FRACTALS IN PATTERN RECOGNITION

Transcription

1 FRACTALS IN PATTERN RECOGNITION b Wtold Dzwnel Fractals We consder a mathematcal set F to be fractal, we thnk of t as havng (some) of the followng propertes : F has detal at ever scale. F s (eactl, appromatel, or statstcall) self-smlar. There s a smple algorthmc descrpton of F. Fractal bult-up procedures: geometrcal decomposton of space teratve transformaton wth a pre-defned rule set of teratve transformaton 1

2 Fractals geometrc procedure Fractals capact dmenson d(f) Let F R n. Cover F b hpercubes of sze ε Let N(ε) s the mnmum number of these hpercubes N(ε)~(1/ε) D D=lm ε (logn(ε))/log(1/ε)) Eamples: Cantor set =.639 Mandelbrodt (edge) = 2 Koch curve = 1,26 Serpnsk carpet = 1,89 2

3 Hausdorff dmenson D(F) Let F R n. Coverage I of F set of spheres A I coverng F, Permeter of I the largest sphere permeter α(d, ε) = nf I Σ A I dam(a) d (nf s computed from all the possble coverages of permeters smaller than ε) α(d) = lm ε α(d, ε) d ; f d>d α(d) =, f d<d α(d) = D(F)<=d(F) D(F) s nvarant on dfeomorphc transformaton.e., D(A)=D(f(A)) Topologcal dmenson Defnton Let X be a subset of a metrc space. Topologcal dmenson n (.e., mnmal nducton dmenson) dm top X s defned as follows: dm top X=-1 X= dm top X=n X U V ; 1. { V U } 2. dm top (δv X)<=n-1, where δv s the edge of V 3. n s the smallest natural number for whch (2) s fulflled. n s smaller than Hausdorff dmenson (n(cantor set)=; n(koch curve)=1; n(edge of Mandelbrodtset)=1) 3

4 Fractal defnton b Mandelbrodt Fractal s defned as a set, whch topologcal dmenson s dfferent (smaller) than Hausdorff dmenson not fractals: Smooth geometrcal fgures Cantor and homeomorphc sets Mandelbrodt set IFS (some of them) Fractals teratve procedure Mandelbrot set: z Z(Comple) {z {=},z 1, z 2, z 3, }; z n1 =z n2 c (c-parameter) We defne c comple space. For each c, z n can be or cannot be bounded. The c ponts resultng n bounded soluton bult-up fractals Jula set: z Z(Comple) {z,z 1, z 2, z 3, }; used for fndng solutons of f(z)=z n -1 teratvel b usng Newton method z n1 =z n -f(z n )/f (z n ) We defne z n comple space. z resultng n bounded soluton bult-up fractal (each z producng varous roots can be panted b varous colors makng Jula set more mpressve) 4

5 Eamples of Mandelbrodt and Jula sets Banach theorem Defnton : A transformaton t () s sad to be contractve f for an two ponts 1, 2, s (,1); the dstance for some s < 1. d(t( 1 ),t( 2 )) < sd( 1, 2 ), Theorem: Let us assume a complete metrc space (X,d). In ths space a contractve transformaton t(.) has a sngle fed pont (t( )= ). The lmt of {, 1, 2,, } sere ests where X, and n1 =t( n ). Then one can estmate that: d( n, ) s n /(1-s) * d(, 1 ) and s (,1) 5

6 Banach theorem When t(,z), where z Z (Z,dz), X(X,d), the soluton depends on z parameters. Let us assume that: s (,1) (1,2 X && za,zb Z) the Lpschtz condton s fulflled.e.,: d(t(1,za),t(2,zb)) s d(1,2) α dz(za,zb) for α>= then z Z 1 (z); whch s the soluton of =f(,z) d((za),(zb)) α /(1-s) * dz(za,zb) (close values of z parameters correspond to close values of (z)) Fractals affne transformaton Affne IFS transformaton : Each affne transformaton t(.) Z Z, a t = c b e d f can skew, stretch, rotate, scale and translate an nput mage; n partcular, t alwas maps squares to parallelograms. 6

7 Hausdorff dstance d(a,b) A A,B H(X) - the space of compact and non-empt subsets of space X d(b,a) B A, B, d(,b)=mn (d(,); d(a,b)=ma d(,b); d(b,a)=ma d(,a); h(a,b) = ma {d(a,b), d(b,a)} Fractal space and set of teratve functons Fractal space s defned as (H(X),h). In ths space we defne contractve IFS (Iterated Functon Sstem) {X, t 1,t 2, t k } consstng of affne functons. Contracton coeffcent of ths sstem s the largest s value for =1,..k Let us defne n the fractal space the followng operaton: f A H(X); W(A) = t 1 (A) t 2 (A) t k (A) so: h(w(a),w(b)) s ma h(a,b) where s ma = ma k {s,s 1,s 2,,s k } A H(X) the sere {A,A 1, } where A n1 =W(A n ) s bounded n nfnt and lm s a sngular soluton of the followng equaton: W(A )= A A s called the ATTRACTOR of IFS W(.). Dfferent transformatons lead to dfferent attractors. 7

8 Attractors of IFS Another attractors 8

9 Image fractal compresson The ntal mage does not affect the fnal attractor; Onl the poston and the orentaton of the copes that determnes what the fnal mage wll look lke. To determne the fnal result we onl descrbe (fnd) these transformatons. Fern an eample The Barsle fern can be represented b four affne transformatons. Each affne transformaton t s defned b 6 numbers, a b c d e and f. The can be stored n 4 transformatons 6 numbers/transformatons 32 bts/number = 768 bts. Storng mages as collectons of transformatons leads to mage compresson. 9

10 Gre-scale mages Consder the gre-scale mages. One more dmenson than bnar mages s needed. That s, {(,,z ) z =f(, ) s the gre-level at poston (, )}. The contractve propert must hold for both dstance and gre-level. Contractve requrement for dstance s ver naturall accomplshed b algorthm desgn. Codng strateg focuses on makng grelevels closer. Parttonng Fed pont (attractor) s the decoded mage. Note that the Barsle fern has a whole mage self-smlart. The gre-level mages, however, do not appear to contan affne transformatons to themselves. Gre-level mages do contan a dfferent sort of self-smlart. Rather than havng the mages be formed of copes of ts whole self, here the mages wll be formed of copes of properl transformed parts of themselves. 1

11 11 Parttonng Parttoned IFS ß ø º Ø ß ø º Ø ß ø º Ø = ß ø º Ø o f e z s d c b a z t ß ø º Ø ß ø º Ø ß ø º Ø = ß ø º Ø f e d c b a v The affne transformatons b PIFS : where s controls the contrast and o the brghtness of the transformaton. It s convenent to wrte :

12 Parttoned IFS t (.) coordnates To reduce the computaton load, v s usuall restrcted to one of the followng eght smple transformaton : 1) Rotate b 2) Rotate b 9 3) Rotate b 18 4) Rotate 27 5) Flp over the vertcal mddle lne 6) Flp over the horzontal mddle lne 7) Flp over the 45 lne 8) Flp over the 135 lne Parttoned IFS gre level Let a 1, a 2, a 3,, a n be the pels from sub-sampled, transformed D j, D j, and b 1, b 2, b 3,, b n be the pels from R. Then s and o n g are selected as : n n n n n Ø ø Ø 2 s = n ( akbk ) - ( ak ) ( bk ) n ak - ( ak ) º k = 1 k = 1 k = 1 ß º k = 1 k = 1 o Ø = º n b - s It can be proved that such s and o wll mnmze the followng error measure : n a k k k = 1 k = 1 ß ø n ø ß 2 n R = ( sa k= 1 k o b k ) 2 12

13 Fractal codng Image can be coded n a set of equatons. These equatons are usuall affne transformatons that transform a sub-mage, called a doman block, nto another sub-mage, called a range block. An mage s dvded nto non-overlappng range blocks, and a search for a best matchng doman block s performed for each range block. Doman blocks are usuall larger than range blocks, and are smlar to one another under that affne transformaton. Fractal codng transformaton For each range block R, onl one doman block Dj s transformed b t, not the whole mage. Let Dj s be doman blocks from the ntal mage. Then T ( f ) = t ( D ) t ( D ' ' 1 j 2 j 2 )... t ( D 1 N Snce the sze of range block s small, the dstorton won t be large to represent t b fractal. Both Dj and R are from the same mage : Frst, we use Dj s from f (orgnal mage) to get R s; then R s together form f1, we then get the new Dj s from f1 and compute the new R s, the new R s together form f2, and so on. Durng encodng the best appromated-range block s found for each range block b searchng and transformng from a pool of doman blocks. N ' j ) 13

14 Encodng Step 1 For each range block R of the orgnal mage to be encoded do Step 2 and Step 3; Step 2 Compute the varance V of R; Step 3 If V<V t then transmt the mean of R else search for t k, Dj such that d(t k (Dj ), R) s mnmzed; transmt t k and the locaton of Dj; Decodng Step 1 for each block R do Step 2; Step 2 f t s a mean value then put t to R else put t k (D j ) to R 14

15 Speed-up compresson Doman blocks D j onl selected from the neghborhood regon of R, nstead of the whole mage. Use the quad-tree technque. Onl blocks of large actvt are fractal encoded. Use sub-band (wavelet) codng to reduce the search range, Search range can be reduced to below 2% of the orgnal. Speed-up compresson (quad-tree) LH 2 LH 1 LH R 2 R 1 R D 2 D 1 D 15

16 Regular, quad-tree, HV Fractal dentfcaton mplementaton 1 ()The range blocks are non-overlappng unform square blocks of sze 4 b 4 pels. ()The heght and wdth of the doman blocks are twce as large as the heght and wdth of the range blocks. ()The doman blocks overlap b half n the vertcal and horzontal drectons. Havng overlappng doman blocks ncreases encodng accurac as the probablt of locatng the optmal doman block that matches a gven range block ncreases. (v) The mappng of doman blocks to appromated-range blocks are affne transformatons descrbed before. Isometrc transformatons such as reflectons and rotatons of the doman block are not used so that the amount of search requred s reduced. 16

17 Fractal dentfcaton mage parttonng Fractal dentfcaton mplementaton 2 (v) The Eucldean norm s used for dstance measures. In ths case the dstance between an two gven mages, sa p and q wth heght I h and wdth I w,s defned as the root mean square (RMS) dfference between those mages: (v)there are no search restrctons on the doman pool. All possble doman blocks are searched for the one that mnmzes d(p (nr) I, τ n (p )) where p (nr) denotes the n th range block n the mage p. (v) The value of a n s fed constant (v) 17

18 Fractal dentfcaton mplementaton 3 An eample 18

19 19 Sound recognton In ths approach, the speech waveform s ( s the ampltude as) s dvded onto N equal ntervals (, 1 ) (=1..N) n tme doman and then the followng affne transformatons W are performed for all the ntervals separatel: (1) assumng that the followng boundar condtons are defned: (2) = f e d c a W = W = 1 1 N N W more Assumng addtonall that b = smplfes the mappng problem, whle d from equaton (2) represents a vertcal scalng factor. The equatons (2) and (3) can be rewrtten as follows: The foregong sstem conssts of four equatons wth fve unknowns. Therefore, to solve such a problem one of unknown values has to be selected as a free varable. = = = = N N N f d c f d c e a e a * * * * * * 1 1

20 more The vertcal scalng factor s a parameter, whch doman s known. We can assume that ts absolute value belongs to (,1) nterval to make IFS contractve. The man problem s to select the best ntal value for d factor to obtan the fnal result n the most effcent wa. Intal guess of d value where: d F ma F mn = 2* F ma where F ma and F mn are mamum and mnmum ampltudes of the sgnal, respectvel. The Fma s the mamum ampltude n the entre tme doman of the waveform. B usng such the ntal guess, IFS for N ntervals appromated b N transformatons can be obtaned. d s selected to much the sgnal the best. Fnall, the waveform can be easl reconstructed Polsh sound a Orgnal mage Attractor - Polsh letter "a" Letter "a" Attractor 2

21 Fractal dmenson method The fractal dmenson (FD) provdes an objectve value for comparng dfferent fractal structures. Intutvel, the fractal dmenson represents the roughness of an object. The value of FD allows for comparng fractal patterns from the real world to those artfcall generated b IFS attractors. The sldng wndow algorthm s one of the smplest methods to fnd the value of FD for a waveform. The dea of sldng wndow s based on the calculaton of the fractal dmenson for ts fragment (wndow) of a gven wdth. Whle movng ths nterval teratvel along the tmescale, the sequence of numbers s generated. These numbers correspond to the sequence of fractal dmenson values. The valles and dps of the plot reflect utterance s endponts and word or sllable boundares. An eample Sklep jest nedaleko NAFD Sklep jest nedaleko -3-4 Wndow number 21