Image Segmentation on Spiral Architecture

Transcription

1 Image Segmetatio o Spiral Architecture Qiag Wu, Xiagjia He ad Tom Hitz Departmet of Computer Systems Uiversity of Techology, Sydey PO Box 3, Broadway Street, Sydey 007, Australia {wuq,sea,hitz}@it.uts.edu.au Abstract Spiral Architecture is a relatively ew ad powerful approach to geeral purpose machie visio system. It cotais very useful geometric ad algebraic properties. Two algebraic operatios, Spiral Additio ad Spiral Multiplicatio, have bee defied o it. This paper presets a way to segmet the object(s) i a image uiformly by Spiral Multiplicatio. Namely, a umber of aalogous small copies of the origial object(s) are made durig segmetatio. A algorithm is also developed i this paper to compute the scalig factor or the umber of the small copies, so image segmetatio ca be doe flexibly ad quatitatively accordig to the specific applicatio. The research results are very beeficial to image segmetatio i parallel image processig ad distributed image processig. Keywords: Image rotatio, image segmetatio, distributed image processig Itroductio The algorithm to be preseted i this paper is based o a special data structure, Spiral Architecture (Sherida, Hitz ad Moore 99) which is ispired from aatomical cosideratios of the primate s visio (Schwartz 980). From the research about the geometry of the coes o the primate s retia it ca be cocluded that the coes distributio has iheret orgaizatio ad is featured by its potetial powerful computatio abilities. The coes with the shape of hexagos are arraged i a spiral cluster. This cluster cosists of the orgaizatioal uits of visio. Each uit is a set of seve-hexago (Sherida, Hitz ad Alexader 000) as show i Figure. I the traditioal rectagular image architecture, a set of 3 3rectagles is used as the uit of visio ad each pixel has eight eighbour pixels (See Figure ). I Spiral Architecture ay pixel has oly six eighbour pixels which have the same distace to the ceter hexago of the seve-hexago uit of visio. So the Spiral Architecture has the possibility to save time for global ad local processig. Figure. Uit of visio o rectagular image architecture A atural data structure that emerges from geometric cosideratio of the distributio of photo receptors o the primate s retia has bee called the Spiral Hoeycomb Mosaic (SHM) ad is preseted i details i (Sherida ad Hitz 999). SHM is made up of the hexagoal lattices, which are idetified by a desigated positive umber idividually. The umbered hexagos form the cluster of size 7. The hexagos tile the plae i a recursive modular maer alog the spiral directio (Alexader 99). A example of a cluster with size of 7 ad the correspodig addresses are show i Figure 3. Figure. Seve-hexago uit of visio o Spiral Architecture Copyright 00, Australia Computer Society, Ic. This paper appeared at the Pa-Sydey Area Workshop o Visual Iformatio Processig (VIP00), Sydey, Australia. Cofereces i Research ad Practice i Iformatio Techology, Vol.. David Daga Feg, Jesse Ji, Peter Eades, Hog Ya, Eds. Reproductio for academic, ot-for profit purposes permitted provided this text is icluded. Figure 3. SHM with size of 49 SHM cotais very useful geometric ad algebraic properties, which ca be iterpreted i terms of the mathematical object, Euclidea rig. Two algebraic operatios have bee defied o SHM, Spiral Additio

2 ad Spiral Multiplicatio. After image is projected oto SHM, each pixel o the image is associated with a particular hexago ad its SHM address. The these two operatios metioed above ca be used to defie two trasformatios o SHM address space respectively, which are traslatio of image ad scalig rotatio of image. This paper deepes the research work of Spiral Multiplicatio to achieve uiform image segmetatio. Moreover, such segmetatio ca be measured exactly ad quatitatively. It is very useful to distributed image processig (Bharadwaj 000). By uiform image segmetatio, workig load ca be balaced amog all the odes i the distributed processig system. So system performace is improved very much. There is o doubt that this algorithm lights a gateway for the applicatio of Spiral Architecture i image processig. The orgaizatio of this paper is as follows. I order to simplify our presetatio, the relative kowledge about Spiral Multiplicatio will be explaied briefly i Sectio accordigly. Sectio 3 shows a way to compute scalig factor or the umber of small copies durig image segmetatio. The experimet results are demostrated i Sectio 4. Coclusio ca be see i Sectio. Spiral Multiplicatio SHM is a subset of the complex plae. Spiral Multiplicatio is a arithmetic operatio with closure properties defied o SHM addressig system so that the resultig product will be SHM address i the same fiite set o which the operatio is performed (Sherida 996). I additio, Spiral Multiplicatio icorporates a special form of modularity. I order to achieve Spiral Multiplicatio, a scalar form of Spiral Multiplicatio is defied i Table I Table I: Scalar Spiral Multiplicatio Table α( a) = ( αa a = ( a a αa αa ) a ) for a i where { 0,,,6} (.) If the address i Spiral Multiplicatio is ot a scalar, α, but a commo address like, the { 0,,,6} b = ( bb b ) for bi (.) a b = i= a b i i 0 (.3) where deotes Spiral Additio ad deotes Spiral Multiplicatio. Surely carry rule is required i Spiral Additio to hadle the additio of umbers composed of more tha oe digit. I order to guaratee that all the pixels are still located withi the origial Spiral area after Spiral Multiplicatio, a modular multiplicatio o SHM is defied. Furthermore the trasformatio through Spiral Multiplicatio defied o SHM is a bijective mappig. That is each pixel i the origial image maps oe-to-oe to each pixel i the output image after Spiral Multiplicatio. Modular Multiplicatio is show as follows, Let p be the product of two elemets a, b. That is, p = a b, a, b SHM (.4) If p (modulus), the if a is a multiple of 0 map p to ( + ( p (modulus))) mod(modulus) otherwise, map p to p (.) p mod(modulus), where modulus = 0 (.6) Here, it is assumed that the umber of hexago i spiral area is 7. I additio, aother poit relative to Spiral multiplicatio is the existece of iverse multiplicative. Give a elemet a SHM, there exists a iverse value b SHM, such that a b = (Spiral Multiplicatio), deoted by a, i.e., b = a. Two cases must be cosidered to fid out the iverse value for oe SHM address. Multiplicatio of address a by the scalar α ( α { 0,,,6}) is obtaied by applyig scalar multiplicatio to the compoets of a accordig to the above scalar form, ad deoted by, CASE a is ot a multiple of 0. Let us assume = ( aa a) for ai { 0,,,6} iverse value b ( b b b) for b { 0,,,6} a ad the = i. I geeral, we ca get the iverse values for the basic SHM addresses,, 3, 4, ad 6. They are, 6,, 4, 3 ad respectively.

3 So the iverse value, b, ca be costructed successfully by the followig formula, b = a b = ( a b ) b b = ( ( i= 0 a i b CASE a is a multiple of 0, i.e., a = k 0 m modulus = 0 (m < ) i+ )) b (.7) (.8) segmetatio if the multiplier is ot a multiple of 0 (See Figure ). A ovel algorithm to compute the scalig factor is developed as follows for the multiplier beig a arbitrary Spiral Address. STEP Refie Spiral Architecture I order to segmet the objects ito ay umber of parts, the relatio betwee the multiplier ad the umber of segmets after Spiral Multiplicatio must be foud. I this paper, Spiral Architecture is refied ad is show i Figure 6. Here, three parameters are itroduced to locate each pixel o Spiral Architecture. We ca get k by formula (.7). So, a = k 0 (Spiral Multiplica tio) (.9) m Here we assume that SHM address 0 has o valid iverse value. 3 Scalig Factor Computatio While Image Segmetatio After a Spiral image is multiplied by a specific Spiral Address, x, this image will rotate by agle θ which is determied by vector 0 ad vector 0 X (He, Hitz ad Sherida 996) (See Figure 4). a) Multiplier = x θ Figure 4. Rotatio agle is determied by multiplier But this rotatio is ot a pure rotatio. It is accompaied by a scalig segmetatio. It is disadvatageous to image rotatio, but it ispires us with a good idea to segmet the objects i a image uiformly. Ufortuately, it is kow oly how to segmet the object ito 7 parts if there are m 7 ( m > ) pixels o the Spiral Architecture. This ca be m doe by Spiral Multiplicatio with a multiplier 0 directly. There has ot bee a way yet to compute scalig factor or the umber of segmets durig image b) Multiplier = 63 Figure. Image segmetatio by Spiral Multiplicatio. Origial image is a up-right arrow with 6807 pixels o Spiral Architecture. r = r =3 r = r =4 i =0 r =6 r = Figure 6. Subdivided Spiral Architecture i = l =0 l =

4 The origial Spiral Architecture is divided ito 6 regios, which is deoted by r =,,, 6. I each regio, the pixels are grouped i differet levels deoted by l = 0,, alog the radial directio. O each level, each pixel is regarded as a item deoted by i, where i = 0,,, l clockwise as show i Figure 4. So each pixel ca be located by these three parameters, (r, l,, uiquely i additio to the Spiral Address o Spiral Architecture. parameter (,, ) (See Figure 8). The the scalig factor is /3 by formula (3.) (See Figure 9). I Figure 9, we fid that three ear copies are created after the above multiplicatio. This is due to the fact that each copy results from a uique samplig of the iput image. Each sample is mutually exclusive ad the collectio of all such samples represets a partitioig of the iput image. As the scalig i effect represets the viewig of the image at a lower resolutio, each copy has less iformatio. STEP Locate iverse value of multiplier i Figure 6 Suppose the origial image is multiplied by Spiral Address x, its correspodig iverse Spiral Address, y = x, ca be obtaied by the way metioed i sectio. The the locatio parameters, ( r, l, of y ca be gotte accordig to STEP. STEP3 Compute scalig factor Our key cotributio is to develop a formula for computig the scalig factor durig image segmetatio o Spiral Architecture. This formula is show, Scale ( r, l, r =,,...,6; l =,,...; ad i = 0,,..., l. = [ l i( l ) + i( i )] where, (3.) It is foud that scale value is oly determied by the parameters, l ad i. That meas the fial scalig factor is determied by the level umber ad the item umber of the iverse value of multiplier. The oly differeces amog the images derived from Spiral Multiplicatio with the differet multipliers are the rotatio agles. The agle differece is the multiple of 60 degrees. Usig the above algorithm with Spiral Multiplicatio, a image ca be segmeted as required to may parts which are the codesed copies of the origial image o Spiral Architecture. We ca also uiformly separate the origial image to sub-images of certai size based o the requiremet of precisio ad the capacity of processig odes o the etwork for distributed image processig. The experimet results are show i Sectio 4. 4 Experimet Results As a simplified illustratio of our algorithm without loss of geerality, a image cotaiig a up-right arrow ad a image cotaiig a toy duck (See Figure 7) are used here. There are totally 6807 = 7 hexagoal pixels i the Spiral Architecture area idividually. Suppose that the origial image is multiplied by Spiral Address 63. Accordig to Spiral Multiplicatio priciple the iverse value of 63 is. O the refied Spiral Architecture, Spiral Address is located by the r = r =3 (r,l,=(,,) Figure 7. Up-right arrow ad toy duck r = r =4 i =0 r =6 i = r = Figure 8. Parameter, ( r, l,, of iverse value of multiplier, l =0 l = However, as oe of the idividual light itesities have bee altered i ay way, the scaled image still holds all of the iformatio cotaied i the origial. This meas that the computatioal complexity has bee both reduced ad

5 icely partitioed without givig away ay iformatio if distributed processig system is implemeted here. With the help of the algorithm metioed above for computig scalig factor we ca partitio the origial image correctly ad quatitatively o Spiral Architecture based o the practical distributed system performace ad detail requiremets i the real situatio. It is also foud that image rotatio accompaies with image segmetatio, but this kid of rotatig effect will ot affect image processig if image segmetatio here is for object recogitio as the fial goal. Fidig object represetatio ivariat to affie trasformatio ca avoid such rotatig effects. Near copy Near copy Near copy 3 Figure 9. Scalig factor after Spiral Multiplicatio by Spiral Address 63 Coclusio This paper presets the deep research work about Spiral Multiplicatio o Spiral Architecture. A ew way is developed to measure image segmetatio quatitatively o Spiral Architecture by Spiral Multiplicatio. From the experimetal results we see the objective is achieved successfully. It successfully improves the Spiral Architecture s usage i image processig, ad especially i image segmetatio for distributed image processig. 6 Refereces SHERIDAN, P., HINTZ, T., AND MOORE, W. (99): Spiral Architecture i Machie Visio, Australia Occam ad Trasputer Coferece. IOS Press, T. Bossamier, Editor, Amesterdam. SCHWARTZ, E.(980): Computatio Natomy ad Fuctioal Architecture of Striate Cortex: a Spatial Mappig Approach to Perceptual Codig. Visio Research 0, pp SHERIDAN, P., HINTZ, T. AND ALEXANDER, D. (000): Pseudo-ivariat Image Trasformatios o a Hexagoal Lattice. Image ad Visio Computig 8 (): SHERIDAN, P. AND HINTZ, T. (999): Primitive Image Trasformatios o a Hexagoal Lattice. Charles Sturt Uiversity, Bathurst, NSW, Tech. Rep., December. ALEXANDER, D. (99): Recursively Modular Artificial Neural Network. PhD thesis. Macquire Uiversity, Australia. BHARADWAJ, V., LI, XIAOLIN AND CHUNG KO, CHI. (000): Efficiet Partitioig ad Schedulig of Computer Visio ad Image Processig Data o Bus Networks Usig Divisible Load Aalysis. Image ad Visio Computig 8 : SHERIDAN, P. (996): Spiral Architecture for Machie Visio. PhD thesis. Uiversity of Techology, Sydey. HE, X., HINTZ, T. AND SHERIDAN, P. (996): Object Recogitio with Spiral Architecture. Proc. 3 rd Australasia Cof. o Parallel ad Real-time Systems, Brisbae, Australia, HE, X. (999): D-object Recogitio with Spiral Architecture. PhD thesis. Uiversity of Techology, Sydey. Australia. REMARK I order to make Spiral Architecture practically workable o the existig image capture device, mimic Spiral Architecture is used i the research work (He 999). Due to a few differeces betwee real Spiral Architecture ad mimic Spiral Architecture, a little distortio is itroduced ito the image durig image rotatio (See Figure 7, Figure 9). But it does ot affect the theoretical research about Spiral Architecture. Surely, this distortio will be resolved with the developmet of image capture hardware device.