Image Segmentation and Compression using Neural Networks

Transcription

1 Image Segmentaton and Compresson usng Neural Networks Constantno Carlos Reyes-Aldasoro, Ana Laura Aldeco Departamento de Sstemas Dgtales Insttuto Tecnológco Autónomo de Méxco Río Hondo No. 1, Tzapán San Angel, Méxco D.F. ABSTRACT Kohonen [1] has developed an algorthm wth self-organsng propertes for a network of adaptve elements. These elements receve an nput sgnal and the sgnal representatons are automatcally mapped onto a set of output responses so that these responses acqure the same topologcal order as the nput sgnal. Images can be used as nput sgnals and the networks can adjust to extract certan topologcal features. Image segmentaton can be performed satsfactorly. By emprcal knowledge, t can be supposed that as the number of neurones ncreases, so does the qualty of the segmentaton. Ths paper concentrates on the relatonshp between the qualty of segmentaton and the number of neurones that consttute a Kohonen Neural Network. Several experments were conducted and the Eucldean dstance between adjacent neurones measured the qualty of the segmentaton, whch tended to mantan constant after a certan optmum level. The amount of nformaton of the orgnal set of mages was compared wth the nformaton of the segmented structure and results were presented. Compresson rates hgher than 250:1 were obtaned. Keywords: Neural Networks, Self-Organsng Maps, Image Segmentaton. 1. Introducton Artfcal Neural Networks are software or hardware systems that try to smulate a smlar structure to the one that s beleved the human bran has. Most neural networks n the bran, especally n the cortex, are formed by two-dmensonal layers of cellular modules that are densely nterconnected between them. Ths area of the bran s organsed nto several sensory modaltes such as speech or hearng. The response sgnals of these areas are obtaned n the same topographcal order on the cortex n whch they were receved at the sensory organs. The theoretcal nvestgatons n the self-organsng maps (SOMs) [1] were motvated by the possblty that the representaton of the knowledge n a partcular category of thngs n general mght assume the form of a feature map that s geometrcally organsed over a part of the bran. In ths neural model, each neurone or node s densely nterconnected wth the rest of the neurones. The temporal status of a neurone, as well as the nput sgnal, s represented by ts topologcal poston x, y, z. The nterconnecton of neurones s consdered as a lateral couplng. The functon that defnes the couplng has two dfferent actons: exctatory and nhbtory. The exctatory nteracton exsts n a regon defned by a short range up to a certan radus wth the neurone as a centre, and the nhbtory regon surrounds the exctatory area up to a bgger radus. Outsde the nhbtory range, a weaker, and much bgger exctatory zone exsts. The ntensty of the acton decreases as the dstance from the neurone ncreases. A cluster or bubble, called the neghbourhood, around one partcular node of the network s formed because of the lateral couplng around a gven cell. The prmary nput receved by the network determnes a "wnner" neurone. Around ths wnner neurone, exctatory and nhbtory regons wll form. The wnner node wll adapt to the nput sgnal and then the neurones that le wthn exctatory and nhbtory regons wll adapt

2 themselves accordngly. Ths process of adaptaton wll contnue for a number of teratons untl a certan degree of adaptaton s reached. When the nput s an mage, certan features can be extracted from the fnal adaptaton of the neurones. The remanng of the document s organsed as follows: the next secton descrbes the mplementaton of the algorthm. Secton 3 deals wth the experments over medcal mages, n secton 4 a defnton of the qualty n terms of the number of neurones s presented, secton 5 dscusses the compresson rate of the algorthm. Fnally, conclusons are presented. 2. Implementaton of the Kohonen Algorthm The Self-Organsng algorthm proposed by Kohonen [1] follows two basc equatons: matchng and fndng the wnner neurone determned by the mnmum Eucldean dstance to the nput (1) and the update of the poston of neurones nsde the cluster (2). x( m ( = m n m ( t + 1) = m ( + α( m ( t + 1) = m ( c x( m ( [ x( m ( ] N c N c (1) (2) Where, for tme t, and a network wth n neurones: x s the nput N c s the neghbourhood of the wnner, 1< N c <n α s the gan sequence 0<α<1 m s any node, 1<<n, and s the wnner, m c It should be noted from equaton (2) that the nhbtory regon s not beng consdered and the ntensty of the acton nsde the exctatory regon was consdered constant. The orgnal "Mexcan Hat" functon was reduced to a gate functon wth satsfactory results (see Fgure 1). Acton Lateral dstance Fgure 1 Lateral degree of nteracton: Mexcan Hat and Step Functons Fgure 2 descrbes graphcally the Self-Organsng algorthm. Frst, an nput sgnal x, s receved and the network determnes a "wnner" neurone by calculatng the Eucldean dstances wth equaton 1. The updatng process of equaton 2 s a varaton of the topologcal locaton of the neurone, proportonal to the Eucldean dstance from the wnner node to the nput. The gan sequence α s a value between 0 and 1 that reduces wth tme. In Fgure 2.a only the wnner neurone adapts to the nput sgnal and n Fgure 2.b other neurones that le wthn the

3 neghbourhood of the wnner adapt ther topologcal co-ordnates. Neurones outsde the neghbourhood reman unaltered. Ths process s reterated untl certan crteron s satsfed. (a) N c ( (b) m c ( N c ( m c (t+1) x( m c (t+1) Fgure 2 Graphcal descrpton of Self-Organsng process x( Kohonen [1] stated that the neghbourhood should be shrnkng n tme and α s a lnearly decreasng functon and the process stops when α=0. Ths behavour allows a fast and coarse adaptaton at the begnnng of the process and a fne and slow adaptaton at the end. Fgure 3 shows an n-64 network (8 by 8) adapted to a square nput regon after 4000 teratons. The ntal values of the neghbourhood and the gan sequences and ther varaton wth tme are studed n [2]. Fgure 3. Adaptaton of an 8*8 network to square nput regon. 3. Experments over medcal mages Medcal Images have receved consderable attenton n several areas, beng segmentaton one of the most nterestng ones [3]. The SOM algorthm can be related wth medcal mage segmentaton n the followng way. The nput sgnal receved bye de SOM n fgure 3 was a smple square regon wth equal probablty for every poston. If an mage s to be used as nput sgnal for a self-organsng algorthm, a probablty and a weght can be assgned to each pxel of the mage.

4 Fgure 4 shows a Magnetc Resonance (MR) mage of a human head. Ths mage can be converted to a two dmensonal matrx where for each x, y poston, a z value s assgned accordng to the grey level ntensty of the current pxel. Ths transformaton allows the mage to be used as nput for a SOM. As the map receves the nput, a wnner s selected and then neghbourhood s updated to the mage. Ths process can present nterestng segmentaton results of dfferent structures of the mage. Fgure 4 Human head Magnetc Resonance Image As an example of segmentaton, The mage n fgure 4 s pre-segmented by grey level and then an annular SOM wth 80 neurones s used to segment the surface of the mage. The result s presented n Fgure 5. Fgure 5 Segmentaton of Magnetc Resonance mage of fgure 4. It can be noted, that the number of neurones play a crtcal role n the segmentaton process, and ntutvely, as the number of neurones ncreases, the dstance between them s reduced and therefore, the qualty of the segmentaton s also ncreased. In [4] a MR mages database of a human head s used to extract the border of the mages,.e. the shape of the head. In fgure 6 the segmentaton of 54 slces of a human head MR mages s shown, for each slce, 58 neurones are used n the segmentaton.

5 Fgure 6 Segmentaton of MR mages of a human head If the prevous ponts are used to reconstruct the surface of the head, several problems arse. The algorthm presented n [5] s used to obtan the surface defned by the 3D collecton of ponts depcted n fgure 6. Along the surface, several "holes" are observed due to the lack of nformaton n those specfc postons. Ths lack of nformaton s caused by the absence of a neurone n a crtcal x, y, z poston that could have been avoded f more neurones would have been used n the segmentaton. Fgure 7. Reconstructon of surface defned by the ponts n fgure 6 shown as shaded mage It s obvous that addng neurones to the SOM wll ncrease the computatonal complexty to the process defned by equatons (1) and (2), therefore the number of neurones should not be ncreased at wll. Indeed, the queston arses, s there a lmt n the qualty of the segmentaton as the number of neurones s ncreased? The next secton studes the segmentaton qualty, dependng on the number of neurones. 4. Number of neurones and qualty defnton The algorthm of the SOM that was presented n secton 2 depends on a seres of neurones that wll self adapt to a certan nput sgnal. As the neurones adapt to a sgnal, so does the Eucldean

6 dstance between the neurones. Ths dstance can defne the qualty of the fnal segmentaton as t can be seen on fgure 8. The fgure presents the fnal state of 4 SOMs wth dfferent number of neurones; 10, 20, 40 and 80. It can be observed that as the number of neurones ncreases, the shape defned by the neurones resembles better the shape of the human head as presented n the MR mage of fgure neurones 20 neurones 40 neurones 80 neurones Fgure 8 Segmentaton obtaned by SOM wth dfferent number of neurones The SOM wth 80 neurones s evdently better than the one wth 10, but the dfference between 80 and 40 neurones s not so clear, therefore, an analytc defnton of the qualty should be used. As the network adaptaton s closer to the nput sgnal, and the number of neurones ncreases, the dstance between neurones wll decrease. Ths dstance between adjacent neurones wll be used as a qualty parameter followng: d adj max = max m ( m+ 1( (3) Three dfferent MR mages were used as nput sgnals and equaton (3) was appled to the segmentatons obtaned wth SOMs of dfferent number of neurones. The experment was repeated numerous tmes to obtan an average measurement for each mage. For all the experments the parameters; Iteratons = 15000, α(0) = 0.2, and N c = 0.4 n were constant. In the three cases, the maxmum dstance between adjacent neurones tended to decrease asymptotcally as the number of neurones ncreased. The fnal number of neurones wll depend on the partcular mage, but n all cases a lmt value seemed to be obtaned closer to 300 neurones, after ths regon, the number of neurones make no dfference n the qualty of the segmentaton.

7 Maxmum Adjacent Dstance Neurones Image 1 Image 2 Image 3 Fgure 9 Maxmum dstance between adjacent neurones 5. Compresson Rate The process of segmentaton of a certan structure of the mage, as the external shape of a human head n the past examples, mples a loss of nformaton. The segmentaton delberately looses the nformaton that corresponds to all the nternal structures such as the bran, cerebellum, and ventrcles and retans the poston of the external contour. Nevertheless, through ths process, the amount of nformaton correspondng to the x, y, z postons of the neurones s consderably less than the orgnal mage. 10,000 1, Compresson rate Neurones / slce The mages consst of 512*512 pxels, each wth 256 levels of grey, or 8 bts/pxel. Each slce of the MR mples therefore: I mage = 512*512*8 = 2,097,152 [bts].

8 And the whole set of 54 MR mages of a human head: I head = 512*512*8*54 = 113,246,208 [bts]. The resultng SOM wll requre n turn the x, y, z poston for each neurone or 9 * 3 = 27 bts/neurone. Fgure 6 has 54 SOMs, each wth 58 neurones, 84,564 [bts], a compresson rate closer to 1340:1. Evdently, the orgnal set of mages contan the whole head, but at ths rate, other 1300 smlar structures could be segmented and treated as a sngle database wth the same amount of nformaton as the orgnal set of mages. Fgure 10 presents the relatonshp of the compresson rate and the number of neurones. If a number of 310 neurones s consdered as the optmum for the adjacent dstance, the compresson rate would be 250:1. 6. Conclusons The Kohonen Self-Organsng Algorthm was programmed to use medcal mages as nput sgnals. The use of an annular Self-Organsng Map allowed segmentng the external shape of a human head out of database of Magnetc Resonance mages. Through ths process of segmentaton, the nformaton descrbng certan structures of the mage s dscarded thus compressng the nformaton requred to descrbe the segmented structure. Numerous experments were conducted over dfferent mages to determne the qualty of the segmentaton measured as the maxmum dstance between adjacent neurones of a SOM. These experments showed that as the number of neurones ncreased, the dstance decreased up to a certan level where dstance tend to reman constant. An optmum number of neurones wll depend on the partcular type of mage to be processed. As the number of neurones ncrease, so does the amount of nformaton comprsed by the postons of the SOM, and therefore, the compresson rate decreases. The fnal compresson rate wll be determned by the partcular complexty mage to be segmented ts and the qualty desred, but even wth 310 neurones, the compresson rate would be 250:1. The results encourage the use of SOM for mage segmentaton. 7. Acknowledgements Keth A. Johnson and J. Alex Becker from Brgham and Women's Hosptal, Harvard Medcal School, provded the Magnetc Resonance mages, through The Whole Bran Atlas. The authors are grateful to them. Ths work was supported by CONACYT. 8. References [1] Kohonen T (1988). Self-Organzaton and Assocatve Memory, Sprnger-Verlag, Hedelberg. [2] Reyes-Aldasoro CC (1998). A Non-lnear Decrease Rate to Optmse the Convergence of the Kohonen Neural Network Self-Organsng Algorthm, ROCC99 Acapulco, Mexco, pp [3] Kapur T (1999). Model based three dmensonal Medcal Image Segmentaton, Ph.D. Thess, Artfcal Intellgence Laboratory, Massachusetts Insttute of Technology. [4] Reyes Aldasoro, CC, Algorr Guzmán, M.E. (2000) "A Combned Algorthm for Image Segmentaton usng Neural Networks and 3D Surface Reconstructon usng Dynamc Meshes", V IBERO-AMERICAN SYMPOSIUM ON PATTERN RECOGNITION, LISBON, Portugal, September [5] Algorr, M.E., F.Schmtt, (1996) ''Surface Reconstructon from Unstructured 3D Data'', Computer Graphcs Forum, 15(1), pp