnd of of h=zeros(256,1,b); the the ime ime for for intensity intensity Store Store tht tht vlue vlue in in the the 256x1xB 256x1xB historm historm % rne throuh the intensity t t h(+1,1,b) h(+1,1,b) vlues for =0:255 h(+1,1,:) = sum(sum(==)); % ccumulte end sum(sum(==)) sum(sum(==)) computes computes one one number number f f B==3, B==3, then then h(+1,1,:) h(+1,1,:) contins contins 3 return; for for ech ech bnd bnd in in the the ime ime numbers: numbers: the the number number of of pixels pixels in in bnds bnds 1, 1, 2, 2, & 3 tht tht hve hve intensity intensity 15 16

5 Point Ops vi Functionl Mppins me: Pixel: nput (r, Φ, Φ, point point opertor opertor function, function, ff Output (r, = Φ[ ] f nd then = f ( ) = = The trnsformtion of ime into ime is ccomplished by replcin ech input intensity,, with specific output intensity,, t every loction (r, where (r, = The rule tht ssocites with is usully specified with function, f, so tht f () = Point Ops vi Functionl Mppins One-bnd me = f ( ), for ll pixels loctions Three-bnd me c, b ) = f c, b ), or ( ) c, b) = f ( c, b) ), b for b = 1,2,3 nd ll Point Ops vi Functionl Mppins Point Opertions usin Loo-up Tbles One-bnd me Either Either ll ll 3 bnds bnds re re mpped mpped throuh throuh the the sme sme function, function, f, f, or or = f ( ), for ll pixels loctions Three-bnd me c, b ) = f c, b ), or ( ) c, b) = f ( c, b) ), b for b = 1,2,3 nd ll ech ech bnd bnd is is mpped mpped throuh throuh seprte seprte functiontion, ff b func- b A loo-up tble (LUT) implements functionl mppin f = f ( ) for = 0, K,255, ndif tes on vlues in, { 0, K,255}, K To remp 1-bnd ime,, to : then the LUT tht implements f is 256x1 rry whose ( +1) th vlue is = f () = LUT ( +1) 19 20

then ) ) ech bnd is is mpped seprtely usin the

bnd is is mpped usin different LUTs one for

(:,:, b) + 1) for b = 1,2,3 ) or b) b output

106 input vlue 64 68 69 70 70 71 output 21 22

exmple: Let = 2 Let x σ { 0, K,255} 255 1+ e (

with this thisvlue cell index 0 0 64 32 128

mpped to to this thisvlue Or in Mtlb: = 2; x =

6 Point Opertions usin Loo-up Tbles Point Opertions = Loo-up Tble Ops f f is is 3-bnd, then ) ) ech bnd is is mpped seprtely usin the the sme LUT for for ech bnd or or b) b) ech bnd is is mpped usin different LUTs one for for ech bnd = LUT( + 1), (:,:, b) = LUT ( (:,:, b) + 1) for b = 1,2,3 ) or b) b output vlue input vlue E: index input vlue output Loo-Up Tbles How to Generte Loo-Up Tble For exmple: Let = 2 Let x σ { 0, K,255} e ( x; ) = ( x 127) / 32 input pixel pixel with with this thisvlue cell index contents output is is mpped to to this thisvlue Or in Mtlb: = 2; x = 0:255; LUT = 255 / (1+exp(-*(x-127)/32)); This is is just one exmple 23 24

Point Processes: ncrese Contrst Point Processes: Decrese Contrst Let T = [ 127] + 127, where > 1 0 0, if T < 0, =

7 Point Processes: ncrese Brihtness Point Processes: Decrese Brihtness, if + < 256, if + > 255 { 1, 2 } is the bndindex + = nd, 3 trnsform mppin 0 nd, 3 if 0, if { 1, 2 } is the bndindex 0, < = trnsform mppin Point Processes: ncrese Contrst Point Processes: Decrese Contrst Let T = [ 127] + 127, where > 1 0 0, if T < 0, = T, if 0 T 255, 255, if T > 255 { 1, 2, 3} trnsform mppin T = [ 127] where 0 < , nd { 1,2,3 } trnsform mppin 27 28

Point Processes: Contrst Stretch historms

Let m = min Then, [ ], M = mx[ ], [ ], M =

m 0 m 127 M 255 trnsform mppin lo-c hi-c 29

Point Processes: ncresed Gmm ori ori ori lo-c

hih-contrst restored restored difference

between oriinl nd restored oriinl nd restored

difference hih-contrst hih-contrst 1/ γ (, )

8 Point Processes: Contrst Stretch historms nformtion Loss from Contrst Adjustment ori Let m = min Then, [ ], M = mx[ ], [ ], M = mx[ ] ( ) ( ) r, c = M m m = min m + m M m M m 0 m 127 M 255 trnsform mppin lo-c hi-c nformtion Loss from Contrst Adjustment Point Processes: ncresed Gmm ori ori ori lo-c lo-c rest hi-c rest hi-c lo-c diff hi-c diff bbrevitions: bbrevitions: oriinl oriinl low-contrst low-contrst hih-contrst hih-contrst restored restored difference difference difference between difference between oriinl nd restored oriinl nd restored low-contrst low-contrst difference between difference between oriinl nd restored oriinl nd restored hih-contrst hih-contrst 1/ γ (, ) 255 r c = for γ > trnsform mppin 31 32

Point Processes: Decresed Gmm Gmm Correction: Effect on Historm 1/ γ (, ) 255 r c = for γ < 10 255 0 m 127 M 255 trnsform mppin 33

ime of pixels in is the number Tht is if R rows by C columns then A = R C of pixels in ) with vlue, is This is is the probbility

Probbility Density Function of n me p bnd bnd (+1) is is the the frction of of pixels in in ( ( specific bnd of) of) n n ime tht

Wheres the the sum of of the the historm h bnd bnd (+1) over ll ll from 1 to to 256 is is equl to to the the number of of pixels

9 Point Processes: Decresed Gmm Gmm Correction: Effect on Historm 1/ γ (, ) 255 r c = for γ < m 127 M 255 trnsform mppin The Probbility Density Function of n me Let A = 255 = 0 ( + 1) h Note tht since h (the th color A is the number ( +1) bnd of ime of pixels in is the number Tht is if R rows by C columns then A = R C of pixels in ) with vlue, is This is is the probbility tht n n rbitrry pixel from hs vlue Then, 1 p ( + 1) = h ( 1) + A is the rylevel probbility density function of pdf [lower cse] The Probbility Density Function of n me p bnd bnd (+1) is is the the frction of of pixels in in ( ( specific bnd of) of) n n ime tht hve intensity vlue p bnd bnd (+1) is is the the probbility tht pixel rndomly selected from the the iven bnd hs intensity vlue Wheres the the sum of of the the historm h bnd bnd (+1) over ll ll from 1 to to 256 is is equl to to the the number of of pixels in in the the ime, the the sum of of p bnd bnd (+1) over ll ll is is 1 1 p bnd bnd is is the the normlized historm of of the the bnd 35 36

10 The Probbility Distribution Function of n me The Probbility Distribution Function of n me Let q = [q 1 q 2 q 3 ] = (r, be the vlue of rndomly selected pixel from Let be specific rylevel The probbility tht q is iven by PDF [upper cse] Let q = [q 1 q 2 q 3 ] = (r, be the vlue of rndomly selected pixel from Let be specific rylevel The probbility tht q is iven by Also Also clled clled CDF CDF for for Cumultive Distribution Function P ( + ) = p ( γ + 1) = h ( γ + 1) ( γ + 1) 1 γ = 0 1 = 255 γ= 0 A γ= 0 h γ + γ = 0 h ( 1), P ( + ) = p ( γ + 1) = h ( γ + 1) ( γ + 1) 1 γ = 0 1 = 255 γ= 0 A γ= 0 h γ + γ = 0 h ( 1), where h (γ +1) is the historm of the th bnd of This is is the probbility tht ny iven pixel from hs vlue less thn or or equl to to where h (γ +1) is the historm of the th bnd of This is is the probbility tht ny iven pixel from hs vlue less thn or or equl to to A A Cumultive Distribution Function The Probbility Distribution Function of n me P bnd bnd (+1) is is the the frction of of pixels in in ( ( specific bnd of) of) n n ime tht hve intensity vlues less thn or or equl to to P bnd bnd (+1) is is the the probbility tht pixel rndomly selected from the the iven bnd hs n n intensity vlue less thn or or equl to to P bnd bnd (+1) is is the the cumultive (or (or runnin) sum of of p bnd bnd (+1) from 0 throuh inclusive P bnd bnd (1) (1) = p bnd bnd (1) (1) nd P bnd bnd (256) = 1; 1; P bnd bnd (+1) is is nondecresin Note: the Probbility Distribution Function (PDF, cpitl letters) nd the Cumultive Distribution Function (CDF) re exctly the sme thins Both PDF nd CDF will refer to it However, pdf (smll letters) is the density function Point Processes: Historm Equliztion Ts: remp ime so tht its historm is s close to constnt s possible ( γ ) Let +1 P be the cumultive (probbility) distribution function of Then hs, s closely s possible, the correct historm if, [ + 1] ( r = 255 P The CDF itself is used s the LUT ll ll bnds processed similrly 39 40

Point Processes: Historm Equliztion Luminosity Historm EQ pdf

for for remppin before CDF = 255 P ( + 1) fter 41 42 Historm EQ

11 Point Processes: Historm Equliztion Luminosity Historm EQ pdf The The CDF CDF (cummultive distribution) is is the the LUT LUT for for remppin before CDF = 255 P ( + 1) fter Historm EQ pdf The The CDF CDF (cummultive distribution) is is the the LUT LUT for for remppin Historm EQ pdf The The CDF CDF (cummultive distribution) is is the the LUT LUT for for remppin LUT LUT 43 44

12 Historm EQ Point Processes: Historm Equliztion Ts: remp ime with min = m nd mx = M so tht its historm is s close to constnt s possible nd hs min = m nd mx = M ( γ ) Let P +1 be the cumultive (probbility) distribution function of Then hs, s closely s possible, the correct historm if Usin intensity extrem ( r = ( M m ) [ + 1] P ( m + 1) 1 P ( m + 1) P + m, Point Processes: Historm Mtchin Ts: remp ime so tht it hs, s closely s possible, the sme historm s ime Becuse the imes re diitl it is not, in enerl, possible to me h h Therefore, p p Q: How, then, cn the mtchin be done? A: By mtchin percentiles Mtchin Percentiles ssumin 1-bnd 1-bnd ime ime or or sinle sinle bnd bnd of of color color ime ime Recll: The CDF of ime is such tht 0 P ( ) 1 P ( +1) = c mens tht c is the frction of pixels in tht hve vlue less thn or equl to 100c is the percentile of pixels in tht re less thn or equl to To To mtch percentiles, replce ll ll occurrences of of vlue in in ime with the the vlue,,, from ime whose percentile in in most closely mtches the the percentile of of in in ime 47 48

Mtchin Percentiles ssumin ssumin 1-bnd 1-bnd ime ime or or

Mtchin Alorithm ssumin ssumin 1-bnd 1-bnd ime ime or or

n ime, K, from ime such tht K hs nerly the sme CDF s ime

P ( ) > P ( -1) AND P ( ) P ( ) P ( ) P ( ) Exmple:

056 P (10) (10) = 067 067 K(r, K(r, = 10 10 [R,C] =

P ( + 1) < 1 AND P ( + 1) < P ( + 1) +1; = end K = K + = =

+ ) P 1 : CDF of P 1 : CDF of m = min, M = mx, m = min, M

Exmple: Historm Mtchin Exmple: Historm Mtchin me pdf me

13 Mtchin Percentiles ssumin ssumin 1-bnd 1-bnd ime ime or or sinle sinle bnd bnd of of color color ime ime Historm Mtchin Alorithm ssumin ssumin 1-bnd 1-bnd ime ime or or sinle sinle bnd bnd of of color color ime ime So, to crete n ime, K, from ime such tht K hs nerly the sme CDF s ime do the followin: f (r, = then let K(r, = where is such tht P ( ) > P ( -1) AND P ( ) P ( ) P ( ) P ( ) Exmple: Exmple: (r, (r, = 55 P (5) (5) = P (9) (9) = P (10) (10) = K(r, K(r, = [R,C] = size(); K = zeros(r,c); = m ; for = m to M while < 255 AND P ( + 1) < 1 AND P ( + 1) < P ( + 1) +1; = end K = K + = = end [ ( )] This This directly mtches ime to to ime ( + ) ( + ) P 1 : CDF of P 1 : CDF of m = min, M = mx, m = min, M = mx Better to to use use LUT See See slide Exmple: Historm Mtchin Exmple: Historm Mtchin me pdf me with with intensity vlues me CDF CDF () * * Cumultive Distribution Function, CDF 51 52

Exmple: Historm Mtchin Exmple: Historm Mtchin Tret pdf Tret with with 16 16 intensity vlues Tret CDF CDF () * * Cumultive Distribution Function, CDF 53 54 LUT

thn remppin ech pixel in the ime seprtely, one cn crete tble tht indictes to which tret vlue ech input vlue should be mpped Then K = LUT[+1] n Mtlb if the LUT is

14 Exmple: Historm Mtchin Exmple: Historm Mtchin Tret pdf Tret with with intensity vlues Tret CDF CDF () * * Cumultive Distribution Function, CDF LUT Cretion Historm Mtchin with Looup Tble The lorithm on slide 49 mtches one ime to nother directly Often it is fster or more verstile to use looup tble (LUT) Rther thn remppin ech pixel in the ime seprtely, one cn crete tble tht indictes to which tret vlue ech input vlue should be mpped Then K = LUT[+1] n Mtlb if the LUT is mtrix with vlues from 0 to 255 nd if ime is of type uint8, it cn be rempped with the followin code: K = uint8(lut(double()+1)); me CDF 10 LUT Tret CDF 55 56

15 Loo Up Tble for Historm Mtchin nput & Tret CDFs, LUT nd Resultnt CDF LUT = zeros(256,1); = 0; for = 0 to 255 while P ( ) < P ( + 1) +1; end LUT end = ( + ) = ; AND < 255 This This cretes loo-up tble which cn cn then be be used to to remp the the ime ( + ) ( + ) ( + ) P 1 : CDF of, P 1 : CDF of, LUT 1 : Loo- Up Tble P ( ) LUT ( ) nput nput LUT LUT P ( ) P K ( ) Tret Tret Result Result Exmple: Historm Mtchin Probbility Density Functions of Color me oriinl tret rempped Atls-Mercury Atls-Mercury red pdf reen pdf blue pdf luminosity pdf 59 60

luminosity pdf 61 62 Cumultive Distribution Functions (CDF) Remp n me to hve the Lum CDF

16 Cumultive Distribution Functions (CDF) Probbility Density Functions of Color me red CDF reen CDF blue CDF luminosity CDF TechnoTrousers TechnoTrousers red pdf reen pdf blue pdf luminosity pdf Cumultive Distribution Functions (CDF) Remp n me to hve the Lum CDF of Another red CDF reen CDF blue CDF luminosity CDF oriinl tret luminosity rempped 63 64

CDFs nd the LUT Effects of Luminnce Remppin on pdfs Atls-Mercury Luminosity CDF TechnoTrousers Luminosity CDF LUT (Luminosity) Atls-Mercury to TechnoTrousers

17 CDFs nd the LUT Effects of Luminnce Remppin on pdfs Atls-Mercury Luminosity CDF TechnoTrousers Luminosity CDF LUT (Luminosity) Atls-Mercury to TechnoTrousers Atls-Mercury Rempped Luminosity CDF Before After Effects of Luminnce Remppin on CDFs Remp n me to hve the rb CDF of Another Before After oriinl tret R, G, & B rempped 67 68

CDFs nd the LUTs Effects of RGB Remppin on pdfs Atls-Mercury Red PDF Atls-Mercury Green PDF

LUT (Red) Atls-Mercury to TechnoTrousers LUT (Green) Atls-Mercury to TechnoTrousers LUT (Blue)

PDF Atls-Mercury RGB Rempped Blue PDF Before After 69 70 Effects of RGB Remppin on CDFs Remp n

18 CDFs nd the LUTs Effects of RGB Remppin on pdfs Atls-Mercury Red PDF Atls-Mercury Green PDF Atls-Mercury Blue PDF TechnoTrousers Red PDF TechnoTrousers Green PDF TechnoTrousers Blue PDF LUT (Red) Atls-Mercury to TechnoTrousers LUT (Green) Atls-Mercury to TechnoTrousers LUT (Blue) Atls-Mercury to TechnoTrousers Atls-Mercury RGB Rempped Red PDF Atls-Mercury RGB Rempped Green PDF Atls-Mercury RGB Rempped Blue PDF Before After Effects of RGB Remppin on CDFs Remp n me: To Hve Two of its Color pdfs Mtch the Third Before After oriinl G & B R B & R G R & G B 71 72

7 - Continuous random variables

7 - Continuous random variables 7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin