A DIMENSIONALITY REDUCTION ALGORITHM OF HYPER SPECTRAL IMAGE BASED ON FRACT ANALYSIS

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1 A DIMENSIONALITY REDUCTION ALGORITHM O HYPER SPECTRAL IMAGE BASED ON RACT ANALYSIS SU Junyng a, *, Shu Nng a a School of Reote Sensng and Inforaton Engneerng, Wuhan Unversty, Luoyu Road 9#, Wuhan, Hube, P.R. Chna, jysu_sjy@sna.co Cosson VII, WG VII/3 KEY WORDS: Hyper spectral, densonaty reducton, spectral doan, fractal analyss, feature age ABSTRACT: A densonaty reducton algorth based on feature extracton of the spectral curve usng fractal analyss whch consderng both the spatal characterstc and spectral characterstc of hyper spectral reote sensng age s proposed A spectral doan feature analyss based on fractal easureent technque s desgned for hyper spectral ages. ractal characterstc of spectral curve s dscussed. A bref descrpton s gven to explan the nonnear echans resultng n fractal of spectral curve. And the spectral curves of sae objects are presented to show the self slar. And soe coputatonal results are gven to show exponental relaton between the total length of spectral curves and the dfferent easureent unt. A fractal denson calculaton algorth of hyper spectral curve s desgned. A nose reove algorth based on wavelet transforaton s done before the fractal analyss of spectral curve. Then a feature analyss procedure n spectral doan based on fractal easureent s also proposed to reduce densonaty of hyper spectral ages. The fractal denson value s taken as the feature of spectral curve and the fractal denson feature age s proposed to represent the densonaty reducton result of hyper spectral age. Experents of fractal denson value of dfferent objects spectral curve show fractal can be used to represent the spectral feature to reduce densonaty of hyper spectral age. nally, the appcaton of fractal easureent of spectral doan feature analyss s brefly dscussed.. INTRODUCTION The characterstcs of hyper spectral reote sensng data are nuerous channels, hgh spectral resoluton and large aounts of data, whch akes t easy to dscrnate objects n the scene, however, the vast aounts of data not only akes t dffcult for transsson and storage, but also feature extracton and classfcaton. Therefore, t s very portant to reduce the denson n the hyper spectral age analyss (C. Lee,993; A. Jan,997). As hyper spectral sensors acqure ages n very close spectral bands, the resultng hgh-densonal feature sets contan redundant nforaton. Consequently, the nuber of features gven as nput to a classfer can be reduced wthout a consderable loss of nforaton. Densonaty reducton can general fall nto feature extracton and band selecton. eature selecton technques generally nvolve both a search algorth and a crteron functon whle feature selecton s usually done n the age spatal space and feature transforaton and feature extracton s used to reduce the age denson such as the prncpal coponent analyss, absorpton features extracton and spectral statstcal analyss. Band selecton s usually based on the spectral curve of hyper spectral age whch can convert hyper spectral vector nto low densons or one denson. Due to ther cobnatoral coplexty, band selecton algorths cannot be used when the nuber of features s larger than a few tens (L. Bruzzone, 995; L. Bruzzone, 000; C. Lee,993; C.I Chang,006; A. Plaza,005). And densonaty reducton algorths based on feature extracton and band selecton can not cobned spectral and spatal characterstc. All these densonaty reducton algorths have the dsadvantages of less consderng the spectral nforaton and low effcency. ractal theory s used n any appcatons for the advantage to solve the non-near syste of the coplex phenoenon. Thus ractal s usually used to solve the coplex non-near syste analyss. ractal theory has been used n the reote sensng research whle t does not obtan the full appcaton (Qu H L; Weng Q, 003). Coonly, fractal denson of hyper spectral reote sensng age s calculated by the spatal characterstc and then the fractal denson s used for the band selecton. And the fractal denson value s calculated wth spectral curve to unfy the spectral nforaton to the spatal dstrbuton feature age thus t can be used to reduce the densonaty of hyper spectral reote sensng age. In ths paper, a densonaty reducton algorth based on feature extracton of the spectral curve usng fractal analyss whch consderng both the spatal characterstc and spectral characterstc of hyper spectral reote sensng age s proposed A spectral doan feature analyss based on fractal easureent technque s desgned for hyper spectral ages. ractal characterstc of spectral curve s dscussed. A bref descrpton s gven to explan the nonnear echans resultng n fractal of spectral curve. And the spectral curves of sae objects are presented to show the self slar. And soe coputatonal results are gven to show exponental relaton between the total length of spectral curves and the dfferent easureent unt. A fractal denson calculaton algorth of hyper spectral curve s desgned. A nose reove algorth based on wavelet transforaton s done before the fractal analyss of spectral curve. Then a feature analyss procedure n spectral doan based on fractal easureent s also proposed to reduce densonaty of hyper spectral ages. The fractal denson value s taken as the feature of spectral curve and the fractal denson feature age s proposed to represent the densonaty reducton result of hyper spectral age. Experents of fractal denson value of dfferent objects * Correspondng author. SU Junyng, PhD, School of Reote Sensng and Inforaton Engneerng, Wuhan Unversty, P.R. Chna, jysu_sjy@sna.co,

2 The Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B7. Bejng 008 spectral curve show fractal can be used to represent the spectral feature to reduce densonaty of hyper spectral age. nally, the appcaton of fractal easureent of spectral doan feature analyss s brefly dscussed. structure, correlaton. The spectral curve has the self-slar n statstcs whch ndcate t has the fractal characterstc. The self-slar property of spectral curve can be represented n the SPOT age and TM ultple age as fgure shown:. DATA ANALYSIS. Spectral feature of hyper spectral age Spectral feature s the an dfference of the hyper spectral age and the coon reote sensng age (Peter, 00; Shu N,00). Pxel value n each band whch constructs the spectral curve can represent the object nforaton for age classfcaton. Wth the hgh resoluton of spectral band, the spectral curve can be used for feature extracton, band selecton and classfcaton. Spectral atchng ethod s used to dentfy the object wth the spectral brary supportng whle t s te consung wth the vast ount of spectral data. In order to full use the spectral nforaton of each pxel n hyper spectral age, the feature analyss can be done to each spectral curve to obtan feature and for the feature age of hyper spectral data. Thus the classfcaton can be done wth the spectral curve feature age. Densonates reduce and feature analyss s done at the sae te whch can ncrease the processng effcency. The key proble of spectral feature analyss s to extract the feature fro the spectral curve.. ractal characterstc of spectral curve ractal s a tool to analyss the spatal structure and spatal coplex and t obtans fast progress n the reote sensng appcaton. ractal denson s used to present the spatal structure thus the fractal research focus on the age spatal fractal analyss (Qu H L; Weng Q, 003). or the hyper spectral age, both the spatal doan and spectral doan has fractal characterstc. The fractal characterstc n spectral doan s fro the spectral curve as the followng tes: ) Hyper spectral curve represent the object spectral agng course s non-near Hyper spectral curve represent the object spectral agng course. And the object spectral agng course s a non-near. As the reote sensng physcal prncple, spectral agng odel s: ' L = K [ τ ( N sn θρdω + We ε ) + b ] () Where K s the spectral response coeffcent of the sensor. τ s the atosphere spectral transttance. N s the solar ncdent spectral energy. θ s the solar alttude angle. ρ s ' the object spectral reflectvty. Ω s the solar azuth. W e s the black body spectru radaton flux densty. ε s the object spectru essve. b s the energy of atospherc scatterng and radaton. As the equaton (), object spectral agng course s a coplex non-near syste. Non-near s the an characterstc of fractal phenoenon. Thus we can conclude that the spectral curve has the characterstc of fractal as the spectral agng odel. ) Spectral curve has soe statstcal self-slar property As the fractal defnton of Mandelbrot n 986, fractal s a odel whch the partal s slar to the total object. Thus the fractal has the portant property that the local part of fractal odel s slar to the whole odel n soe sdes such as the gure Self-slar property n spectral age gure s one of the SPOT age and TM age n Wuhan cty. The two ages are slar. SPOT age has the total nforaton of the vsble spectral bands and t can be taken as the total odel. TM age s just one band of the total 7 bands and t can be taken as a local partal odel whle t s qute slar to the SPOT age. Thus we can conclude that the local partal spectral s slar to the whole spectral and t s one of portant characterstc of fractal. As the spectral self-slar property, the spectral curve has the characterstc of fractal odel. 3) The length of spectral curve under dfferent easureent unt shows exponental relaton Dfferent objects of 30 bands MAIS ages are selected to easure the length of spectral curve under dfferent band wdth. The result s shown as table : Road Tree Water Band wdth length Band wdth length Band wdth length Table Spectral Curve length under dfferent spectral wdth 98

3 The Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B7. Bejng 008 As table shown, the easureent length of spectral curve decrease wth the ncrease of spectral band wdth and the trend of the decrease s saller and saller. And ths relatonshp shows the easureent length of spectral curve has the exponental relaton to the easureent band wdth. Thus the spectral curve can be descrbed wth the fractal denson. Take dfferent easureent spectral wdth as ε, the length of spectral curve as N, the statstcs curve between log(ε ) and log(n) as fgure : gure Log relaton between spectral curve length and wdth As fgure shown, the log(ε ) - log(n) has obvous near relaton. The result of ne fttng, t can reaze the level ofα < 0. 0, thus the spectral curve has the characterstc of fractal and the fractal denson value can be used to present the spectral feature of spectral curve to each pxel. 3. METHODLOGY The densonaty reducton algorth can be explaned nto the followng three steps. rstly, nose reoval processng s done to the hyper spectral curve for the densonaty reducton. Wavelet transforaton s used to flter the nose of spectral curve of the hyper spectral age. Wth the ultple resoluton analyss of wavelet transforaton, the spectral curve whch can be constructed by the pxel vector n spectral denson s decoposed nto sooth coponent and nose coponent. The hgh frequency nose coponent s reoved before the nverse wavelet transforaton to obtan the nose reoval spectral curve. Secondly, fractal denson value s calculated to the nose flterng spectral curve. The fractal denson calculaton algorth s desgned to the spectral curve. nally, the densonaty reducton s done wth the fractal denson feature of the spectral curve. Spatal spectral data cube of hyper spectral reote sensng age s fored by the fractal denson value of spectral value, whch can obtan spectral dstrbuton age n spatal space. The spatal spectral data cube age can cobne spectral and spatal texture characterstc together. gure 3 gves the densonaty reducton procedure of hyper spectral age. gure 3 Densonaty reducton wth fractal analyss 3. Spectral curve flterng Spectral curve nose wll affect the result of spectral feature analyss. A non-near strength wavelet flterng algorth (nlw) s proposed to spectral curve flterng. rst the level wavelet decoposton s done to the spectral curve wth Morlet flter. The devaton of low frequency s selected as the nose threshold. Hgh frequency coeffcent under nose threshold s set zero and the coeffcent above the nose threshold s nonnear strength. Wth the wavelet reconstructon, we can obtan the spectral curve after nose reoval. ollowng s the detal procedure of spectral curve flterng. Step : Deterne Morlet flter and flter wndow sze Morlet wavelet flter wth the wndow sze of 3 s selected as the wavelet flter as equaton () and (3): h[] = { , , , () , ,0.956, ,0.956, , , , , } g[] = { , , , , (3) , ,0.956, ,0.956, , , , , } Where h [] s the low pass flter of Morlet wavelet and g [] s the hgh pass flter of Morlet wavelet. Step : Cycle expand of spectral curve as the flter wndow sze as equaton (4). l, =, l + = l (4) N N =,,3,4,5,6 l s the spectral curve and N s the feature pont Where, nuber or band nuber. Step 3: Two level wavelet decoposton of spectral curve as equaton (5). { LLl, HL, LH, HH, N} l (5) Step 4: Non-near strength of nose reoval Nose s central at the hgh frequency coeffcent after wavelet transforaton. The coon nose reoval ethods s to select nose threshold and set the coeffcent under threshold wth zero to reove nose fro the orgnal sgnal (Pan Quan,007,998; Jansen M,00; Wu C. Q,004). The nose threshold can be deterned fro the orgnal spectral curve nose level. And the nose level of orgnal spectral curve can be calculated fro the low frequency coeffcent thus the devaton of low frequency coeffcent can be taken as the nose level as equaton (6). σ 0 = σ{ LLl, N} (6) Thus the nose level of each decoposton coeffcent can be calculated as the nose expands theory. 0 ( H ) = 0 σ = σ G (7) Where H s the ourer transforaton of low pass flter of. G s the ourer transforaton of hgh pass flter of h [] g []. represents convoluton, H s the expanson of H, G scale s the scale expanson of G, s the nor. If the scale s =, the nose level of each decoposton coeffcent s, σ σ H G (8) = 0 99

4 The Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B7. Bejng 008 Consderng H G s tend to, the nose level of low frequency can approxately present nose level of hgh frequency coeffcent thus t can be taken as the nose threshold. If the coeffcent of hgh frequency s under 3σ, t can be done as equaton (9). HL, LH, HH } = 0 { HL, LH, HH } 3σ (9) { If the coeffcent of hgh frequency s above3σ, t should be non-near strength as equaton (0) and (). sg( c *( y / yax b)) sg( *( y / yax + b)) f ( y) = * y (0) ax sg( c *( b)) sg( *( + b)) sgc ( *( y/ yax b)) sg( *( y/ yax + b)) ( y / yax )* d () f ( y) = * yax*exp sgc ( *( b)) sg( *( + b)) Where sg( t) = () t + exp y s the hgh frequency coeffcent of wavelet decoposton. c, d s to adjust the strength coeffcent. b s to adjust the ntal value of the strength detal and t s the threshold of the strength processng. In ths paper, b s taken as the threshold of flterng that eans b = 3σ. c s taken as the characterstc of age between 0 and 40. In ths paper, c s 30. d s ranged aong and Step 5: Wavelet reconstructon Mallat reconstructon s done to wavelet coeffcent after flterng and strength to obtan the fltered spectral curve wth nose reoval. A spectral curve flterng processng experent s done wth ean flter soothng, least squares soothng and the nonnear strength wavelet flterng algorth (nlw). gure 4 gves the dfferent curve flterng result. gure 4 Dfferent flterng result of spectral curve As fgure 4 shown, nlw flterng algorth of spectral curve can obtan reasonable result n two typcal nose cases (as the rectangle n fgure 4) copared wth ean flter flterng, least squares flterng. Mean flterng s over soothng and soe detal s lost whle the least square flterng s coordnate to the orgnal spectral curve. Sgnal nose rato and nose level of spectral curve are selected to further assess dfferent spectral curve flterng algorths. Table gves the dfferent spectral curve flterng algorths to dfferent objects of the 30 bands MAIS ages and 8 bands OMIS ages. Sensor MAIS OMIS Object Nose paraeter Orgnal Spectral Curve Mean flterng Least square flterng nlw Water Nose level SNR Tree Nose level SNR Resdent Nose level Area SNR Spare Nose level Area SNR Road Nose level SNR Tree Nose level SNR Resdent Nose level Area SNR eld Nose level SNR Table Dfferent spectral curve flterng results As table shown, dfferent object n dfferent sensor hyper spectral age has dfferent nose level and sgnal nose rato. nlw algorth obtans hgher SNR and lower nose level than orgnal spectral curve. And ts result s better than the ean flterng and least square flterng algorth. 3. ractal denson calculaton algorth In ths paper, a step easureent ethod of fractal denson calculaton algorth consderng the spectral curve characterstc s proposed. Curve length L(r) under dfferent step easureent unts s deterned by the step length N (r) and steps r as equaton (3). L( r) = N( r) r (3) And the curve length L(r) can be represented as equaton (4) under the defnton of fractal. μ L( r) = q r (4) Where r s the step nuber under dfferent step easureent unt. μ = D s the render denson, D s the fractal denson value of the spectral curve, q s the coeffcent to deterne. Thus we can obtan, log( L ( r)) = ( D) log r + C (5) Where C s the coeffcent to deterne. Wth the dfferent step easureent unt, we can obtan pont array of. Thus we can calculate the slope of the (log( L ( r)),log r) ne K( orμ) whch s the fractal denson value of spectral curve. D = K (6) The detal procedure of step easureent fractal denson calculaton can be descrbed as followng steps: Take d as the ntal step unt, calculate the dstance d between the frst pont P and the second pont P of spectral curve. f d > d, nterpolate one pont p between P and P to ake the dstance fro p to P as d. 3 f d < d, to calculate the dstance d 3 between P and P 3, f d 3 > d, to nterpolate one pont p between P and P 3 to ake the 300

5 The Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B7. Bejng 008 dstance fro p to P as d fro the curve. f d 3 < d,consderng P4 as step 3 untl the last pont of the spectral curve. 4 suarze the step nuber n under the step easureent unt d. Then the length of spectral curve under d s, L ( d = n d (7) ) 5 Change step easureent unt to obtan spectral curve length as equaton (8). L ( d ) = nd,..., L( d ) = nd (8) Thus we can calculate the fractal denson value of spectral curve as equaton (5). 3.3 Densonaty reducton wth fractal feature age Wth the fractal denson calculaton of spectral curve for each pxel, t can be taken as the feature of spectral curve. Ths feature value s the result of densonaty reductons of spectral curve. The fractal feature of spectral curve cab be used for hyper spectral age segent and classfcaton for t can transfor the hyper spectral nforaton nto one denson fractal feature age. Thus the fractal feature age analyss can reaze densonaty reducton and ncrease the effcency of data processng for t can full use the age analyss algorth n spatal doan. The fractal denson value s taken as the feature of spectral curve and the fractal denson feature age s proposed to represent the densonaty reducton result of hyper spectral age. The densonaty reducton procedure based on the fractal analyss has been descrbed as fgure 3. gure 5 gves the densonaty reducton feature age of MAIS age. gure 5(a) s one of band ages. gure 5(b) s the fractal feature age. gure5 (b) Spectral fractal denson feature age of MAIS gure 5 Densonaty reducton result based on spectral fractal analyss As fgure 5, the fractal feature can cobne the spectral nforaton and spatal nforaton together and reaze the densonaty reducton through the spectral feature transforaton. The fractal feature age can represent the spectral nforaton of hyper spectral age and obtan better detal representaton and t s a new ethod of spectral feature analyss of hyper spectral age. 4. EXPERIMENTS AND CONCLUSIONS Hyper spectral texture code s taken as the portant hyper spectral age analyss technque[]. In order to verfy the densonaty reducton algorth based on fractal analyss, the author select dfferent object texture unt to calculate the fractal denson value of spectral curve of each pxel together wth the correlaton of the centre pxel. The texture unt s 3 3. The result s shown as table 3. gure 5(a) Orgnal band age of MAIS resdent area tree water ractal denson correlaton ractal correlato ractal correlato denson n denson n ax n devaton range Table 3 ractal denson and correlaton of spectral curve n texture unt 30

6 The Internatonal Archves of the Photograetry, Reote Sensng and Spatal Inforaton Scences. Vol. XXXVII. Part B7. Bejng 008 As table 3 shown, the correlaton of spectral curve aong the centre pxel and each pxel n texture unt has obvous dfference especally for the resdent texture unt. And dfferent object has very slar correlaton whch wll lead the pxel confuse for the pxel classfcaton. The fractal denson value can obtan better result for the classfcaton. The dfferent object has dfferent fractal denson value of spectral curve n texture unt. The resdent object has the fractal denson value ranged wth.099~.07, the tree ranged wth.05~.00 and water ranged wth.005 ~.076. Thus the densonaty reducton result usng the fractal feature of spectral curve can reaze better texture code atchng whch s very useful for age classfcaton. or the Huges phenoenon, there are stll soe of dfferent object fractal denson feature whle the fractal denson of the centre pxel s qute dfference. The densonaty reducton based on spectral curve fractal analyss can cobne the spectral nforaton and spatal texture nforaton together to reaze the feature analyss and t can obtan better processng effcency of hyper spectral data. The densonaty reducton based on spectral curve fractal analyss provde a new ethod to dffer the confuse pxel such as the feature analyss cobned the spatal fractal analyss and spectral fractal. ACKNOWLEDGEMENTS The research work s supported by the Naton Scence ounder of Chna. (NSC: ) and 973 Project of Chna (006CB70303). REERENCES C. Lee and D. A. Landgrebe,993. Analyzng hgh-densonal ult-spectral data, IEEE Trans. Geosc. Reote Sensng, vol. 3, pp Qu H L, La N S N, Quattroch D A, 99. ractal characterzaton of hyper spectral agery[j]. Photograetrc Engneerng and Reote Sensng, 65: 63~7 Weng Q., 003. ractal analyss of satelte-detected urban heat sland effect [J]. Photograetrc Engneerng and Reote Sensng, 69: 555~566 Peter, Toy L, Catherne O., 00. Statstcal analyss of hyper spectral data fro two Swedsh lakes[j].the Scence of the Total Envronent,68:55~69. Shu N., 00.Edge Extracton fro Mult-spectral Iages and Densty Analyss of Super-densonal Spectral Space. Proceedngs of SPIE,4 550:63~66 Lu Juan and Moun P. 00. Inforaton-theoretc analyss of nter-scale and ntra-scale dependences between age wavelet coeffcents [J].IEEE Transacton on Iage Proc., 0():647~658. Berkner K and Wells R O.,00. Soothness estates for softthreshold denosng va translaton nvarant wavelet transfors [J].Apped and Coputatonal Haronc Analyss, ():~ 4. Jansen M, 00. Nose reducton by wavelet threshold [M].Sprnger Verlag, Lecture notes n Statstcs (6). Wu C. Q., 004. Adaptve flterng n spatal doan based on spectral nforaton [J]. J. Reote Sensng, 004.8():5~55 Pan Quan, Da Guanzhong, Zhang Hongca,998. A Threshold selecton ethod for hard-threshold lterng Algorth [J]. Acta Electronc Scence,,6():5~7. Pan Quan, Meng J.L., Zhang L., Cheng Y.M., Zhang H.C.,007. Wavelet lterng Method and Its Appcatons [J]. J. Electronc and Inforaton Technology, 9(): 36~4 A. Jan and D. Zongker, 997. eature selecton: evaluaton, appcaton, and sall saple perforance, IEEE Trans. Pattern Anal. Machne Intell.,vol. 9, pp L. Bruzzone,. Ro, and S. B. Serpco,995. An extenson to ultclass cases of the Jeffreys-Matusta dstance, IEEE Trans. Geosc. Reote Sensng, vol. 33, pp L. Bruzzone and S. B. Serpco, 000. A technque for feature selecton n ultclass cases, Int. J. Reote Sensng, vol., pp C. Lee and D. A Landgrebe Analyzng hgh-densonal ultspectral data, IEEE Trans. Geosc. Reote Sens., vol. 3, pp C.I Chang and S. Wang, 006. Constraned band selecton for hyperspectral agery, IEEE Trans. Geosc. Reote Sens., vol. 44, pp A. Plaza,P. Martnez, J. Plaza and R.005. Perez Densonaty reducton and classfcaton of hyper spectral age data usng sequences of extended orphologcal transforatons, IEEE Trans. Geosc. Reote Sens., vol. 43, pp