Absolute chain codes. Relative chain code. Chain code. Shape representations vs. descriptors. Start

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1 Shape representatons vs. descrptors After the segmentaton of an mage, ts regons or edges are represented and descrbed n a manner approprate for further processng. Shape representaton: the ways we store and represent the obects Permeter Interor Shape descrptors: methods for characterzng obect shapes. The resultng feature values should be useful for dscrmnaton between dfferent obect types. Start Absolute chan codes Search drecton: look to the left frst and check neghbors n clockwse drecton Chan code n clockwse drecton: INF 43 INF 43 Chan code The chan code depends on the startng pont. It can be normalzed for start pont by treatng t as a crcular/perodc sequence, and redefne the startng pont so that the resultng number s of mnmum magntude. We can also normalze for rotaton by usng the frst dfference of the chan code: (drecton changes between code elements Code: 33 Frst dfference (counterclockwse: Mnmum crcular shft of frst dfference: To fnd the dfference, look at the code and count 3 counterclockwse drectons. Treatng the curve as crcular we add the 3 for the frst pont. Ths nvarance s only vald f the boundary tself s nvarant to rotaton and scale. INF 43 3 Relatve chan code Drectons are defned n relaton to a movng perspectve. Example: Orders gven to a blnd drver ("F, "B, "L, "R". The drectonal code representng any partcular secton of lne s relatve to the drectonal code of the precedng lne segment. Why s the relatve code: 3 R,F,F,R,F,R,R,L,L,R,R,F? Note: rotate the code table so that s The absolute chan code for the trangles are forward from your 4,,7 and, 5, 3. poston The relatve codes are 7, 7,. (Remember Forward s Invarant to rotaton, as long as startng pont remans the same. Start-pont nvarance by Mnmum crcular shft. To fnd the frst R, we look back to the end of the contour. INF 43 4

2 Sgnature representatons A sgnature s a D functonal representaton of a D boundary. It can be represented n several ways. Smple chose: radus vs. angle: Boundary segments from convex hull The boundary can be decomposed nto segments. Useful to extract nformaton from concave parts of the obects. Convex hull H of set S s the smallest convex set contanng S. The set dfference H-S s called the convex defcency D. If we trace the boundary and dentfy the ponts where we go n and out of the convex defcency, these ponts can represent mportant border ponts charaterzng the shape of the border. Border ponts are often nosy, and smoothng can be appled frst. Smooth the border by movng average of k boundary ponts. Use polygonal approxmaton to boundary. Smple algorthm to get convex hull from polygons. Invarant to translaton. Not nvarant to startng pont, rotaton or scalng. INF 43 5 INF 43 6 Descrptors extracted from the Convex Hull Useful features for shape characterzaton can be e.g.: Area of obect and area of convex hull (CH CH soldty aka convexty = (obect area/(ch area = The proporton of pxels n CH also n the obect Better than extent = (obect area/(area of boundng box Number of components of convex defcency Dstrbuton of component areas Relatve locaton of ponts where we go n and out of the convex defcency. ponts of local maxmal dstance to CH. Skeletons The skeleton of a regon s defned by the medal axs transform: For a regon R wth border B, for every pont p n R, fnd the closest neghbor n B. If p has more than one such neghbor, t belong to the medal axs. The skeleton S(A of an obect s the axs of the obect. The medal axs transform gves the dstance to the border. INF 43 7 INF 43 8

3 Introducton to Fourer descrptors Suppose that we have an obect S and that weareableto fndthelengthoftscontour. The contour should be a closed curve. We partton the contour nto M segments of equal length, and thereby fnd M equdstant ponts along the contour of S. Travelng ant-clockwse along ths contour at constant speed, we can collect a par of waveforms (=coordnates x(k and y(k. Any D sgnal representaton can be used for these. If the speed s such that one crcumnavgaton of the obect takes, x(k and y(x wll we perodc wth perod. y INF 43 9 x Remnder: complex numbers a+b b a+b a INF 43 Contour representaton usng D Fourer transform The coordnates (x,y of these M ponts are then put nto a complex vector s : s(k=x(k+y(k, k[,m-] We choose a drecton (e.g. ant-clockwse We vew the x-axs as the real axs and the y- axs as the magnary one for a sequence of complex numbers. The representaton of the obect contour s changed, but all the nformaton s preserved. We have transformed the contour problem from D to D. x = y = s( 3 s( s(3 3 3 s(4 3 4 INF 43 7 Start Fourer-coeffcents from f(k We perform a D forward Fourer transform a( u M M k uk s( kexp M M M k uk s( k cos M uk sn, M Complex coeffcents a(u are the Fourer representaton of boundary. a( contans the center of mass of the obect. Exclude a( as a feature for obect recognton. a(, a(,...,a(m- wll descrbe the obect n ncreasng detal. These depend on rotaton, scalng and startng pont of the contour. For obect recogntons, use only the N frst coeffcents (a(n, N<M Ths corresponds to settng a(k=, k>n- INF 43 u, M

4 Approxmatng a curve wth Fourer coeffcents n D Orgnal sgnal of length N=36 Take the Fourer transform of the sgnal of length N Keep only the M (<N/ frst Fourer coeffcents (set the others to n ampltude. Compute the nverse D Fourer transform of the modfed sgnal. Dsplay the sgnal correspondng to M coeffcents. Approxmatng n ncreasng detal m= m=3 m=4 m=8 Reconstructed sgnal usng only Fourer coeffcents m= m=3 m=5 Orgnal INF 43 3 INF 43 4 Look back to D Fourer spectra (3 Fourer Symbol reconstructon Inverse Fourer transform gves an approxmaton to the orgnal contour N ˆ( uk s k a( uexp, k, M k M We have only used N features to reconstruct each component of sˆ ( k. The number of ponts n the approxmaton s the same (M, but the number of coeffcents (features used to reconstruct each pont s smaller (N<M. Most of the energy s concentrated along the lowest frequences.we can reconstruct the mage wth an ncreasng accuracy by startng wth the lowest frequences and addng hgher...3 INF 43 5 Use an even number of descrptors. The frst -6 descrptors are found to be suffcent for character descrpton. They can be used as features for classfcaton. The Fourer descrptors can be nvarant to translaton and rotaton f the coordnate system s approprately chosen. All propertes of D Fourer transform pars (scalng, translaton, rotaton can be appled. INF 43 6

5 Fourer descrptor example Fourer coeffcents and nvarance Image, 6x pxels Boundary coeffcents 4 coeffcents 6 coeffcents 8 coeffcents coeffcents Matlab DIPUM Toolbox: b=boundares(f; b=b{}; %sze(b tells that the contour s 65 pxels long bm=boundm(b,6,; %must tell mage dmenson z=frdescp(b; znv=frdesc(z,; zm=boundm(znv,6,; mshow(zm; Translaton affects only the center of mass (a(. Rotaton only affects the phase of the coeffcents. Scalng affects all coeffcents n the same way, so ratos a(u /a(u are not affected. The start pont affects the phase of the coeffcents. Normalzed coeffcents can be obtaned, but s beyond the scope of ths course. See e.g., Ø. D. Trer, A. Jan and T. Taxt, Feature extracton methods for character recognton a survey. Pattern Recognton, vol. 9, no. 4, pp , 996. INF 43 7 INF 43 8 Run Length Encodng of Obects Sequences of adacent pxels are represented as runs. Absolute notaton of foreground n bnary mages: Run = ;<row, column, runlength >; Relatve notaton n graylevel mages: ;(graylevel, runlength ; Ths s used as a lossless compresson transform. Relatve notaton n bnary mages: Start value, length, length,, eol, Start value, length, length,, eol,eol. Ths s also useful for representaton of mage bt planes. RLE s found n TIFF, GIF, JPEG,, and n fax machnes. INF 43 9 Gray code Is the conventonal bnary representaton of graylevels optmal? Consder a sngle band graylevel mage havng b bt planes. We desre a mnmum complexty n each bt plane Because the run-length transform wll be most effcent. Conventonal bnary representaton gves hgh complexty. If the graylevel value fluctuates between k - and k, k+ bts wll change value: example: 7 = whle 8 = In Gray Code only one bt changes f graylevel s changed by. The transton from bnary code to gray code s a reversble transform, whle both bnary code and gray code are codes. INF 43

6 Gray Code transforms Learnng goals - representaton Bnary Code to Gray Code :. Start by MSB n BC and keep all untl you ht. s kept, but all followng bts are complemented untl you ht 3. s complemented, but all followng bts are kept untl you ht 4. Go to. Gray Code to Bnary Code :. Start by MSB n GC and keep all untl you ht. s kept, but all followng bts are complemented untl you ht. 3. s complemented, but all followng bts are kept untl you ht. 4. Go to. Chan codes Absolute Frst dfference Relatve Mnmum crcular shft Polygonzaton Focus on recursve Sgnatures Convex hull Skeletons Thnnng Fourer descrptors Run Length Encodng INF 43 INF 43 Bayes rule for a classfcaton problem Suppose we have J, =,...J classes. s the class label for a pxel, and x s the observed feature vector. We can use Bayes rule to fnd an expresson for the class wth the hghest probablty: p( x P( P( x p( x pror probablty posteror probablty lkelhood normalzng factor P( s the pror probablty for class. If we don't have specal knowledge that one of the classes occur more frequent than other classes, we set them equal for all classes. (P( =/J, =.,,,J. Eucldean dstance vs. Mahalanobs dstance Eucldean dstance between pont x and class center : T x x x Mahalanobs dstance between x and : r T x x INF 43 3 INF 43 4

7 Dscrmnant functons for the normal densty We saw that the mnmum-error-rate classfcaton can computed usng the dscrmnant functons g ( x ln p( x ln P( Wth a multvarate Gaussan we get: t g ( x ( x μ d ( x μ ln ln ln P( Let ut look at ths expresson for some specal cases: INF 43 5 Case : Σ =σ I An equvalent formulaton of the dscrmnant functons: t g ( x w x w t where w μ and ln ( w μ μ P The equaton g (x=g (x canbe wrttenas t w ( x x where w μ -μ and x P( μ -μ ln μ -μ μ -μ w= - s the vector between the mean values. Ths equaton defnes a hyperplane through the pont x, and orthogonal to w. If P( =P( the hyperplane wll be located halfway between the mean values. P( INF 43 6 A smple model, Σ =σ I The dstrbutons are sphercal n d dmensons. The decson boundary s a generalzed hyperplane of d- dmensons The decson boundary s perpendcular to the lne separatng the two mean values Ths knd of a classfer s called a lnear classfer, or a lnear dscrmnant functon Because the decson functon s a lnear functon of x. If P( = P(, the decson boundary wll be half-way between and INF 43 7 Case : Common covarance, Σ = Σ If we assume that all classes have the same shape of data clusters, an ntutve model s to assume that ther probablty dstrbutons have the same shape By ths assumpton we can use all the data to estmate the covarance matrx Ths estmate s common for all classes, and ths means that also n ths case the dscrmnant functons become lnear functons T g ( x ( x μ Σ ( x μ ln Σ ln P( T T T ( x Σ x μ Σ x μ Σ μ ln Σ ln P( Common for all classes, no need to compute Snce x T x s common for all classes, g (x agan reduces to a lnear functon of x. INF 43 8

8 Case : Common covarance, Σ = Σ An equvalent formulaton of the dscrmnant functons s t g ( x wx w where w Σ μ t and w μ Σ μ ln P( The decson boundares are agan hyperplanes. Because w = Σ - ( - s not n the drecton of ( -, the hyperplane wl not be orthogonal to the lne between the means. Case 3:, Σ =arbtrary The dscrmnant functons wll be quadratc: t t g ( x x W x w x w where W Σ, w Σ μ t and w μ Σ μ ln Σ ln P( The decson surfaces are hyperquadrcs and can assume any of the general forms: hyperplanes hypershperes pars of hyperplanes hyperellsods, hyperparabolods hyperhyperbolod INF 43 9 INF 43 3 Use few, but good features To avod the curse of dmensonalty we must take care n fndng a set of relatvely few features. A good feature has hgh wthn-class homogenety, and should deally have large between-class separaton. In practse, one feature s not enough to separate all classes, but a good feature should: separate some of the classes well Isolate one class from the others. If two features look very smlar (or have hgh correlaton, they are often redundant and we should use only one of them. Class separaton can be studed by: Vsual nspecton of the feature mage overlad the tranng mask Scatter plots Evaluatng features as done by tranng can be dffcult to do automatcally, so manual nteracton s normally requred. INF 43 3 Exhaustve feature selecton If for some reason you know that you wll use d out of D avalable features, an exhaustve search wll nvolve a number of combnatons to test: D! n D d! d! If we want to perform an exhaustve search through D features for the optmal subset of the d m best features, the number of combnatons to test s m D! n D d! d! d Impractcal even for a moderate number of features! d 5, D = => n = INF 43 3

9 Suboptmal feature selecton k-nearest-neghbor classfcaton Select the best sngle features based on some qualty crtera, e.g., estmated correct classfcaton rate. A combnaton of the best sngle features wll often mply correlated features and wll therefore be suboptmal. More n INF 53 Sequental forward selecton mples that when a feature s selected or removed, ths decson s fnal. Stepwse forward-backward selecton overcomes ths. A specal case of the add - a, remove - r algorthm. Improved nto floatng search by makng the number of forward and backward search steps data dependent. Adaptve floatng search Oscllatng search. A very smple classfer. Classfcaton of a new sample x s done as follows: Out of N tranng vectors, dentfy the k nearest neghbors (measure by Eucldean dstance n the tranng set, rrespectvely of the class label. k should be odd. Out of these k samples, dentfy the number of vectors k that belong to class, :,,...M (f we have M classes Assgn x to the class wth the maxmum number of k samples. k must be selected a pror. INF INF About knn-classfcaton If k= (NN-classfcaton, each sample s assgned to the same class as the closest sample n the tranng data set. If the number of tranng samples s very hgh, ths can be a good rule. If k->, ths s theoretcally a very good classfer. Ths classfer nvolves no tranng tme, but the tme needed to classfy one pattern x wll depend on the number of tranng samples, as the dstance to all ponts n the tranng set must be computed. Practcal values for k: 3<=k<=9 Supervsed or unsupervsed classfcaton Supervsed classfcaton Classfy each obect or pxel nto a set of k known classes Class parameters are estmated usng a set of tranng samples from each class. Unsupervsed classfcaton Partton the feature space nto a set of k clusters k s not known and must be estmated (dffcult In both cases, classfcaton s based on the value of the set of n features x,...x n. The obect s classfed to the class whch has the hghest posteror probablty. The clusters we get are not the classes we want. INF INF 43 36

10 K-means clusterng Note: K-means algorthm normally means ISODATA, 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 x k for these ponts.. Assgn each obect/pxel x n the mage to the closest cluster center usng Eucldean dstance. Compute for each sample the dstance r to each cluster center: T r x k x k x k Assgn x to the closest cluster (wth mnmum r value 3. Recompute the cluster centers based on the new labels. 4. Repeat from untl #changes<lmt. Learnng goals from classfcaton Be able to use and mplement Bayes rule wth a n- dmensonal Gaussan dstrbuton. Know how s and s are estmated. Understand the -dmensonal case where a covarance matrx s llustrated as an ellpse. Be able to smplfy the general dscrmnant functon for 3 cases. Have a geometrc nterpretaton of classfcaton wth features. ISODATA K-means: splttng and mergng of clusters are ncluded n the algorthm INF INF Learnng goals contnued Understand how dfferent measures of classfcaton accuracy work: Be famlar wth the curse of dmensonalty and the mportance of selectng few, but good features Understand knn-classfcaton Understand the dfference between supervsed and unsupervsed classfcaton Understand the Kmeans-algorthm. Be able to solve the prevous exam questons on classfcaton INF 43 39