Image Segmentation. Image Segmentation

Transcription

1 Image Segmentaton REGION ORIENTED SEGMENTATION Let R reresent the entre mage regon. Segmentaton may be vewed as a rocess that arttons R nto n subregons, R, R,, Rn,such that n= R = R.e., the every xel must be n a regon; R s a connected regon, =,, n; R R j = φ,, j,.e., the regons are dsjonts; P ( R ) = TRUE, for =,,..., n ; (e.g., all xel wthn a regon have the same ntensty);, j, j, P ( R R j ) = FALSE (e.g.., ntenstes of xel n dfferent regons are dfferent) where P(R) s a logcal redcate defned over the onts n the set R, and Ø s the null set. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens Image Segmentaton REGION GROWING BY PIXEL AGGREGATION Start wth a seed xel (or a set of seed xels); Aend to each xel n the regon those of ts 4-connected or 8-connected neghbors that have smlar roertes (gray level, color, texture, etc); Sto when the regon cannot be grown any further. Examle: (b) absolute dfference less than 3; (c) absolute dfference less than 8. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens

2 Image Segmentaton (a) orgnal mage showng seed ont; (b) early stage of regon growth; (c) ntermedate stage; (d) fnal regon. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 3 Image Segmentaton Dffculty: results deend uon selecton of seed xels, and measure of smlarty (ncluson crtera). Possble soluton: our mult-tolerance regon growng rocedure! Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 4

3 Image Segmentaton REGION SPLITTING AND MERGING Assumng the mage to be square, subdvde the entre mage R successvely nto smaller and smaller quadrant regons such that, for any regon R, P ( R ) = TRUE. In other words, f P(R) = FALSE, dvde the mage nto quadrants; f P s FALSE for any quadrant, subdvde that nto subquadrants, and so on... Ths slttng technque may be reresented as a quadtree. As the slttng rocedure could result n adjacent regons that are smlar, aly a mergng ste: merge two adjacent regons R and Rk f P( R Rk ) = TRUE Sto the rocedure when no further slttng; or mergng s ossble. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 5 Image Segmentaton Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 6 3

4 Image Segmentaton (a) orgnal mage; (b) result of slt and merge rocedure; (c) result of thresholdng. P(R) = TRUE f at least 80% of the xels n R have the roerty z j m σ z j : the gray level of j th xel m : mean gray level of R Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 7 Image Segmentaton The use of moton n segmentaton d, j f f ( x, y, t ) f ( x, y, t j ) > T ( x, y ) = 0 otherwse Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 8 4

5 Image Segmentaton Accumulatve dfferences Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 9 External characterstcs: Boundary or contour morhology, Boundary roughness, Boundary comlexty. It s desrable that boundary descrtors are nvarant to translaton, scalng, and rotaton. Internal characterstcs: Gray level, Color, Texture, Statstcs of xel oulaton. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 0 5

6 Descrtons of (ds)smlarty; Dstance measures, Correlaton coeffcent. Relatonal descrtons: Placement rules, Strng, tree, and web grammars, Structural descrtons, Syntactc analyss. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens CHAIN CODES Chan codes are used to reresent a boundary by a connected sequence of straght lne segments of secfed length and drecton. 4-connectvty or 8-connectvty may be used. As the chan code deends uon the startng ont, t may be normalzed by redefnng the startng ont such that the code forms an nteger of mnmum magntude. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 6

7 resamlng (c) (d) Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 3 Polygonal aroxmatons Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 4 7

8 SIGNATURES A sgnature s a -D functonal reresentaton of a boundary. An examle s a lot of the dstance from the centrod of the regon to the boundary as a functon of angle (or to each boundary xel). Another examle s to reresent the coordnates (x,y ) of each boundary xel as a comlex varable z = ( x + jy ), = 0,,..., N, where N s the number of boundary xels (closed loo). z may then be analyzed as a erodc sgnal. Sgnatures reduce boundary descrton from a D roblem to a -D roblem. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 5 Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 6 8

9 SKELETONIZATION The skeleton of a regon may be obtaned by a thnnng algorthm: Assume that the regon has been bnarzed, wth the regon xels beng and the background xels beng 0. A contour ont s any xel wth value havng at least one 8-connected neghbor valued 0. Defne the ndexng of xels n an 8-connected neghborhood as below, where s the xel beng rocessed: Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 7 Ste : Flag a contour ont for delecton f the followng are true (reeat for all border onts): ( a) N ( ) 6; ( b) ( c) ( d ) S ( ) = ; = 0; = 0; Ste : Delete all flagged xels (change to 0). Ste 3: Do the same as Ste, but relace (c) and (d) wth: ( c ) = 0; ( d ).. = N( ) s the number of nonzero neghbors of,.e, N( ) = S( ) s the number of 0- transtons n the sequence, 3,, 9. Ste 4: Delete all flagged xels. Iterated stes -4 untl no further xels are deleted. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 8 9

10 (a) result of ste of the thnnng algorthm durng the frst teraton through a regon; (b) result of ste ; (c) fnal result. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 9 Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 0 0

11 SHAPE FACTORS The shae comlexty of a regon s boundary may be descrbed n terms of ts Comactness Moments of dstances to the centrod Fourer descrtors of the sgnature Chord-length statstcs Coyrght RMR / RDL PEE Processamento Dgtal de Imagens COMPACTNESS The common defnton of comactness s C = / a, where s the ermeter and a s the area of the regon. Comactness s a measure of the effcency of a contour n contanng maxmal area. A crcle has the mnmum comactness value of 4π. The comactness of a square s 6. Comactness may redefned as 4π a C = whch s normalzed to the range (0,), wth 0 for a crcle. Comactness s nvarant to translaton, scalng, and rotaton, and has been useful n classfyng breast tumors and calcfcatons as bengn or malgnant. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens

12 CHORD-LENGTH STATISTICS A chord s a lne jonng a boundary xel to another boundary xel. For a boundary contour wth N onts, K = N(N - )/ dstnct chords exst. Statstcal measures of the dstrbuton of the chord-lengths L, =,,..., K may be useful n dfferentatng between some tyes of boundary shaes. Mean: M K K c = = L, Varance: M K ( ) c = L M c K =, Skewness: Kurtoss: M M K ( ) 3 c3 = 3 L M c M c = K 4 c4 = 4 ( L M c) M c K =, Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 3 MOMENTS OF DISTANCES TO THE CENTROID Let z, =,,..., N reresent the dstances from the centrod of the regon to each of the N boundary xels. Moments of varous orders of the z S may assst n dstngushng between contours of dfferent tyes: the varance wll be zero for a crcle, and large for a comlex shae wth a "rough" boundary. The shae factor MF 3 = F F3 has been useful n classfyng tumors and calcfcatons n mammograms as bengn or malgnant, where N m = z, N N N F = ( z m), F3 = m N m N = 4 [ ( z m ) ] 4. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 4 = =

13 FOURIER DESCRIPTORS Consder the reresentaton of each boundary xel as a comlex varable z = ( x + jy ), = 0,,..., N. where N s the number of boundary xels. The DFT of may then be comuted as A( u) = N N k = 0 z k.ex [ jπuk / N ], u = 0,,..., N. Normalzed Fourer descrtors may then be defned as NFD(k) = z 0; k = 0 A( k ) / A (); k A( k + N ) / A (); k =,,..., N / =,,..., N / +. Note that the boundary may have to be resamled to have samles, k beng an nteger, for the sake of FFT comutatons. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 5 A dgtal boundary and ts reresentaton as a comlex sequence. The onts (x o, y o ) an (x, y ) are (arbtrarly) the frst two onts n the sequence. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 6 3

14 nversa sˆ( k) = M u= 0 A( u).ex [ jπuk / N], k = 0,,..., N. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 7 As rough contours wll lead to ncreased hgher-frequency comonents, we could comute a shae factor wth ncreasng weghts for hgher-frequency comonents. However, ths could lead to senstvty to nose and errors n boundary reresentaton. A better aroach s to use a decreasng weght for hgherfrequency comonents, and then subtract t from : FF N / k = N / + = N / k = N / NFD ( k ) NFD ( k ) FF s lmted to the range (0,), and ncreases wth roughness. FF has been useful n classfyng breast tumors and calcfcatons as bengn or malgnant. / k. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 8 4

15 REGION EDGE DEFINITION AND ACUTANCE Acutance s a measure of change of densty from a regon of nterest (ROI) to ts background. Acutance has been defned n D as the mean-squared gradent along a knfe-edge sread functon: A = f ( b) f ( a ) a b df ( x) dx dx, where f(x) s the sread functon, and a and b are the endonts of the sread functon. Acutance has been shown to be a well correlated wth subjectvely erceved edge sharness n mages. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 9 ADAPTIVE COMPUTATION OF DIMAGE EDGE PROFILE ACUTANCE Polygonal aroxmaton of the boundary; Adatve comutaton of dfferences along normals at boundary xels. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 30 5

16 Ste : Comute the sum of the dfferences along the normal at each boundary xel j. D j f ( ) b ( ) d ( j ) = = where f() and b(), I=,,,D j, are xels along the normal nsde and outsde the ROI; j = 0,,,..., N- reresent the boundary xels. Ste : Comute the normalzed root mean-squared dfference over all boundary xels: A = N d N max = 0 d ( j). D j d max s a normalzaton factor such that A s lmted to (0, ) and deends uon the gray level dynamc range and max{d j } Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 3 APPLICATION OF ACUTANCE TO TUMOR ANALYSIS Features of Bengn Masses: Smooth, round, or oval shae; Crcumscrbed, Shar, well-defned edges. Features of Malgnant Tumors: Rough, sculated, or stellate shae; Fuzzy or blurred boundares. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 3 6

17 Database CB CM SB SM Total MIAS Calgary Combned Table : Numbers of dfferent tyes of masses and tumors n the database used n the study. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 33 Combned database Clasfcaton usng A only # of # of correctly cases classfed cases Bengn Malgnant % correct Bengn Malgnant Total Table 3 : Detals of the best bengn / malgnant classfer for the combned database. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 34 7

18 Table 4: Bengn / malgnant classfcaton rates for varous combnatons of features. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 35 Table 7: Crcumscrbed / sculated classfcatons rates for varous combnatons af features. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 36 8

19 Combned # of Clasfcaton usng A only FF # of correctly classfed cases database cases CB CM SB SM % correct CB CM SB SM Total Table 9: Detals of the best four-grou classfer for the combned database. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 37 Table 0: Four-grou classfcatons rates for varous combnatons of features. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 38 9

20 OBSERVATIONS ON TUMOR CLASSIFICATION MF 3 Comactness, FF, and rovde good crcumscrbed/ sculated classfcaton; Acutance rovdes excellent bengn/ malgnant classfcaton; Four-grou classfcaton requres acutance and shae factors. Coyrght RMR / RDL PEE Processamento Dgtal de Imagens 39 0