Novel Artificial Neural Networks For Remote-Sensing Data Classification

Transcription

1 ovel Artfcal eural etwors For Remote-Sensng Data Classfcaton Xaol Tao * and Howard E. chel ξ Unversty of assachusetts Dartmouth, Dartmouth A 0747 ABSTRACT Ths paper dscusses two novel artfcal neural networ archtectures appled to mult-class classfcaton problems of remote-sensng data. These approaches are a spng-neural-networ model for the parttonng of data nto clusters, and a neuron model based on complex-valued weghts (CV. In the former model, the learnng process s based on the Spe Tmng-Dependent lastcty rule under the Hebban Learnng framewor. Wth temporally encoded nputs, the synaptc effcences of the delays between the pre- and post-synaptc spes can store the nformaton of dfferent data clusters. Wth the encodng method usng Gaussan receptve felds, the model was appled to the remote-sensng data. The result showed that t could provde more useful nformaton than usng tradtonal clusterng method such as -means. The CV model has proved to be more powerful than tradtonal neuron models n solvng the XOR problem and mage processng problems. Ths paper dscusses an mplementaton of the complex-valued neuron n RBF neural networs to mprove the RBF structure. The complex-valued weghts are used n the supervsed learnng part of an RBF neural networ. Ths classfer was tested wth satellte mult-spectral mage data and results show that ths neural networ model s more accurate and powerful than the conventonal RBF model. eywords: ormalzed Radal Bass Functon (RBF, Spng eural etwor, Remote-sensng data classfcaton.. ITRODUCTIO The study of varous technques for the classfcaton of remote-sensng data s an mportant topc n dgtal mage processng. Usng these methods, people can obtan thorough understandng of the types of land coverage n the area of study. Ths s sgnfcantly valuable to mprove the envronmental condton and preserve natural resource. However, the conventonal statstcal models are not suffcent to descrbe the remote sensng data because of ther complcated structures and propertes. Wdely used as powerful learnng machnes, neural networs offer extraordnary opportuntes to study the mult-dmensonal satellte mage data. For such reasons, we bult a collecton of neural networ models to descrbe the nature of satellte mages and appled them to the classfcaton of these mages. Ths paper s a summary of these mathematcal methods we developed on such modelng and classfcaton wor. We explored varous deas to deal wth the lmtatons of the exstng neuron models. We not only proposed an unsupervsed neural networ model but also a supervsed neural networ model to address the classfcaton problems. We frst proposed an unsupervsed spng neural networ model to study the remote sensng data. Ths model represents nformaton n the artfcal neuron usng explct neuron frng tmes rather than frng rate. It s a smplfed model of the bologcal neuron, and the descrpton s more realstc than tradtonal networ models. In ths model, the nternal characterstcs of nput patterns are stored n the connectng effcences for each delay between the presynaptc neuron and the post-synaptc neuron. In the system dentfcaton stage, we harnessed the wdely used STD rule. Ths rule provded a way to represent the nternal relatonshp between the delays and the nput patterns. It ensured that those delays whch carry the characterstc of the nput pattern have large effcences. Usng the overlappng Gaussan receptve felds n the encodng of the nput patterns, the spng neural networ was tested on the satellte mage data. The classfcaton results demonstrated that the spng neural networ can acqure more accurate nsghtful nformaton of the nput patterns than the tradtonal -means method. In summary, ths new model provdes an effcent approach to nterpret and classfy the data n a straghtforward manner. * g_xtao@umassd.edu; phone ; FAX ; Electrcal and Computer Engneerng Department, Unversty of assachusetts Dartmouth, 85 Old Westport Road, orth Dartmouth, A 0747; ξ hmchel@umassd.edu; phone ; FAX ; Electrcal and Computer Engneerng Department, Unversty of assachusetts Dartmouth, 85 Old Westport Road, orth Dartmouth, A 0747;

2 The classfcaton algorthms can be supervsed f the prototype samples are labeled by vrtue of ground truth. Based on the paper, we proposed a new supervsed neural networ model for classfcaton. The tradtonal L (ult-layer erceptron model has been used n the past to do the classfcaton wor. However, t nvolves a complcated bacpropagaton tranng process. To avod ths problem, we proposed to use the RBF neural networ as the basc supervsed networ model n our wor. A RBF neural networ has also a two layer feed-forward structure, whle the tranng process s dvded nto two stages. Ths new approach greatly decreases the computatonal complexty. The RBF neural networ not only eeps the propertes of the RBF neural networ (for example, the effcency, but also converts the localzed behavor of the radal bass neuron to a non-localzed one. Based on the structure of RBF neural networ, we proposed a new complex-valued neuron model. The power of ths complex-valued neuron was frst demonstrated n solvng the XOR problem wth only a sngle neuron. In our complex-valued neural networ model, complex weghts were mplemented between the hdden layer and the output layer. Compared wth the real-valued neuron, the sngle complex-valued neuron obvously has a more powerful structure wth only a slght ncrease n the computatonal cost. Consderng that the nput and output are all real values n our wor, we added another layer to compute the magntudes of the output from the RBF neural networ. The expermental results showed that the complex-valued RBF neural networ can sgnfcantly mprove the classfcaton performance of the real-valued one.. REOTE SESIG DATA Landsat T records data n seven dfferent bandwdths that are broen nto dfferent spectral regons of vsble, nfrared, and thermal nfrared. Table s the band descrptons. Band Wavelength ( µ m Resoluton (meters Spectral Regon Vsble Blue Vsble Green Vsble Red Reflectve Infrared d-infrared Thermal Infrared d-infrared Table T Band Descrpton Snce 97 satelltes have provded hgh-resoluton multspectral magery usng hgh technology. The T bands have been selected to maxmze ther capabltes for detectng dfferent types of Earth resources. The study area used n our wor s a satellte mage of ew England. Ths data was obtaned through Landsat 7 ET+ sensors on July 7 th, 999. The number of pxels of ths data set s 6600*6000 for band -7 except band 6 whose number s 3300*3000 because of ts dfferent resoluton. In the followng experment, we use all of the bands except band 6. We extracted the tranng samples and the testng samples from the data set based on remote sensng expert nowledge. The number of the tranng samples used s 50. The number of the testng samples used s 448. In order to avod over-fttng, these samples are extracted from the dfferent postons for each cluster. 3. Archtecture of Spng eural etwor 3. SIIG EURAL ETWOR In ths subsecton, we brefly revew the archtecture of a spng neural networ and the temporally encodng of real nput data. A spng neural networ s a one layer feed-forward neural networ. We chose the smplest spe response model (SR as shown n Fgure (a. In order to smplfy the model and mae t easy to understand, we assumed that before a neuron generates a spe, t has been at ts restng state for a suffcently long tme so that the bac propagaton acton potental s neglgble. Also, n one learnng cycle, the spe neuron can fre at most once. Therefore, the

3 electrcal membrane potental of a post-synaptc neuron s the lnear weghted combnaton of the ernel functon ε (s whch models the spe response to the pre-synaptc spes. The weghts ndcate the effcacy of each synapse. When the membrane potental s over the threshold, the neuron wll fre a spe, and then be reset to ts restng potental. (a Smplfed SR odel. (b Enlargement of Each Connecton Fgure Archtecture of Delayed Spng eural etwor Fgure and Equaton ( represents one of the most popular mathematcal spe response models. s / τ exp( s / τ f s > 0 ε ( s = ( no response else Epson Functon 0.8 embrane otental Tme [ms] Fgure Spe Response Functon, ost-synaptc otental s Exctatory (ES The nternal state of a post-synaptc neuron can be descrbed as n Equaton (, where s the number of the presynaptc neurons. u ( t = w ε ( t t ( = Ths spng neural networ s a delayed neural networ as ntroduced by atschläger for the unsupervsed clusterng n hs wor. 3 Between each par of pre-synaptc and post-synaptc neurons, there are multple tme delays for each connecton advsed by Hopfeld. 4 Fgure (b s an enlargement of ths connecton. The delay ndcates the dfference

4 between the frng tme of the pre-synaptc spe and the arrval tme of the spe to the post-synaptc neuron. Each delay has ts own synapse effcacy. Therefore the nternal state of a post-synaptc neuron can be descrbed as n Equaton (3. T = = u ( t = w ε ( t t d (3 In order to encode the nput patterns temporally we can lmt the value of the nput varables n a range [0, T] whch s also referred to as the codng nterval. The value of the delay for each connecton s n the same range and the step + ncrease s d d =. Then, each nput pattern can be encoded wth the frng tme of the nput neurons as t = T x. In ths model, the number of the pre-synaptc neurons s the dmenson of the nput vector. Each postsynaptc neuron ndcates one cluster of the nput patterns. 3. System Identfcaton ethod Our goal s that after the system dentfcaton (.e. learnng all the parameters n the model, the spng neural networ can do clusterng by usng the frng tme of post-synaptc neurons assocated wth each nput pattern. Each postsynaptc neuron represented one cluster and fres earler than other post-synaptc neurons for nput patterns belongng to ths cluster. We adopted a wnner-tae-all approach for the post-synaptc neurons to mae sure only one spng neuron fres n each nput pattern. For each nput vector, as soon as a post-synaptc neuron generates a spe, all the others wll be nhbted. oreover, only the synapse effcacy related to ths post-synaptc neuron ust fred wll be changed accordng to the learnng algorthm. Here, we chose Hebban Learnng rule through Spe Tmng-Dependent lastcty (STD as our learnng algorthm. 5,6 Ths learnng method, proposed by Donald Hebb n hs boo enttled The Organzaton of Behavor'', s referred to hereafter as the Hebb's Learnng Rule". It s the earlest and smplest learnng rule for a neural networ. It can be smply stated as When an axon of cell A s near enough to excte a cell B and repeatedly or persstently taes place n frng t, some growth process or metabolc change taes place n one or both cells such that A s effcency, as one of the cells frng B, s ncreased. 7 In our STD learnng method, we ncorporated synaptc depresson n order to weaen the synaptc effcacy where the pre-synaptc neuron fres after the post-synaptc neuron. t d t pre t t post Fgure 3 Tme Chart for Spng In Equaton (4, G represents a generalzed learnng equaton, where t s the frng tme of the pre-synaptc neuron, t post s the frng tme of the post-synaptc neuron and d s the delay tme. Fgure 3 s the tme chart for the spng neuron. d dt w ( t = G( t, t, d (4 post re-synaptc neuron emts a spe at t wth delay and generates a spe at t post d, the post-synaptc neuron receves the synapse at t = t + d pre

5 Fgure 4 Learnng Functon W ( t where t = t post t pre Under the STD rule (Fgure 4, the synaptc effcacy tends to change n two drectons, ether toward zero or toward the maxmum weght. 5 If the synapse occurs before the post-synaptc spe, t wll be ncreased; otherwse, decreased. Ap exp( t / τ p W ( t = Ad exp( t / τ d f t < 0 f t > 0, (5 where t = t + d t post, W ( t s a general representaton of how the connectng weght changes. In Equaton (5, A p and A d are the maxmum changes caused by strengthenng or weaenng synaptc effcacy n every learnng cycle. In our system dentfcaton, the learnng wndow s asymmetrc as n B s wor. 8 It mples that τ p andτ d have dfferent values. In ths model, the asymmetrc property s essental snceτ p andτ d ndcate the range of how synaptc effcacy strengthenng or weaenng taes place. STD learnng rule ensures that only those synapses that reach the post-synaptc neuron shortly before t emts a spe are greatly ncreased. Those synapses that reach t much earler merely ncrease nsgnfcantly or stay the same, and those that reach t after are decreased. Addtonally, the range of the weghts must be bounded; otherwse the model wll not be stable. In order to constran the weghts, we can choose ether hard-bound or soft-bound wth certan saturaton functons. After each learnng cycle, only a few delays wll obtan large weghts, whle most of them wll be close to the mnmum weght or stay the same. Therefore, for each spng neuron, all the delays wll be changed n dfferent drectons adaptng to the propertes of the nput patterns that actvated ths neuron. These changes can preserve the propertes of each data cluster for future use. The model was frst tested n some smulaton tests and obtaned good results. 9 But we found that the basc spng neural networ could not perform well f the nput dmenson s too small. Ths suggests that the hgh dmensonal data may be more approprate to ths networ model. However, the remote sensng data only have 7 bands n total. We excluded band 6 n our analyss because of ts relatvely low resoluton. ractcally, the reflectance data may not be avalable for all bands n each recordng. In order to study the classfcaton performance of dfferent band combnatons, subsets of the 6 bands need to be tested. Ths could lead to the nsuffcent dmenson problem for the networ model. Here, we utlzed the encodng method smlar to the Gaussan mxture model n Fgure 5. We used several overlappng Gaussan receptve felds to encode each nput value to a vector 0, so the total dmenson of the nput vector s the sum of all the numbers of the nput data multplyng the number of the Gaussans.

6 p = {0,0,3,4,6,4,0,0,0,0} Fgure 5 Encodng a Value to a Vector wth Overlappng Gaussans Accordng to Fgure 5, we could easly encode an nput value to a vector. For example, we have an nput value p (shown n the fgure. The dashed lne ndcates ts responses for all Gaussans. We assumed those responses whch are below a predefned threshold are not gong to fre n the spng neural networ. Therefore, t s relatvely easy to get the encoded vector = {0,0,3,4,6,4,0,0,0,0} for the nput value p. The number of Gaussans s up to the necessty. The more Gaussans there are, the more expensve the computaton wll get, whle they can provde more nformaton wth a low nput dmenson. We appled ths encodng method to the remote sensng data. Here, we normalzed the orgnal data to the range [0, C] by usng the followng equaton, X X mn( X = C, (6 max( X 443 mn( X 443 where s the band number, s the number of total samples, s n the range {,, L, }, and C s a postve constant. In order to show that ths spng neural networ performs better, we compared ther classfcaton results wth tradtonal -means clusterng method, and recently developed spectral clusterng method. Table shows the partal result wth dfferent unsupervsed classfcatons. We can see that the spng neural networ, though not perfect, can partally classfy the data. In the table, Answer represents the number for each class extracted from the samples. Result represents the number of samples that belong to ths class. The shaded area shows the spng neural networ acheved a better classfcaton than the other two methods. 4. RBF neural networ 4. COLEX-VALUED RBF EURAL ETWOR RBF neural networs have a strong bologcal bacground. In the feld of the bran cortex, local regulated and folded receptve feld s the characterstc of the reflecton of the bran. Based on ths characterstc, oody and Daren, proposed a new neural networ structure, whch s referred to as RBF neural networ. Fgure 6 shows the basc topologcal structure of RBF neural networ.

7 means-band3457 Answer Answer Answer3 Answer4 Answer5 Result Result Result Result Result Spectral-band3457 Answer Answer Answer3 Answer4 Answer5 Result Result Result Result Result Spng-Band3457 Answer Answer Answer3 Answer4 Answer5 SIE EURO SIE EURO SIE EURO SIE EURO SIE EURO Table artal Result of Three Technques Fgure 6 Topologcal Structure of RBF eural etwors Based on the RBF neural networ structure and the chosen radal bass functon, f f l ( X s the lth output of the output layer, and φ X s the output of th radal bass functon, then the whole networ forms a mappng: ( l = r f ( X φ = l ( X λ, (7 where X s an -dmensonal feature vector; Y s the actual output vector correspondng to the nput vector X ; r s the number of the hdden unts and λ s the connecton weght between the th hdden unt and lth output unt. l Ths weght shows the contrbuton of the hdden unt to the correspondng output unt. In some approaches, the output of the hdden layer s normalzed by the sum of all the radal bass functon components ust as n the Gaussan-mxtures estmaton model. 3 The followng equaton shows the normalzed radal bass functons:

8 Φ ( X = φ r = ( X c φ ( X c (8 A networ obtaned usng the normalzed form for the bass functons s called a normalzed RBF neural networ (RBF and has some mportant propertes. Ths normalzed form bounds the hdden output n the range between 0 and, whch can be nterpreted as probablty values to ndcate whch hdden unts s most actvated n classfcaton applcaton. 4 oreover, we modfed the localzed behavor to non-localzed behavor whch provdes the rght decson to all nput vectors wth ths normalzaton. Usng the above normalzed form, we let fl ( X be the lth output of the output layer, and Φ ( X be the normalzed radal bass functons. Therefore, the equaton ( can be changed to the followng one: 4. Complex-valued RBF neural networ l = r = l ( X f ( X λ Φ (9 Here, we demonstrate that the complex-valued RBF neural networ can be appled to the classfcaton of the remote sensng data. As n paper, the output of each class s the lsted Table 3: Class Output Vegetaton Sol (sparse vegetaton Urban area Deep water Shallow water Table 3 Output Table of Each Class An advantage of the complex-valued RBF networ s that the output of the hdden layer s always a postve number. Snce we have normalzed the nput vector, the nput vector s always wthn the range [0, ]. As for the output, snce we used the defnton as n Table 3, t s more approprate to use the magntude of the output from the hdden layer to represent the output of the networ. It s necessary to add another actvaton functon n the output layer as n Fgure 7. Re(x +Im(x Re(x +Im(x φ φ Re(λ Im(λ Im(λ Re(λ Re( f (X g ( X = f ( X Re(λ r Im(λ r Im( f (X Re(x +Im(x φ Fgure 7 Schematc of Improved Complex-valued RBF

9 The frst stage of the complex-valued tranng s the same as the tradton RBF neural networ. In our experments, the -means clusterng method was tested at ths stage. Gaussan functons are fast decayng functons so that not all bass functon unts contrbute sgnfcantly to the networ output. In our wor, they were used as the radus bass functon (RBF. After the output of the hdden unts s obtaned, only the connectng weghts between the hdden layer and the output layer should be traned. Ths modfcaton of the structure changed the lnear relatonshp between the hdden layer and the output layer to non-lnear. Assume we have a group of nput vectors { X X C, =,, L } values { Y Y R, =,, L } functon as error functon: and ther mappngs are real, where s the dmenson of the nput vector. We defned the followng energy E = = ( Y f ( X, (0 where f ( X s the output from the networ of each correspondng nput vector X. As we have dscussed, a varable n the complex doman ncludes a magntude and a phase. In the complex-valued RBF neural networ, the normalzed output of the hdden unts for each nput vector X can be denoted as follows (for general use, we eep the hdden output n a complex-valued formula n case t s necessary to use the complex-valued number n the hdden layer. : ϕ Φ ( X = λ e ( Addtonally, the weghts between the hdden layer and the output layer can be denoted as θ W = e, where the magntude of the weght s because the output s usually normalzed. Ths defnton shows one advantage of the complex-valued weghts because durng the learnng process, only the phase s changed. Ths mples that we do not need any boundary lmtaton rule for the weghts whch s necessary n real artfcal neural networ. Usng the gradent descent method to mnmze the error sgnal, we can change the phase of each connectng weght through dervaton: ( E θ = η = η From the archtecture of RBF neural networ, we had the detaled expresson of the output before usng the actvaton functon. ( X = Φ ( X = f W = = ϕ ( Y θ f ( X ( θ ( λ e e = λ e, (3 = = where s the number of the hdden unt n the networ, and η s the learnng step. Expandng (3, we had (4 f = ( λ cos( θ + λ sn( θ = ( X (4 luggng (4 nto (, we had Let E θ = η = η { ( Y [ λ cos( θ ] + [ = = = λ sn( θ ] }. (5

10 Θ = [ cos( θ ] + [ = = λ λ sn( θ ]. (6 Substtutng (6 n (5, E θ = η = η = η = η = = ( Y ( Y = ( Y ( Y Θ Θ ( Θ Θ Θ Θ, (7 where ( Θ = = ([ = λ λ cos( θ cos( θ ] + [ = [ λ ( sn( θ ] + = = λ sn( θ ] λ sn( θ [ λ cos( θ ] Fnally, we had the followng full expresson formula for the change of phase for each connecton weght: E θ = η = η ( = η = = λ cos( θ Θ λ ( m( f Θ λ ( sn( θ + = = Y Y Θ Θ ( X cos( θ re( f λ sn( θ ( X sn( θ (8 λ cos( θ The least mean square error algorthm does not guarantee convergence to globally optmal networ parameters. However, t does appear to converge to reasonable solutons n practce. (9 4.3 Smulaton Results In ths subsecton, we explored ths complex-valued RBF model n the classfcaton of the remote sensng data. We obtaned the tranng and testng accuracy from both real-valued and the complex-valued RBF neural networs. The results are lsted n Table 4. Intutvely, we also showed them n Fgure 8. Here, we compared the classfcaton results of these two methods. We used the same collecton of subsets of all the bands as the nput for the neural networs. From Table 4, we can see that the complex weghted RBF neural networ can mprove the classfcaton result n most cases. umercally, n the randomly selected 3 combnatons, the complex-valued obtaned hgher accuracy n, and lower accuracy n. To statstcally compare the results, we used the sgn test by tang the null hypothess rob(the complex-valued method are more accurate = 0.5. Under ths hypothess, we got p-value = 5.5x0 -. The null hypothess wth such a low p-value should be reected. Ths statstcal result demonstrated that the complex-valued RBF model performs sgnfcantly better than ts real-valued counterpart. ote that for a few band combnatons, for example, bands, 3 or bands 4, 5, 7, the complex-valued method even provded evdently hgher classfcaton accuracy.

11 00 95 c y(% ra A ccu Complex-valued meanstestng means Testng Test Tmes Fgure 8 Real-valued RBF and Complex-valued RBF Table 4 Real-valued RBF and Complex-valued RBF

12 5. COCLUSIOS In ths paper, we developed a collecton of new clusterng methods usng several neural networ models. These methods are focused on the practcal applcatons of classfcaton of remote sensng data. Our wor focused on the classfcaton of a satellte mage of ew England. Ths s a very mportant research feld n envronment management and protecton. The dstrbutons of all land coverage are very heterogeneous, and t s hard to study ther propertes under any parametrc framewor. In our wor, we proposed to descrbe the dstrbutons usng these powerful artfcal neural networ models. We frst descrbed the development of a spng neural networ model whch encodes the nput data n temporal space. Ths s a new unsupervsed classfcaton technque. Usng tmng nformaton, ths model s a more bologcal accurate neural system than tradtonal artfcal neurons. The parameters n the model were estmated usng the STD rule whch s a robust learnng method converged effcently. We appled ths spng neural networ to the classfcaton of the remote sensng data. Expermental results showed that t can provde more useful nformaton than tradtonal clusterng methods. Secondly, we descrbed the Complex-valued RBF. The complexvalued RBF neural networ utlzed the characterstcs of complex numbers and represented the weghts n the complex doman. We demonstrated that these complex weghted RBF neural networs provded sgnfcant advantages n processng the remote sensng data. It not only ept the structure and effcency of the RBF neural networ but also mproved the computaton power. Ths suggested that ths new model s an effcent and effectve method for classfcaton of mult-spectral satellte mage data. Our quanttatve study showed that the proposed methods obtaned more accurate classfcaton than the tradtonal/exstng methods. It can also overcome the computaton complexty problem that has occurred n other wor. All these wors can be drectly appled to varous research areas. They can also be mplemented n other neural networ structures. REFERECES. X.. Tao, and H. E. chel, "Classfcaton of ult-spectral Satellte Image Data Usng Improved RBF eural etwors," roceedngs of SIE Vol. #567-4, Robert A. Schowengerdt, Remote Sensng: odels and methods for mage processng. Academc ress, T atschläger, & B. Ruf (998. Spatal and temporal pattern analyss va spng neurons, etwor: Computaton n eural Systems, 9(3: J. J. Hopfeld (995. attern recognton computaton usng acton potental tmng for stmulus representaton, ature, 376: S. Song,. D. ller, & L. F. Abbott ( 000. Compettve Hebban learnng through spe tmng-dependent synaptc plastcty, ature euroscence. 3: W Gerstner, & W.. stler (00. athematcal formulaton of Hebban learnng, Bologcal Cybernetcs. 87: D. Hebb, (949. The organzaton of behavor. ew Yor: Wley. 8. G. Q. B, &.. oo (998. Synaptc modfcatons n cultured hppocampal neurons: dependence on spng tmng, synaptc strength, and post synaptc cell type, Journal of euroscence. 8: X.Tao, and H. E. chel, "Data Clusterng Va Spng eural etwors Through Spe Tmng-Dependent lastcty," roceedng IC-AI'04, June -4, 004, Las Vegas, evada, USA. 0. S.. Bohte, H. L. outre, and J.. o (00 Unsupervsed Clusterng Wth Spng eurons by Sparse Temporal Codng and ult Layer spe eural etwor. IEEE Transactons on eural etwors, Vol: 3 Issue:, arch 00 age(s: oody J, Daren C. Fast Learnng n etwors of Locally-turned rocessng Unts eural Computaton, 989 (: 8-94, oody J, Daren C. Learnng wth localzed Receptve Feld roc 988 Connectonst odels Summer school, Dtouretzy, G Hnton, and T Senows(Eds. Carnege ellon Unversty, organ aufmann ublshers, Cha, I., assam, S. A. RBF restoraton of nonlnearly degraded mages IEEE Trans. On Image rocessng, vol. 5, no. 6, pp , Robert J. Howlett and Lahm C. Jan. Radal Bass Functon etwors /, hysca-verlag Hedelberg 00.