INF 4300 Digital Image Analysis REPETITION

Transcription

1 INF 4300 Dgtal Image Analyss REPEIION Classfcaton PCA and Fsher s lnear dscrmnant Morphology Segmentaton Anne Solberg 406 INF 4300 Back to classfcaton error for thresholdng - Background - Foreground P error P error, d P error p d In ths regon, foreground pels are msclassfed as background In ths regon, background pels are msclassfed as foreground INF 4300

2 Mnmzng the error P error P error, d P error p d When we derved the optmal threshold, we showed that the mnmum error was acheved for placng the threshold or decson boundary as we wll call t now at the pont where P = P hs s stll vald INF Dscrmnant functons he decson rule Decde f P P, for all can be wrtten as assgn to f g g he classfer computes J dscrmnant functons g and selects the class correspondng to the largest value of the dscrmnant functon Snce classfcaton conssts of choosng the class that has the largest value, a scalng of the dscrmnant functon g by fg wll not effect the decson f f s a monotoncally ncreasng functon hs can lead to smplfcatons as we wll soon see INF

3 Equvalent dscrmnant functons he followng choces of dscrmnant functons gve equvalent decsons: he effect of the decson rules s to dvde the feature space nto c decson regons R,R c If g >g for all, then s n regon R he regons are separated by decson boundares, surfaces n features space where the dscrmnant functons for two classes are equal INF ln ln P p g P p g p P p P g INF he condtonal densty p s Any probablty densty functon can be used to model p s A common model s the multvarate Gaussan densty he multvarate Gaussan densty: If we have d features, s s a vector of length d and and s a dd matr depends on class s s s the determnant of the matr s, and s - s the nverse s s t s s n s p μ Σ μ Σ / / ep nn nn n n n S ns s s S 3 Σ μ Symmetrc dd matr s the varance of feature s the covarance between feature and feature Symmetrc because =

4 he covarance matr and ellpses In D, the Gaussan model can be thought of as appromatng the classes n D feature space wth ellpses he mean vector =[, ] defnes the the center pont of the ellpses, the covarance between the features defnes the orentaton of the ellpse and defnes the wdth of the ellpse S he ellpse defnes ponts where the probablty densty s equal Equal n the sense that the dstance to the mean as computed by the Mahalanobs dstance s equal he Mahalanobs dstance between a pont and the class center s: r he man aes of the ellpse s determned by the egenvectors of he egenvalues of gves ther length INF Eucldean dstance vs Mahalanobs dstance Eucldean dstance between pont and class center : Ponts wth equal dstance to le on a crcle Mahalanobs dstance between and : r Ponts wth equal dstance to le on an ellpse INF

5 Dscrmnant functons for the normal densty We saw last lecture that the mnmum-error-rate classfcaton can be computed usng the dscrmnant functons Wth a multvarate Gaussan we get: Let ut look at ths epresson for some specal cases: INF ln ln P p g ln ln ln t P d g μ μ INF Case : Σ =σ I he dscrmnant functons smplfes to lnear functons usng such a shape on the probablty dstrbutons ln ln ln ln ln ln P I d I P I d I g μ μ μ μ μ Common for all classes, no need to compute these terms Snce s common for all classes, an equvalent g s a lnear functon of : ln P μ μ μ

6 he dscrmnant functon when Σ =σ I that defnes the border between class and n the feature space s a straght lne he dscrmnant functon ntersects the lne connectng the two class means at the pont 0 = - / f we do not consder pror probabltes he dscrmnant functon wll also be normal to the lne connectng the means 0 Decson boundary Case : Common covarance, Σ = Σ An equvalent formulaton of the dscrmnant functons s t g w w where w Σ μ and w he decson boundares 0 are agan hyperplanes he decson boundary has the equaton: 0 t μ Σ μ ln P w 0 w 0 ln P / P 0 Because w = Σ - - s not n the drecton of -, the hyperplane wll not be orthogonal to the lne between the means INF 4300

7 Case 3:, Σ =arbtrary he dscrmnant functons wll be quadratc: t t g W w w where W Σ, t and w μ Σ μ 0 w Σ μ ln Σ ln P 0 he decson surfaces are hyperquadrcs and can assume any of the general forms: hyperplanes hypershperes pars of hyperplanes hyperellsods, Hyperparabolods, he net sldes show eamples of ths In ths general case we cannot ntutvely draw the decson boundares ust by lookng at the mean and covarance INF Dstance measures used n feature selecton In feature selecton, each feature combnaton must be ranked based on a crteron functon Crtera functons can ether be dstances between classes, or the classfcaton accuracy on a valdaton test set If the crteron s based on eg the mean values/covarance matrces for the tranng data, dstance computaton s fast Better performance at the cost of hgher computaton tme s found when the classfcaton accuracy on a valdaton data set dfferent from tranng and testng s used as crteron for rankng features hs wll be slower as classfcaton of the valdatton data needs to be done for every combnaton of features INF

8 Method - Sequental backward selecton Select l features out of d Eample: 4 features,, 3, 4 Choose a crteron C and compute t for the vector [,, 3, 4 ] Elmnate one feature at a tme by computng [,, 3 ], [,, 4 ], [, 3, 4 ] and [, 3, 4 ] Select the best combnaton, say [,, 3 ] From the selected 3-dmensonal feature vector elmnate one more feature, and evaluate the crteron for [, ], [, 3 ], [, 3 ] and select the one wth the best value Number of combnatons searched: +/d+d-ll+ INF Method 3: Sequental forward selecton Compute the crteron value for each feature Select the feature wth the best value, say Form all possble combnatons of features the wnner at the prevous step and a new feature, eg [, ], [, 3 ], [, 4 ], etc Compute the crteron and select the best one, say [, 3 ] Contnue wth addng a new feature Number of combnatons searched: ld-ll-/ Backwards selecton s faster f l s closer to d than to INF

9 k-nearest-neghbor classfcaton A very smple classfer Classfcaton of a new sample s done as follows: Out of N tranng vectors, dentfy the k nearest neghbors measured by Eucldean dstance n the tranng set, rrespectvely of the class label Out of these k samples, dentfy the number of vectors k that belong to class, :,,M f we have M classes Assgn to the class wth the mamum number of k samples k should be odd, and must be selected a pror INF K-means clusterng Note: K-means algorthm normally means ISODAA, but dfferent defntons are found n dfferent books K s assumed to be known Start wth assgnng K cluster centers k random data ponts, or the frst K ponts, or K equally spaces ponts For k=:k, Set k equal to the feature vector k for these ponts Assgn each obect/pel n the mage to the closest cluster center usng Eucldean dstance Compute for each sample the dstance r to each cluster center: r k Assgn to the closest cluster wth mnmum r value 3 Recompute the cluster centers based on the new labels 4 Repeat from untl #changes<lmt k ISODAA K-means: splttng and mergng of clusters are ncluded n the algorthm k INF

10 INF 4300 Lnear feature transforms Anne Solberg oday: Feature transformaton through prncpal component analyss Fsher s lnear dscrmnant functon 406 INF Defntons: Correlaton matr vs covarance matr s the covarance matr of E R s the correlaton matr of R E R = f =0 INF

11 Prncpal component or Karhunen-Loeve transform Let be a feature vector Features are often correlated, whch mght lead to redundances We now derve a transform whch yelds uncorrelated features We seek a lnear transform y=a, and the y s should be uncorrelated he y s are uncorrelated f E[yy ]=0, If we can epress the nformaton n usng uncorrelated features, we mght need fewer coeffcents INF 4300 Varance of y Varance along drectons from 0 to 80 degrees INF 4300

12 Varance of y cont Assume mean of s subtracted he sample covarance matr / scatter matr; R Called σ w on some sldes INF Crteron functon Goal: Fnd transform mnmzng representaton error We start wth a sngle weght-vector, w, gvng us a sngle feature, y Let Jw = w Rw = σ w Now, let s fnd As we learned on the prevous slde, mamzng ths s equvalent to mnmzng representaton error INF

13 Prncpal component transform PCA Place the m «prncple» egenvectors the ones wth the largest egenvalues along the columns of A hen the transform y = A gves you the m frst prncple components he m-dmensonal y have uncorrelated elements retans as much varance as possble gves the best n the mean-square sense descrpton of the orgnal data through the «mage»/proecton/reconstructon Ay Note: he egenvectors themselves can often gve nterestng nformaton PCA s also known as Karhunen-Loeve transform INF PCA and rotaton and «whtenng» If we use all egenvectors n the transform, y = A t, we smply rotate our data so that our new features are uncorrelated, e, covy s a dagonal matr Note: Uncorrelated varables need not appear round/sphercal: If we as a net step scale each feature by ther σ -, y = D -/ A t, where D s a dagonal matr of egenvalues e, varances, we get covy=i We say that we have «whtened» the data INF

14 Eample cont: Inspectng the egenvalues he mean-square representaton error we get wth m of the N PCA-components s gven as E N m N ˆ m Plottng s wll gve ndcatons on how many features are needed for representaton INF PCA and classfcaton Reduce overfttng by detectng drectons/components wthout any/very lttle varance Sometmes hgh varaton means useful features for classfcaton: and sometmes not: INF

15 Intro to Fsher s lnear dscrmnant PCA unsupervsed Fsher s LDA supervsed INF Crteron functon - a frst attempt o fnd a good proecton vector for classfcaton, we need to defne a measure of separaton between the proectons hs wll be the crteron functon Jw A nave choce would be proected mean dfference,, st w = hs crteron does not consder varance n y μ w smply becomes a scaled dfference n means μ -μ Optmal only when cov = σ I for all classes then vary does not change wth w μ Decson lne not optmal! INF

16 A crteron functon ncludng varance Fsher s soluton: Mamze a functon that represents the dfference between the means, scaled by a measure of the wthnclass scatter Defne classwse scatter scaled varance s wthn class scatter Fsher s crteron s then We look for a proecton where eamples from the same class are close to each other, whle at the same tme proected mean values are as far apart as possble INF Scatter matrces M classes Wthn-class scatter matr: S S w M P S E Between-class scatter matr: Weghted average of each class sample covarance matr S b M M P P Sample covarance matr for the means Fsher crteron n terms of wthn-class and between-class scatter matrces: INF

17 Solvng Fsher more drectly Alternatvely, you can notce that s a «generalzed Raylegh quotent» and look up the soluton for ts mamum, whch s the prncpal egenvector of S w - S b he followng solutons orthogonal n S w, e, w S w w =0, for are the net prncpal egenvectors Note that the obtaned ws are dentcal up to scalng to those from the two-step procedure from the prevous sldes INF Computng Fshers lnear dscrmnant For l=m-: Form a matr C such that ts columns are the M- egenvectors of Set S w S b yˆ C hs gves us the mamum J 3 value hs means that we can reduce the dmenson from m to M- wthout loss n class separablty power but only f J 3 s a correct measure of class separablty Alternatve vew: wth a Bayesan model we compute the probabltes P for each class =,M Once M- probabltes are found, the remanng P M s gven because the P s sum to one INF

18 Computaton: Case : l<m- Form C by selectng the egenvectors correspondng to the l largest egenvalues of S w S b We now have a loss of dscrmnatng power snce J 3, yˆ J3, INF Lmtatons of Fsher s dscrmnant Its crteron functon s based on all classes havng a smlarly-shaped Gaussan dstrbuton Any devance from ths could lead to problems / suboptmal or poor solutons It produces at most M- meanngful feature proectons One could «overft» S w It wll fal when the dscrmnatory nformaton s not n the mean but n the varance of the data falng to meet that stated n the frst bulletpont! INF

19 INF 4300 Dgtal Image Analyss MORPHOLOGICAL IMAGE PROCESSING Anne Solberg 3006 INF Openng Eroson of an mage removes all structures that the structurng element cannot ft nsde, and shrnks all other structures Dlatng the result of the eroson wth the same structurng element, the structures that survved the eroson were shrunken, not deleted wll be restored hs s called morphologcal openng: f S f θ S S he name tells that the operaton can create an openng between two structures that are connected only n a thn brdge, wthout shrnkng the structures as eroson would do INF

20 Closng A dlaton of an obect grows the obect and can fll gaps If we erode the result wth the rotated structurng element, the obects wll keep ther structure and form, but small holes flled by dlaton wll not appear Obects merged by the dlaton wll not be separated agan Closng s defned as f S f Ŝ θ Sˆ hs operaton can close gaps between two structures wthout growng the sze of the structures lke dlaton would INF Gray level morphology We apply a smplfed defnton of morphologcal operatons on gray level mages Grey-level eroson, dlaton, openng, closng Image f,y Structurng element b,y May be nonflat or flat Assume symmetrc, flat structurng element, orgo at center ths s suffcent for normal use Eroson and dlaton then correspond to local mnmum and mamum over the area defned by the structurng element INF

21 Gray level openng and closng Correspondng defnton as for bnary openng and closng Result n a flter effect on the ntensty Openng: Brght detals are smoothed Closng: Dark detals are smoothed f S f θ S S mamn f f S f S θ S mnma f INF op-hat transformaton Purpose: detect or remove structures of a certan sze op-hat: detects lght obects on a dark background also called whte top-hat op-hat mage mnus ts openng: f f b Bottom-hat: detects dark obects on a brght background also called black top-hat Bottom-hat closng mnus mage: f b f Very useful when correctng for uneven llumnaton or obects on a varyng background INF

22 Eample top-hat Orgnal, un-even background Global thresholdng usng Otsu s method Obects n the lower rght Corner dsappear Msclassfcaton of background n upper rght corner Openng wth a 4040 structurng element removes obects and gves an estmate of the background op-hat transform orgnal openng op-hat, thresholded wth global threshold INF Watershed the dea A gray level mage or a gradent magntude mage or some other feature mage may be seen as a topographc relef, where ncreasng pel value s nterpreted as ncreasng heght Drops of water fallng on a topographc relef wll flow along paths to end up n local mnma he watersheds of a relef correspond to the lmts of adacent catchment basns of all the drops of water F INF

23 Watershed segmentaton Can be used on mages derved from: he ntensty mage Edge enhanced mage Dstance transformed mage eg dstance from obect edge hresholded mage From each foreground pel, compute the dstance to a background pel Gradent of the mage Most common bass of WS: gradent mage F INF Watershed algorthm cont he topography wll be flooded wth nteger flood ncrements from n=mn- to n=ma+ Let C n M be the set of coordnates of ponts n the catchment basn assocated wth M, flooded at stage n hs must be a connected component and can be epressed as C n M = CM [n] only the porton of [n] assocated wth basn M Let C[n] be unon of all flooded catchments at stage n: C[ n] R C n M and C[ma] C M F INF R

24 Dam constructon C[n-] Stage n-: two basns formng separate connected components o consder pels for ncluson n basn k n the net step after floodng, they must be part of [n], and also be part of the connected component q of [n] that C n- [k] s ncluded n Use morphologcal dlaton teratvely Dlaton of C[n-] s constraned to q he dlaton can not be performed on pels that would cause two basns to be merged form a sngle connected component q C n- [M ] C n- [M ] Step n- Step n F INF Watershed algorthm cont Intalzaton: let C[mn+]=[mn+] hen recursvely compute C[n] from C[n-]: Let Q be the set of connected components n [n] For each component q n Q, there are three possbltes: qc[n-] s empty new mnmum Combne q wth C[n-] to form C[n] qc[n-] contans one connected component of C[n-] q les n the catchment basn of a regonal mnmum Combne q wth C[n-] to form C[n] F INF

25 Over-segmentaton or fragmentaton Image I Gradent magntude mage g Watershed of g Watershed of smoothed g Usng the gradent mage drectly can cause fragmentaton because of nose and small rrelevant ntensty changes Improved by smoothng the gradent mage or usng markers F INF Soluton: Watershed wth markers A marker s an etended connected component n the mage Can be found by ntensty, sze, shape, teture etc Internal markers are assocated wth the obect a regon surrounded by brght pont of hgher alttude Eternal markers are assocated wth the background watershed lnes Segment each sub-regon by some segmentaton algorthm F INF

26 How to fnd markers Apply flterng to get a smoothed mage Segment the smooth mage to fnd the nternal markers Look for a set of pont surrounded by brght pels How ths segmentaton should be done s not well defned Many methods can be used Segment smooth mage usng watershed to fnd eternal markers, wth the restrcton that the nternal markers are the only allowed regonal mnma he resultng watershed lnes are then used as eternal markers We now know that each regon nsde an eternal marker conssts of a sngle obect and ts background Apply a segmentaton algorthm watershed, regon growng, threshold etc only nsde each watershed F INF