Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 10

Introductory View of Anomalous Change Detection
 μˆ ( i, j ) 
μˆ 0 ( i, j , l , p ) =  y

μˆ z ( i, j , l , p ) 
 Γˆ y ( i, j )
Γˆ yz ( i, j , l , p ) 

Γˆ 0 =  T
 Γˆ yz ( i, j , l , p ) Γˆ z ( i, j , l , p ) 
 Γˆ y ( i, j )

0

Γˆ 1 = 
 0
Γˆ z ( i, j , l , p ) 

(31)

In the case of difference vector model based algorithms (SDHACD
and SDACD), T ( i, j , r , p ) has the form:
T
T ( i, j , r , p ) = e ( i, j , r , p ) ⋅ C ( i, j , r , p ) ⋅ e ( i, j , r , p )

(32)

e ( i, j , r , p ) = y ( i, j ) − z ( i + r , j + p )

with C(i, j, r, p) obtained as Γˆ 0−1 ( i, j , r , p ) − Γˆ 1−1 ( i, j , r , p ) for the
SDHACD and Γˆ 0−1 ( i, j , r , p ) for the SDACD and
Γˆ 0 ( i, j , l , p ) = Γˆ y ( i, j ) + Γˆ z ( i, j , l , p )
−  Γˆ yz ( i, j , l , p ) + Γˆ Tyz ( i, j , l , p ) 


Γˆ 1 ( i, j , l , p ) = Γˆ y ( i, j ) + Γˆ z ( i, j , l , p )

(33)

(

(

)

T

⋅ Γˆ −z 1 ( i, j , r , p ) ⋅ y ( i, j ) − μˆ z ( i, j , r , p )

)

(34)

where μˆ z ( i, j , r , p ) and Γˆ z ( i, j , r , p ) are those defined in (29). It is
worth noting that the SVACD does not require the estimation of the
joint statistics between the two imperfectly coregistered images.
Thus, it adapts well to the LCRA approach and it is expected to be
less influenced by the RMRE than the other algorithms.
It is worth noting that the LCRA approach has an asymmetric nature, in that it allows the changes in a given direction to be
detected. Particularly, such an approach is suitable for detecting
those objects that appear in the test image and are not present in
the reference image. Of course, one can modify the direction of
detected changes by inverting the test and the reference image in
the expression of each detector [58].

DEALING WITH RADIOMETRIC DIFFERENCES
In general, multitemporal images are acquired in different atmospheric/illumination conditions. Thus, the at-sensor radiance from
a same object is different in the reference and test images. This
may generate pervasive and uninteresting changes that must be
10

y ( i, j ) = R ⋅ ψ ( i, j ) + b
1
−1
R = Γˆ z2 ⋅ Γˆ ψ 2

In this case also, the inaccuracy in the above reported sample estimates might be mitigated by the spatial correlation of the pixels in
each image when the RMRE is on the order of a few pixels.
Finally, as to the SVACD ,the expression of T ( i, j , r , p ) is
T ( i, j , r , p ) = y ( i, j ) − μˆ z ( i, j , r , p )

properly taken into account in the ACD strategy. Several algorithms have been proposed in the past to cope with this problem.
The CC method [50], which is the optimal linear mean square error
predictor, provides a benchmark approach to transform the radiance signatures of the test image Y making them suitable for the
comparison with the reference image Z collected at another time.
Nominally, the CC requires perfect registration because the transformation parameters depend on the cross-covariance matrix between the test and the reference image. Thus, if RMRE affects the
images, the use of such optimal predictor is not appropriate and a
suboptimal method has to be adopted. The whitening/de-whitening
(WD) [34] technique performs a transformation on the test image
by rotating, translating, and scaling the original hyperspectral pixels in order to obtain a transformed hyperspectral image having
the same mean spectrum and covariance matrix as the reference
image. In the framework of change detection applications, WD is
often referred to as CE [49]. Differently from CC, which relies
upon a cross-covariance matrix for prediction, CE is based solely
on the individual covariance matrices of the test and the reference
images and is therefore less sensitive to RMRE.
The CE/WD method estimates the image Y by applying to each
pixel ψ(i, j) of the original test image Ψ the following linear transformation:

(35)

b = μˆ z − Γˆ z ⋅ Γˆ ψ ⋅ μˆ ψ
1
2

− 12

where μˆ ψ , μˆ z and Γˆ ψ and Γˆ z are the estimates of the mean spectrum
and the covariance matrix of Ψ and Z, respectively. Such estimates
are obtained by considering all the pixels of the corresponding image.
Note that both the SACD and the HACD test statistics (TSACD(i,
j) and THACD(i, j)) are invariant to a linear transformation operated
on one or both the vectors y(i, j) and z(i, j) in e(i, j) (see Appendix
II). Thus, the results provided by the two algorithms are the same
independently on the CE transformation. In other words, when
the radiometric differences between the test and the reference images can be modeled as a linear transformation, the SACD and the
HACD do not require any additional processing for RE.
It is important to remark that both CC and CE rely on the assumption that the changes in the illumination and the atmospheric
and the viewing conditions are spatially invariant, i.e., they are
the same all over the considered scene. In practice, there are cases
where this assumption is not valid as, for example, in the presence
of different shadowed areas in the two images. To cope with spatially variant acquisition conditions more sophisticated techniques
need to be applied [16], [21], [52]. For instance, in [21] various
cluster based RE procedures are proposed to account for the variability of the acquisition conditions within the scene.

USEFUL CONSIDERATIONS
In this section we discuss some important issues concerning the presented ACD algorithms. First, we would briefly discuss the issue

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JULY 2017, Part II of II



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