IEEE Geoscience and Remote Sensing Magazine - March 2021 - 118

where I ^ x, y h is a random variable with average intensity
n I and variance s 2I , v ^ x, y h is the RCS with mean n v and
variance s 2v, and n ^ x, y h is the normalized speckle random
variable with mean n n = 1 and variance s 2n = 1/N, where N
is the number of looks.
The autocorrelation function and autocorrelation coefficient of SAR intensity image I are, respectively,

	

R I ^Tx, Ty; N h = < I ^ x, y h I ^ x + Tx, y + Ty h>, (2)
t I ^ Tx , Ty h =

R I ^Tx, Ty h - n I2
, (3)
s I2

where $ denotes taking the expectation, Tx and Ty indicate, respectively, the distances in the azimuth and range
directions. Assuming that v ^ x, y h and n ^ x, y h are independent random variables, the autocorrelation function of I is,
according to this model [20],
	

R I ^Tx, Ty ; N h = R v ^Tx, Ty h R n ^Tx, Ty ; N h, (4)

where R v ^Tx, Ty h is the autocorrelation function of the
RCS v ^ x, y h, and R n ^Tx, Ty; N h represents the autocorrelation function of the speckle n ^ x, y h . It can be seen that the
correlation functions also satisfy the product relationship,
and R I is determined by the autocorrelation functions of
the speckle R n and the RCS R v .
The speckle is white noise, and there is no correlation in
itself. The correlation shown in the SAR image is introduced

Ty
1
Tx
R n ^Tx, Ty; N h = N sinc 2 a rx k sinc 2 c ry m + 1, (5)

	

where rx and ry are the corresponding spatial resolution in
the azimuth and range directions of the sensor. The corresponding autocovariance is
C n ^Tx, Ty; N h = R n ^Tx, Ty; N h - n n2

Ty 
1
Tx
= N sinc 2 a rx k sinc 2 c ry m .

	

In [20], the authors pointed out that, if the pixel interval
(or the sampling interval) is greater than the spatial resolution, the correlation of neighboring pixels can be ignored.
Generally, if the pixel distance is 0.5-1 spatial-resolution
cell (that is, oversampling), two adjacent pixels are correlated, and two pixels with an interval greater than one
pixel are uncorrelated.
Similar to (3), the autocorrelation function of the RCS
can be written as
R v ^Tx, Ty h = s v2 t v ^Tx, Ty h + n v2, (7)

	

where t v ^Tx, Ty h is the autocorrelation coefficient of the
RCS. Substituting (3), (7), and n v2 = n 2I into (4) obtains

Based on
Product Model

Speckle Correlation
Function
RCS Correlation
Function
Birth-Death-Migration Model
Number of
Scatterers

Based on Coherent
Scatterer Model

(6)

Obeys the Negative
Binomial Distribution
Physical Modeling
of a Single Scatterer

Single
Scatterer

Imaging
Impulse
Response

Statistical Modeling
of a Single Scatterer

Correlation Due to
Fluctuation of the
Number of Scatterers

Correlation Due to
Fluctuation of a
Single Scatterer

Gaussian Function
Rectangular Function
Sinc Function

Correlation Due to
Imaging Impulse
Response

Correlation Function of Observed SAR Images

	

during the imaging focus process. The authors of [20] gave
the autocorrelation function of the speckle of the intensity image

Delta Function
Other Functions
FIGURE 2. The context of spatial correlation analysis.

118

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

MARCH 2021
IEEE Geoscience and Remote Sensing Magazine - March 2021

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