Histogram of a population

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1 Histogram of a poplatio Let X be a set of N data samples with qatized scalar vales i { 0, 1,, L 1 } (may be samples from a cotios fctio (t) :, ad X, for example, the height of N adlts, pixels with graylevels from a image =I(x,y), volme desities V(x,y,z), etc.). The vales, with =0,1,..., L 1, may be cosidered as the L possible otcomes of a discrete radom variable. We have time ad spatial freqecies, bt also a evet freqecy f : freqecy = cycles per sec, lies per cm, dots per ich icidece of evet (i a poplatio of size N ) amog L possible otcomes Htz, waves cm, dpi f N The histogram Hist(X) is a graphic plot of f vs. The histogram class itervals or bis are the discretizatio itervals Δ 1 = (L 1) / L, with >0. The th evet is the occrrece of vale, (whe Δ =1), or i geeral: [ 1 ). The histogram of a image, Hist(I), provides a global descriptio of the appearace of the image. A local histogram Hist(ROI), with ROI I describes a regio appearace). Hist(I) is the distribtio of gray level vales withi a image. A sample poplatio of N 20% N Total may well approximate the global distribtio if samples are iformly distribted i the domai poplatio. The relative freqecy distribtio of a poplatio is also its empirical probability distribtio. Ths, Hist(X) = { p( ), =0,1,, L 1 }, is a discrete fctio, give by (spose for example X as a gray level image):

2 The mber of pixels with graylevel The th gray level p( ) = N The total mber of pixels i a ROI X (1) Usally L = 256, ad N Total = pixels. Also Δ =1 ad i 8 bits, gray level images{ 0, 1,, 255 }={0,1,...,255}. The we may write p( ) as p. Let s cosider the gray level vales i a image the vales tae by idepedet ad idetically distribted radom variables (plral refers to each pixel). I this case, the histogram is a approximatio to the probability desity fctio (PDF) for each radom variable I otherwords, p ( ) Prob[ = ], with L 1 p( ) = 1 (2) The approximatio depeds o the mber of samples N N Total ad its distribtio. With iform samplig (or all data is cosidered, we have Σ =N = N Total ), the the PDF is the limit whe N, ad the size of the histogram classes or bis teds to zero: Δ 0. We the have: Prob[ ] = p( ) d 0 0 = 0 max (3) with p( ) d = 1 where max = L 1 as i the discrete versio i eq. (2). Some athors call the histogram a discrete pdf.

3 Potetial Notatioal Coflicts ad Covetios We have data (a sigal): (x) (where domai x may be t) We traditioally plot vs. x (depedet agaist idepedet variables: itesity at poit x) Histogram Hist() = p( ) (or more properly writte as the set { p( )}) is a plot of freqecy (#otcomes) p( ) agaist ; x does ot matter. I images, or other mltidimesioal domai, x > x = ( x, y,...), or x= x = ( x, x, x,...) ad (x) a scalar field whose histogram p( ) is still idimesioal. To simplify, at times the L vales { 0, 1,, L 1 } are directly tae (with Δ =1, for all ) as ==0,1,, L 1. Also idividal freqecies (probabilities) p( ) are oted as p (), or eve p.. Some properties of histograms The histogram loses ay domai iformatio. We do ot write Hist((x)) or p( (x)). As a collorary, very differet datasets ad with differet dimesio (sigal series, images, etc.) may have exactly the same histogram. There is o iverse trasform. Ay represetatio of freqecies of occrrece is called a histogram. A mltidimesioal histogram is a graphic represetatio of joit 2 probability distribtios, p(,v ), where f :, ad f(x)= (,v). See sectio o mltidimesioal histograms. Mode(s) of a histogram mode arg max { p( )} = L 1 = = 0 (maybe ot iqe) ad p max = p( mode ) is the highest icidece (most freqet vale mode ). I a image regio, a arrow histogram (the mai pea) idicates a low cotrast regio. As probabilities, the histogram freqecies p( ) are ormalized to [0,1], by dividig the mber of samples (= pixels) with attribte (itesity) by N (the total of samples, for all ). Some athors also ormalize the

4 vales of the set { 0, 1,, L 1 } to [0,1], by dividig by L 1 (sally L 1 =255 i 8 bit gray level images). Bis Histograms are defied o N bi discrete itervals [, +1 ), with =0,, N bi 1 called bis or classes, where 0 ad Nbi are the miimm ad spremm itesities of I(x,y) (sally 0 ad L=256). If we defie N bi 1 vale of the histogram at some vale, the Hist( ) is the relative freqecy of occrrece of itesity : Hist( ) = p( )= card{ pixels with vale } / (N x N y ). for a N x N y rectagle regio or image. Normalized Histogram Discrete, Empirical Probability Desity Fctio (pdf) Hist( ) of the pixel (or voxel) itesities { } =0 K 1 (K discrete itesity levels). Usally K=256 ad max { Hist( )} = card I(x,y). Note we cold also defie simply Hist() with =0,, 255, bt it may happe that we oly se some (for example the eve vales, or a logarithmic re samplig of [0,256) ). Sampled Histogram: That from a (properly) sampled poplatio related to Mote Carlo estimatio of PDFs. Cmlative Histogram From a discrete ROI of size N x N y : cpdf = 1/( N N ) card( ) = p( ) x y = 0 = 0 (4) (a..a. Cmlative Distribtio Fctio). As a cotios radom variable : cpdf ( ) = p( υ) d υ (5) 0 Histogram Eqalizatio: is the process of applyig the trasfer fctio: i = cpdf ( ) = p( υ) d υ ot i (6) Note that the PDF of the otpt levels (i.e., p( ot )) is iform, that is: 0

5 1 if 0 ot L 1 ( ) = ot 0 otherwise (7) p The trasformatio (6) geerates a image whose itesity levels are eqally liely (ths, itesity level eqalized) ad cover the etire rage [0, 255]. The dyamic rage is icreased, ad will ted to show higher cotrast. Histograms with MATLAB or other compter lagages MATLAB has a histogram fctio (histo); it is sefl to oew how to obtai oeself ay ROI histogram. This code comptes the histogram for a fll image (ay rectaglar ROI ca be easily specified at the mai loops): X = imread('bacteria.tif','tif'); % matrix cotais the image image(x); % display it hold o; % to compare with histogram H=zeros(256,1); % iitialize histogram vector to 0 [m,] = size(x); % obtai dimesios to set loop limits for i=1:m % sca all colms for j=1: % sca all rows H(X(i,j)) = H(X(i,j))+1; % se data as addresses to cot ed % freqecies if occrrece ed plot(h); % ow display it

6 Mltidimesioal Histograms Coocrrece Histogram = 2 d order or Bidimesioal Histogram: 2 Dimesioal pdf Hist( a, b ) (joit probability desities). Two attribtes (e.g. graylevel itesities) a, b are aalyzed simltaeosly, ths the ame coocrrece or cocrrece. Whe a low mber of bis L is sed (v.g., 4, i spatial depedece aalysis of textres) the histogram is called a L L coocrrece matrix. Let a, b be two radom variables represetig attribte vales of poits, either at differet locatios or differet sets of data (examples below) which may be two sigals (or differet parts of a sigle sigal), images, etc. The, as i eqatios (1) ad (2), a secod order joit probability is defied as p(, ) Prob[ =, = v ] = (10) i j a i b j mber of pairs of poits with attribtes =, = total mber of sch pairs of poits i the ROI where for simplicity we se here Δ i = Δ j = 1, for all I ad i, j = 0,1,, L 1. For that simplificatio, the joit probabilities are ofte writte as: p = p(, ),, i j = 0,..., L 1. (11) i j i j Note: do ot cofod i,j with spatial discrete coordiates. Whe poits are pixels, a, b correspod to vales I(x,y) i oe or more images at oe or differet locatios. Coocrrece may be stdied betwee: two pixels at differet locatios of the same image: a =I(x a,y a ), ad b =I(x b,y b ). See spatial depedece aalysis i the chapter o textres. a i b j

7 two pixels at the same locatio of differet images (two adjacet slices i a volme, two video frames, matchig pairs, etc.): : a =I a (x,y), ad b =I b (x,y). Iterpretatio: o if I a (x,y) I b (x,y) the bidimesioal histogram iformatio stays alog the idetity lie (diagoal) formig the 1 D histogram (o histogram vales >0 otside the diagoal). o If I a (x,y) is very similar to I b (x,y), most iformatio clsters arod the diagoal. If the diagoal is ot a straight lie, the images differ i attribte scales by some distorsio, reflectig differet callibratio, for example. I stead of coocrrece, the 1 D histogram of the differetial image (I a (x,y) I b (x,y)) may be sefl. o If I a (x,y) is totally differet to I b (x,y), o iformatio clsters alog the idetity, it spreads o all histogram domai. two pixels at the same locatio of the same image, bt from differet spectral chaels (R ad B, etc), two pixels at almost the same locatio of very similar or idetical images, oe der a geometric trasformatio (stero pairs, rotatio, distorsio or warpig): a =I (x,y), b =I(s,t), with (s,t) = (T(x,y)). correspodece problems. idem, from images of same sbject, with differet imagig modalities (CT, PET, MRI, etc.). a =I CT (x,y), b =I MRI (x,y). Coocrrece, lie histograms, may be sed to stdy higher level featres of regios ad shapes.

8 Tri dimesioal or third order Histogram, either: (1) Whe I(x,y) or V(x,y,z) is a vector valed image (volme), v.g., a color image = (r, g, b), the Hist(r,g,b) is a 3D pdf. (2) Whe we have triads (three poits desiged by some offset rle) i a sigal X( t h ) or set, or three pixels (voxels) I( x h, y h, z h ), h=1,2,3, (or from three images or volmes I h ) their cojoit empirical pdf is a 3D Histogram Hist( 1, 2, 3 ). N Dimesioal PDFs ad Histograms May decisio ad predictio problems may be formlated by exectatio vales of fctio of several radom variables. I oe dimesio, for a scalar radom variable X (ote the chage i otatio, more accordig to litteratre) taig vales χ, we have: p(χ) P( X = χ), (12) 1 χ = χ p(χ) d χ p(χ) d χ ad i geeral: f (χ) = f (χ) p(χ) d χ p(χ) d χ (13) for ay fctio f (χ). By covetio p( ) is ormalized to [0,1], so the deomiator is always p(χ)dχ = 1. Let the set of radom variables X = { X 1,, X }, ow we write the joit probability desities or pdf s: p(χ,..., χ ) P( X = χ,..., X = χ ) (14) f f p d d =... (χ,..., χ ) (χ,..., χ ) χ,..., χ (15) for example, if =I(x,y) is the gray level of each pixel (x,y) i a image, the average vale, or more precisely, the graylevel vale we expect to fid is

9 <>, ad we do ot eed to compte the histogram, bt we obtai it from averagig absoltely all pixel vales. For example, it trs ot that for the fctio defiig the sqared error f()=( <>) 2, its expected vale < f > is the stadard deviatio of. If X 1,, X taes discrete vales χ 1 {χ 1(0), χ 1(1),, χ 1(L 1) }, etc. Eqatio (15) becomes eqatio (16) ad ay of both ca be estimated by a sample average (17): < f > =... f (χ,..., χ ) p (χ,..., χ ) (16) All χ 1 All χ = N N 1 ( ) ( ) f ( x,..., x ) f 1 (17) = 0 Note that strictly speaig, eqatio (16) shold be writte as: L 1 L 1 1 < > = f... f (χ,..., χ ) p (χ,..., χ ) i = 0 i = 0 1 1( i ) ( i ) 1( i ) ( i ) 1

10 It is ofte sefl to se itegers, as with {0,1,,L 1}ad the sm o sbscripts i χ 1

Image Processing tutorial Jorge Márquez Flores - CCADET-UNAM /13. Histogram of a population

Image Processing tutorial Jorge Márquez Flores - CCADET-UNAM /13. Histogram of a population Image Processig ttorial Jorge Márqez Flores - CCADET-UNAM 28 1/13 Histogram of a poplatio Let X be a set of N data samples with qatized scalar vales i {, 1,, L-1 } (may be samples from a cotios fctio (t)