IEEE - Aerospace and Electronic Systems - March 2021 - 62

Automatic Target Recognition on Synthetic Aperture Radar Imagery: A Survey
Table 6.

Attributed Scattering Centers
No

Reference

SAR
chip
size

Pre-processing

Main features

1

Li and Du [53]

64x64

-

L2-norm minimum reconstruction error for
classification

2

Zhang [63]

128x128

-

Hungarian algorithm for classification

3

Ding et al. [66]

128x128

-

D-S theory for classification

4

Ding et al. [65]

variable

morphological operations

SVM for classification

5

Ding et al. [60]

128x128

-

two-step classification: coarse using Hungarian
algorithm and fine using D-S theory

6

Ding et al. [54]

128x128

CFAR, morphological operations,
watershed segmentation

Hungarian algorithm for classification

7

Ding and Wen [64]

128x128

-

feature fusion, score-based classification

algorithm to create a set of low-level feature maps. The
latter are sparsely encoded using a proposed locality constraint discriminative dictionary learning (LcLcDDL) that
is created during the training phase. Then, the low-level
local features are converted into high-level global features
by employing a spatial pyramid matching scheme.

However, advances in the SRC domain extend to several values within the lp À norm space p 2 ð0; 1Þ.
The atoms in a are ideally related to samples that
belong to the same class as y. Hence, during the SRC classification y is assigned to the class that fulfills the following constraint:

SPARSE REPRESENTATION CLASSIFICATION (SRC)

Á
À
classðyÞ ¼ arg minðek Þ ¼ arg min y À X k a Ãk 

It has been proven that a test image y can be adequately
represented by intraclass training templates. Given X ¼
½X
X 1 ; . . . ; X k ; . . . ; X c  2 Rmxn the n templates of size m,
X k ¼ ½x
xk;1 ; x k;2 ; . . . ; xk;nk  2 Rmxnk a template subset
containing nk templates from the kth class with n ¼
P
nk , the SRC theory assumes that if y belongs to the kth
class, it will approximately lie in the linear span of X k ,
y % X k a k , where ak ¼ ½ak;1 ; ak;2 ; . . . ; ak;nk T 2 Rnk is the
coefficient vector with entries being the weight, i.e.,
atoms, of the corresponding training samples in X k , also
known as the dictionary.
Given that, the test image class is not a priori known,
SRC assumes that y is a linear combination of the entire
dataset X , the cardinality of the object classes is large,
and the representation of y is naturally sparse, SRC aims
at finding the sparsest solution to y ¼ X a, which can be
derived by solving the optimization problem

where aÃk is a vector with all entries zero except the ones
associated with class k and ek is the reconstruction error.
In the context of SAR ATR, the various SRC methods utilize different dictionary constructions and lp À norm solutions. An overview of the current methods is presented in
Table 7.
Wei et al. [68] suggested a fast discriminative dictionary learning (FaDDL) method that reduces the execution
time without losing the classification accuracy, by adopting an l1;1 À norm ball as a constraint to replace l0 À
norm and l1 À norm that are typically used with the discriminative dictionary learning dictionaries. For completeness, it is worth noting that the general formulation of the
l1;1 À norm is; considering the projection of a matrix A
to the l1;1 ball to find a matrix B that solves the convex
optimization problem [69]:

a Ã ¼ arg minky À Xa k2F þ g kak0
a

(3)

B;m

where a Ã ¼ ½aaÃ1 ; . . . ; a Ãk ; . . . ; a Ãc , g ! 0 and k Á k0 is the
l0 À norm counting the nonzero entries in a . Provided
that the problem in (3) is NP-hard, compressive sensing
theory [67] suggests that if y is sparse enough the initial
signal can be recovered by substituting the l0 À norm
with the l1 À norm
aÃ ¼ arg minky À Xa k2F þ g kak1 :
a

62

l1;1 : min

(4)

Á2
1XÀ
Bi;j À Ai;j
2 i;j

s:t: 8i; jBi;j mi
P
i mi < C
8i; jBi;j ! 0

(5)

(6Þ

(7Þ

8imi ! 0
where (6) is the Euclidean distance between A and B,
given that B is in the boundary of l1;1 ball of radius C.
Variables mi constrain the coefficients of B as of (7).

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IEEE - Aerospace and Electronic Systems - March 2021

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