Estimation of Higher Order Moments for Compound Models of Clutter by Mellin Transform

1 Estimation of Higher Order Mo ments for Compound Models of Clutter by Mellin T ransform C Bhattacharya DEAL(DRDO), Dehrad un 248001, India email:cbhat 0@ieee.org Abstract — The compound models of clutter statistics are found suitable to describe the nonstationary nature of radar backscat- tering from high-resolution observ ations. In this letter , we sho w that the properties of Mellin transf orm can be utilized to genera te higher order moments of simple and compound models of clutter statistics in a compact manner . Index T erms — Clutter , compound model, Mellin transfo rm, log-cumulants. I . I N T RO D U C T I O N R AD AR backscattering fro m ground or sea surfaces ar e wide-sense stationar y for low-resolution observations as expectations of clu tter statistics o r m oments are assumed to be indepen dent of spatio-tempo ral changes. F or high-reso lution observations, such surfaces reveal heterogen eous structures such as swell in sea waves or winds b lowing over the canopy of grasslands that result in nonstation ary clutter statistics [ 1], [2], [4]. The compound models of probability density functions (pdf) incorpo rate the variation in the parameters of clutter i n such ca ses. T raditionally highe r order m oments of a con tinuous ra ndom variable (rv) X are g enerated from high er order de riv ati ves o f its characteristic function defined a s Φ X ( ω ) = E { exp( j ω x ) } = Z ∞ −∞ f X ( x ) ex p( j ω x ) dx . (1) The continuo us pdf f X ( x ) is fo r − ∞ < x < ∞ . Generation of moments and cu mulants from ( 1) f or the co mpoun d models o f clutter requ ire solutions of inco mplete integrals. The domain of X is 0 ≤ x < ∞ fo r amplitud e and power statistics, and R ∞ 0 f X ( x ) dx = 1 . Prop erties of Me llin transfo rm provid e the formalism to d eriv e higher o rder moments in a com pact manner i n such cases. Some o f these properties wer e used in [3], [6] to derive the momen ts for high-r esolution synthetic aperture rad ar (SAR) clutter statistics. Here, we sh ow that the proper ties o f Mellin transform c an be utilized in an effective manner fo r both simple an d comp ound models of clu tter either in amplitude or in intensity domain. I I . M E L L I N T R A N S F O R M P R O P E RT I E S Mellin transfor m exists for a contin uous fu nction f X ( x ) defined over R + . The tran sform operator is the second kind characteristic function Φ X ( s ) e xpressed a s Φ X ( s ) = M [ f X ( x ); s ] = Z ∞ 0 x s − 1 f X ( x ) dx . (2) Here s = a + jb ∈ C is t he complex Laplace transform v ariable. T raditional moments are generated from (2) with s = n + 1 , n ∈ Z + . m n = M  f X ( x ); s  s = n +1 . (3) Second-k ind m oments or the log-mo ments are gen erated for logarithm of rv X by using the deriv ativ e pr operty of Mellin transform . ˜ m n = M [log ( x ) n f X ( x ); s ]     s =1 = Z ∞ 0 x s − 1 log( x ) n f X ( x ) dx = d n d s n Φ X ( s )     s =1 . (4) Analogou s to the cumu lants derived fro m logarith m of characteristic f unction in ( 1), the n -th order cumu lants of second kind or the lo g - cumu lants are obtained from deriv ati ves of lo garithm o f Φ X ( s ) ; i.e. , Ψ X ( s ) = log (Φ X ( s )) . ˜ k n = d n d s n Ψ X ( s )     s =1 . (5) The lo g-mom ents and the l og-cu mulants are related as ˜ k 1 = ˜ m 1 ˜ k 2 = ˜ m 2 − ˜ m 1 2 ˜ k 3 = ˜ m 3 − 3 ˜ m 1 ˜ m 2 + 2 ˜ m 1 3 ˜ k 4 = ˜ m 4 − 4 ˜ m 1 ˜ m 3 + 6 ˜ m 1 2 ˜ m 2 − 3 ˜ m 1 4 . (6) The und erlying mean of speck le compon ent of clutter vary widely in the c ompou nd mod els o f amplitude or power statistics resulting in lo ng-tailed distributions. Speck le a rises from random ness in the distribution o f backscattering elements in the r esolution cell, the number of such scatterers is no n- stationary for high-resolution ob servations. The pdf of high- resolution clutter is described by takin g into acco unt o f a rv Z signifying ra ndomn ess in the m ean o f clutter . f X ( x ) = Z ∞ 0 f X ( x | z ) f Z ( z ) dz , z > 0 . (7) 2 The compou nd p df m odel in ( 7) is a Mellin co n volution. One nice prop erty of Me llin tr ansform is the p roduct form of th e compon ents o f pdf in the tra nsform d omain [5 ]. M [ f X ( x ); s ] = M [ f X ( x | z ); s ] M [ f Z ( z ); s ] . (8) The log -cumulan ts of the compo nents in (8) are theref ore additive. ˜ k n , x = ˜ k n , ( x , z ) + ˜ k n , z . (9) I I I . M O M E N T S G E N E R AT I O N F O R S I M P L E M O D E L S O F C L U T T E R The shap e and scale par ameters of simple mo dels of p df f or low-resolution cases are station ary . T he usu al p df of spec kle power is a g amma distribution resultin g from conv olution o f L in depend ent expo nential distributions. f V ( v ) = 1 Γ( L )  L µ  L v ( L − 1) exp  − Lv µ  , v ≥ 0 . (10) Here Γ( . ) is the standard gamma function. The shape and s cale of distribution are deter mined by L and µ , mea n value of clut- ter power respectively . Correspon ding amplitude distribution turns o ut to be a Na kagami pd f [2 ], [6] . f N ( r ) = 2 Γ( L )  √ L µ  2 L r (2 L − 1) exp  − Lr 2 µ 2  , r ≥ 0 . (1 1) Mellin tra nsform fo r g amma pdf is Φ G ( s ) = λ L Γ( L ) Z ∞ 0 v ( L+s − 1) − 1 exp( − λ v ) dv (12) with λ = L µ . Using th e transfor m pair M [ x u exp( − λ x ); s ] ⇐ ⇒ λ − ( s + u ) Γ( s + u ) we obtain, Φ G ( s ) =  µ L  s − 1 Γ( s + L − 1) Γ( L ) . (13) The mom ents of first kind for gamma p df are gener ated f rom (13) with s = n + 1 a s m n = Φ G ( s )     s = n +1 =  µ L  n Γ( L + n ) Γ( L ) . (14) As a special case of the re sult in (14), the mo ments of exponential pdf (for L = 1 ) are m n = µ n n ! . Maxwell pdf is th e case for L = 3 . f M ( u ) = 1 σ 3 r 2 π u 2 exp  − u 2 2 σ 2  , u ≥ 0 . (15) W e use th e additiona l M ellin tran sform pair M [exp( − λ x 2 ); s ] ⇐ ⇒ 1 2 ( λ ) − s 2 Γ( s 2 ) with λ = 1 2 σ 2 ; so that Φ M ( s ) = 1 σ r 2 π (2 σ 2 ) s 2 s 2 Γ  s 2  . (16) The m oments of first k ind for Max well pdf are m n = Φ M ( s )     s = n +1 = 1 σ r 2 π (2 σ 2 ) n +1 2  n + 1 2  Γ  n + 1 2  . (17) The moments for amp litude distributions are als o der i ved by Mellin transfo rm. As fo r Nakagam i distribution, with λ = √ L µ Φ N ( s ) = 2 Γ( L ) ( λ 2 ) L Z ∞ 0 r ( s +2 L − 1) − 1 exp( − λ 2 r 2 ) dr =  √ L µ  − ( s − 1) Γ( L + s − 1 2 ) Γ( L ) . (18) The log-cumu lants ar e easier to derive here. In gene ral th e log-cum ulants o f Nak agami distribution are d erived from (5) ˜ k n =  1 2  n Υ ( n − 1 , L ) . (19 ) Here Υ ( . ) is the Digamm a function ; i. e., the first deriv ativ e of ln Γ( s ) at s = 1 . In gen eral Υ ( n − 1 , L ) is th e n th d eriv ati ve of th e Digamma f unction for variable L . One long- tailed p df often used in sea-clu tter amplitude modelling [1] is W eibull distribution. f W ( x ; z , b ) =  b z  x z  b − 1 exp  −  x z  b  , x ≥ 0; b , z > 0 . (20) Here z is the scale para meter a nd b is the sha pe param eter o f distribution. Mellin transform of (20) is Φ W ( s ) = b z b Z ∞ 0 x ( s + b − 1) − 1 exp  −  x z  b  dx . (21) From the Me llin transform pair M [exp( − λ x b ); s ] ⇐ ⇒ b − 1 λ − s b Γ( s b ) , the seco nd ch aracteristic function is Φ W ( s ) = z ( s − 1) Γ  s + b − 1 b  . (22) The m oments of first k ind for W eibull distribution are m n = Φ W ( s )     s = n +1 = z n Γ  n + b b  . (23) 3 The com mon Rayleigh am plitude pdf is a special case of W eib ull distribution with b = 2 . f R ( r ; z ) = 2  r z 2  exp  −  r z  2  , r ≥ 0 . (24) The m oments of first kind for Rayleigh p df are m n = Φ R ( s )     s = n +1 = z n Γ  n + 2 2  . (25) W e show in the next sectio n th e utility of Mellin transfor m for der i ving th e log- moments and th e log- cumulants of com- pound models of clutter in a compact mann er . I V . M O M E N T S G E N E R AT I O N F O R C O M P O U N D M O D E L S O F C L U T T E R The pdf for compound models o f high-resolution clutter have got two compo nents; pd f of speckle c ompon ent, an d pdf of the m odulation in mean amplitude or power of speckle. Considering both to be gamma d istributed r v th e pd f for generalized gamma ( G Γ ) m odel o f clu tter power is [6 ] f V ( v ) = 1 Γ( L )Γ( M )  2 LM < z >  2 LM < z > v  ( L + M − 2 2 ) K M − L  2  LM < z > v  1 2  . (26) The shape param eter for ga mma pdf f Z ( z ) of rv Z ac cording to (7 ) is M , and K M − L ( . ) is the secon d kind modified Bessel function of order ( M − L ) . T he mean estima te of < z > = µ . Assuming speck le an d the m odulation in mean power in the high-r esolution cell to be in depend ent of each other, we h av e by Mellin conv olution p roperty in ( 8) Ψ V ( s ) = ( s − 1 ) log  µ LM  + log Γ( s + L − 1 ) + log Γ( s + M − 1) − lo g Γ( L ) − log Γ( M ) . ( 27) The lo g-cumu lants of G Γ mo del ar e ˜ k 1 = log  µ LM  + Υ ( L ) + Υ ( M )     s =1 ˜ k n = Υ ( n − 1 , L ) + Υ ( n − 1 , M ) . (28) Spikes in high-resolution g round -clutter amplitud e at low grazing ang les are of ten described by the K-distribution mo del [2], [4]. T he co mpoun d K- pdf f N ( r ) = 4 b ( α +1) 2 r α Γ( α ) K α − 1 (2 r √ b ) (29) is a Mellin co n volution of Rayleigh pdf and exponential pdf giv en by , f N ( r ) = 4 rb α Γ( α ) Z ∞ 0 dz z z α − 1 exp  − bz − r 2 z  . (30) Here α is the shape parame ter in the v ariation of mean of sea or g round clutter amplitude , and b is the scale p arameter for associated speckle amplitude. Following deriv ation fo r Nakagami pdf in (19) the second characteristics function for K-pdf is given by , Φ N ( s ) = b − ( s − 1 2 ) µ s − 1 Γ  s 2 + 1 2  Γ( α + s − 1 2 ) Γ( α ) . (3 1) where < z > = µ 2 . The log-cum ulants for K-pdf a re Ψ N ( s ) = −  s − 1 2  log b + ( s − 1 ) log µ + log Γ  s 2 + 1 2  + log Γ  α + s − 1 2  − log Γ( α ) , and ˜ k 1 = − 1 2 log b + log µ + 1 2 Υ ( α ) + 1 2 Υ (1)     s =1 ˜ k n =  1 2  n Υ ( n − 1 , α ) . (32) Here Υ (1) = − 0 . 577 215 is the Euler constant [5]. This shows that th e log-c umulants o f K- distribution are determined by the higher orde r log-cumulan ts of Nak agami distribution in the mean of high-r esolution g round or sea clutter . A more extended case of compo und clu tter model is the scene wh ere variation in the shape of clutter amplitude distri- bution is given by gen eralized W eibull d istribution [ 4]. f WN ( r ; b , c , α ) = 2 cb α Γ( α ) r c − 1 Z ∞ 0 dz z c z 2 α − 1 exp  −  r z  c − bz 2  . (33) This is a Mellin con volution where randomn ess in th e mean amplitude of clutter is described b y Naka gami pdf with the shape p arameter b eing α . f N ( z ; b , α ) = 2 b α Γ( α ) z 2 α − 1 exp( − bz 2 ) , and the clutter amp litude follows a gen eralized W eibull distri- bution w ith the shape param eter being c . f W ( r | z ; c ) = c z c r c − 1 exp  −  r z  c  . Follo wing the transfo rm rule of Me llin conv olution , Φ WN ( s ) = Φ W ( s )Φ N ( s ) =  σ b  s − 1 2 Γ  s + c − 1 c  Γ  α + s − 1 2  Γ( α ) . (34) 4 where < z 2 > = σ . Th e log-moments f or this generalized W eib ull mod el of clutter accord ing to (9) are ˜ k 1 = 1 2 log  σ b  + 1 c Υ (1) + 1 2 Υ ( α )     s =1 ˜ k n =  1 2  n Υ ( n − 1 , α ) . (35) Another comp ound mo del used to describe h igh-reso lution SAR clutter is the Fisher distribution [3 ]. f F ( u ) = Γ( L + M ) Γ( L )Γ( M )  L M µ   L M µ u  L − 1  1 + L M µ u  L + M . (36) Consider L M µ u = λ ; following the Mellin transfor m pair M [(1 + λ ) − b ; s ] ⇐ ⇒ Γ( s )Γ( b − s ) Γ( b ) , with b = L + M the second characteristic functio n for Fisher distribution is Φ F ( s ) =  M µ L  s − 1 1 Γ( L )( M ) Γ( s + L − 1)Γ( M + 1 − s ) . (37) Correspon ding log-cu mulants are ˜ k 1 = log µ + [ Υ ( L ) − log L ] + [ Υ ( M ) − log M ]     s =1 ˜ k n = Υ ( n − 1 , L ) + ( − 1) n Υ ( n − 1 , M ) . (38) One useful ap plication of the log- cumulan ts an d the ir relationship with the log- moments in ( 6) is estimation of parameters of texture. E mpirical data fr om high-r esolution radar backscatterin g x ( t ) follow th e produ ct model, x ( t ) = u ( t ) z ( t ) . (39) Here u ( t ) is the speckle co mpone nt, an d z ( t ) represent texture signifying variation in th e mean parameter . The log-mo ments of ob served data a nd the log-cu mulants of texture can b e estimated for different compoun d models utilizing the relation- ships in (4) an d (6). Parameters fo r texture are der iv ed u sing the log-c umulants of speckle as in (9), and can be verified with the theoretical values deri ved in the paper . F or example, secon d and fou rth ord er log -cumulan ts of textur e compon ent for G Γ model of hig h-resolutio n g round clutter in (28) are estimated in Fig. 1. Second an d fou rth order log-m oments of x ( t ) are derived from the log-cumulan ts of z ( t ) assuming it to be a gamma v ariable, and u ( t ) also follows gamma distribution. The results of simulatio n show that high er o rder log -cumulan ts of texture vanish with incr easing values o f sh ape parameter M . This is expected in the pr esent case as th e texture co mpone nt follows a n early Gau ssian distribution with con stant mean for increa sing v alues of M . F or values of M < 1 , there is presence o f large am ount o f spikes in ob served data sign ifying high values of log -moments an d cumu lants. Lo g-mome nts of 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 8 9 10 shape parameter of texture second kind second order moments 0 2 4 6 8 10 12 14 0 100 200 300 400 500 600 700 shape parameter of texture second kind fourth order moments o−>log moments of data +−>log cumulants of texture *−> log moments of data x−> log cumulants of texture Fig. 1. Log-moments of data and log-cumulan ts of texture for simulation of G Γ model of high-resolut ion clutter . Left: second order moments; right: fourth order mom ents. clutter tend to become con stant with incr easing M sign ifying stationarity of low-resolution observations. V . C O N C L U S I O N The utility o f Mellin tran sform prop erties to generate higher order m oments o f simple and comp ound m odels of clutter in both amplitude and power do main is shown in this letter . The secon d kind ch aracteristic function and its proper ties pr ovide co mpact analytical expressions f or higher order momen ts that are u seful to interpret texture p roperties of h igh-resolu tion clutter . A C K N OW L E D G M E N T The a uthor is than kful to Pro f. D. Muk hopadh aya of Elec- tronics and T ele- Communicatio n En gineering D epartment, Ja- davpur University , India for fruitful discussion and suggestion s on the d raft m anuscript of the p aper . R E F E R E N C E S [1] F . L. Posner , ”Spiky sea cl utter at high range resolutio ns and very lo w grazing angles, ” IEEE T ran s. 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