Image Processing Reference
In-Depth Information
yx yx y of the stochastic signal formation
operator (SFO) S given by (2) defines the signal wavefield formation model. Its mean,
< (
The random functional kernel
S
(,)= (,)+ (,)
S
yx y , is referred to as the nominal SFO in the RS measurement channel
specified by the time-space modulation of signals employed in a particular radar
system/SAR (Henderson & Lewis, 1998), and the variation about the mean
S
,
)> = (
S
,
)
yx =
( y , x ) S ( y , x ) models the stochastic perturbations of the wavefield at different propagation
paths, where ( y , x ) is associated with zero-mean multiplicative noise (so-called Rytov
perturbation model). All the fields
(,)
enu in (2) are assumed to be zero-mean complex
valued Gaussian random fields. Next, we adopt an incoherent model (Henderson & Lewis,
1998), (Shkvarko, 2006) of the backscattered field ()
, ,
e x that leads to the -form of its
correlation function, R e ( x 1 , x 2 ) = b ( x 1 )( x 1 - x 2 ). Here, e ( x ) and b ( x ) = <| e ( x )| 2 > are referred to
as the scene random complex scattering function and its average power scattering function
or spatial spectrum pattern (SSP), respectively. The problem at hand is to derive an estimate
ˆ ()
b x (referred to as the desired RS image) by processing the available finite
dimensional array radar/SAR measurements of the data wavefield u ( y ) specified by (2).
b x of the SSP
()
2.2 Discrete-form uncertain problem model
The stochastic integral-form EO (2) to its finite-dimensional approximation (vector) form
(Shkvarko, 2008) is now presented.
u = S + n = Se + Δe + n ,
(3)
in which the perturbed SFO matrix
S = S + Δ , (4)
represents the discrete-form approximation of the integral SFO defined for the uncertain
operational scenario by the EO (2), and e , n , u are zero-mean vectors composed of the
decomposition coefficients
{ kk
{ M
mm
{ M
m u , respectively. These vectors are
characterized by the correlation matrices: R e = D = D ( b ) = diag( b) (a diagonal matrix with
vector b at its principal diagonal), R n , and R u = <
e
,
n
, and
1
1
1
SR   > p ( Δ ) + R n , respectively, where
<> p ( Δ ) defines the averaging performed over the randomness of Δ characterized by the
unknown probability density function p ( Δ ), and superscript + stands for Hermitian
conjugate. Following (Shkvarko, 2008), the distortion term Δ in (4) is considered as a
random zero mean matrix with the bounded second-order moment 
e
2
||
| Δ . Vector b
is composed of the elements, b k =
e =  e k e k *  = | e k | 2 ; k = 1, …, K , and is referred to as
a K -D vector-form approximation of the SSP, where represents the second-order
statistical ensemble averaging operator (Barrett & Myers, 2004). The SSP vector b is
associated with the so-called lexicographically ordered image pixels (Barrett & Myers, 2004).
The corresponding conventional K y K x rectangular frame ordered scene image B = { b ( k x , k x );
k x , = 1,…, K x ; k v , = 1,…, K y } relates to its lexicographically ordered vector-form representation
b = { b ( k ); k = 1,…, K = K y K x } via the standard row by row concatenation (so-called
lexicographical reordering) procedure, B = L { b } (Barrett & Myers, 2004). Note that in the
()
k
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