2015年IEEE SAR图像目标识别的格拉斯曼流形上稀疏表示方法

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IEEE《应用地球观测与遥感精选主题杂志》第XX卷第XX期，2015年，聚焦于高级技术在合成孔径雷达（SAR）图像中的目标识别。文章标题"基于格拉斯曼流形上单氏信号稀疏表示的SARTargetRecognition"由Ganggang Dong和Gangyao Kuang，IEEE成员共同撰写。他们提出了一种新颖的方法，利用单氏信号——一种新型的矢量值分析信号，以捕捉SAR图像中丰富的频谱信息并保持最大的空间定位。传统的SAR图像特征提取通常局限于欧几里得空间中的单个特征描述符，这种方法可能无法充分利用数据的非线性结构。然而，作者将单氏信号的不同尺度组件视为格拉斯曼流形上的点，这是一种特殊类型的黎曼流形，其几何特性有助于度量两个模式之间的相似性。不同于以往的线性方法，他们引入了隐式复生产核希尔伯特空间（RKHS）来进一步利用这种非线性结构。在格拉斯曼流形上，通过计算两点间的格拉斯曼距离，可以更准确地评估两个模式的匹配程度。这种方法的优势在于能够捕捉到单氏信号的复杂结构，并将其转化为在RKHS中的有效表示，这在处理SAR图像中的目标识别任务时，显著提高了识别的精度和鲁棒性。通过这种创新的稀疏表示和非线性几何分析，本文的工作对遥感领域，尤其是SAR图像分析，具有重要的理论和实践意义。

IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. XX, NO. XX, XX 2015 3

A. Stiefel and Grassmann manifold

The space of d × m-dimensional matrices (R

d×m

, m < d)

with orthonormal columns is a special type of Riemannian

manifold known as a Stiefel manifold S(m, d), where the no-

tion of normality is formulated by an orthonormality condition,

S(m, d) , Y ∈ R

d×m

: Y

Y = I

(1)

for m by m identity matrix I

. Grassmann manifold G(m, d)

is a mathematical object with several similarities to Stiefel

manifold. It can be deﬁned as a quotient manifold of S(m, d)

with the equivalence relation

∼ Y

if and only if Span(Y

) = Span(Y

) (2)

where Span(Y) denotes the subspace spanned by columns of

Y ∈ S(m, d). Thus, a Grassmann manifold G(m, d) is the set

of m-dimensional linear subspaces of R

G(m, d) is a m(d − m)-dimensional compact Riemanian

manifold. An element of G(m, d) can be represented by an

orthonormal matrix Y of size d by m such that Y

Y = I

Thus, we usually pay more attention to Span(Y ) rather than

the speciﬁc values of Y . Since the matrix representation of a

point in G(m, d) is not unique, two matrices Y

and Y

are

considered the same if and only if Span(Y

) = Span(Y

Propositions 1. For Grassmann manifold G(m, d), Y

∼ Y

if and only if Y

= Y

for some orthonormal matrix R

∈

O(m), where O(·) denotes the orthogonal group.

A canonical distance between two subspaces is the Rieman-

nian distance, i.e., the length of geodesic path connecting two

corresponding points on Grassmann manifold. A more intuitive

and computationally attractive way to deﬁne the geodesic

distance derives from the principal angle. If denote by Y

, Y

two orthonormal matrices with d × m in size, the principal

angles 0 ≤ θ

≤ · · · ≤ θ

≤

between two subspaces

Span(Y

) and Span(Y

), are deﬁned recursively by

cos θ

= max

∈Span(Y

)

max

∈Span(Y

)

subject to u

= 1, v

= 1,

= 0, v

= 0, for j = 1, ..., k − 1.

(3)

The ﬁrst principal angle θ

corresponds to the smallest angle

between a pair of unit vectors each from two subspaces, i.e.,

the ﬁrst canonical correlation. Similarly, the k-th principal

angle and canonical correlation can be described recursively.

Riemannian manifold is a special type of differentiable

manifold with a smooth inner product (Riemannian metric)

endowed [28]. For two tangent vectors 4

, 4

at point Y, its

Riemannian metric is deﬁned as

, 4

= tr(4

−

) = tr(4

). (4)

The metric (4) induces a geodesic distance, the length of short-

est curve connecting two points (m-dimensional subspaces),

namely arc length. It is known that the principal angles are

related to the geodesic distance [20],

Arc

, Y

) =

= kΘk

(5)

for Θ = [θ

, θ

, ..., θ

To obtain the principal angles, it is no need to solve the

maximization problem (3). Instead, it can be derived from the

singular value decomposition of the product of two matrices,

= USV (6)

for unitary matrices U = [u

, ..., u

], V = [v

, ..., v

], and

the diagonal matrix S = diag(cos(θ

), ..., cos(θ

)).

B. Distances for subspaces

In this subsection, distance refers to any assignment of

nonnegative values for a pair of points in space Ω. A valid

metric is a distance that satisﬁes the additional axioms.

Propositions 2 (Metric). A real-valued function δ : Ω×Ω 7→ R

is called a metric if

• δ(x

, x

) ≥ 0;

• δ(x

, x

) = 0 if and only if x

= x

;

• δ(x

, x

) = δ(x

, x

);

• δ(x

, x

) + d(x

, x

) ≥ δ(x

, x

);

for all x

, x

∈ Ω.

According to the distance law, it is able to come the

conclusion that the Grassmann distance should be invariant

under different representations.

Propositions 3. For any distance function δ(·, ·) : R

d×m

7→ R, it is a Grassmann distance if δ(Y

, Y

) =

δ(Y

, Y

), for ∀ R

, R

∈ O(m).

In the following, several distances resulting from the prin-

cipal angles for subspaces are provided [20], [23].

1) Projection distance. The projection distance is deﬁned

as the 2-norm of the sine of principal angle,

P j

, Y

) =

j=1

sin

(θ

) = m −

j=1

cos

(θ

)

= m − kY

− Y

(7)

where k · k

denotes the matrix Frobenius norm. It is

a Grassmann distance and a metric too.

2) Binet-Cauchy distance. The Binet-Cauchy distance is

deﬁned as the product of canonical correlations

, Y

) = 1−

j=1

cos

(θ

) = 1−det(Y

) (8)

It is invariant under different representations, and fur-

thermore is a metric.

3) Max correlation. The max correlation distance derives

from the largest canonical correlation (or the smallest

principal angle)

Max

, Y

) = 1 − cos

(θ

) = sin(θ

). (9)

It is a Grassmann distance, yet not a metric.

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