An l
1
2
and Graph Regularized Subspace Clustering Method 53:5
Denition 3 (Frobenius norm of a Matrix). For a matrix X ∈ R
M×N
, the Frobenius norm of X is
expressed as the square root of the sum of the absolute squares of its elements:
X
F
=
M
i=1
N
j=1
|x
ij
|
2
. (3)
Denition 4 (l
2, 1
norm of a Matrix). Let A ∈ R
M×N
,itsl
2, 1
norm can be dened as
A
2, 1
=
N
i=1
M
j=1
|a
ij
|
2
. (4)
2.1 Feature Vector Generation from Superpixels
In image segmentation problem using clustering techniques, nding similar regions is equivalent
to nd the proper number of clusters in that image [6]. For that, an image need to be partitioned
into a particular set of regions called superpixels [4, 5]. Superpixels of an image are a set of homo-
geneous image patches of arbitrary shapes strictly aligned to the edges present in the image [4, 15].
In the proposed method, a given image, A is rst partitioned into a set of superpixels [29]. After-
wards, Local Spectral Histogram (LSH) features such as Color Histogram (CH), Local Binary
Pattern (LBP), Histograms of Oriented Gradient (HOG) [4], and Local Monotonic Pattern
(LMP) of the superpixels are computed. Then, a feature data matrix X = [x
1
, x
2
, x
3
...x
N
] ∈ R
M×N
is created from the LSH features of the superpixels [6, 30]. Each column vector x
i
∈ X is treated
as an M-dimensional feature vector corresponding to the ith superpixel, where N represents the
total number of superpixels. Furthermore, the features that belong to the same coherent region are
linearly dependent on each other [6]. Superpixels conserve the spatial structure as these patches
carry the coherent content of the image [15].
2.2 Sparse Representation Model
According to the sparse representation model, a datapoint can be represented by fewer datapoints
and corresponding sparse coecients with a suitable dictionary. It utilizes the self-expressiveness
property of datapoints [26]. Due to its superior feature representation ability, sparsity theory facili-
tates many research areas such as matrix completion, compresssed sensing, error correction, image
processing, and computer vision [20, 26, 37]. According to sparsity theory, for a matrix X ∈ R
M×N
,
there exists a sparse solution Z ∈ R
N ×N
such that the non-zero entries of Z corresponds to dat-
apoints belong to the same subspace. It can be represented as X = XZ, diag(Z) = 0, where,
diag(Z) = 0 eliminates the chance of getting trivial solution of datapoints [20, 27]. In the above
expression, the matrix X ∈ R
M×N
itself acts as the dictionary of datapoints. Under this scenario,
Z ∈ R
N ×N
can be recovered with a sparsest structure such that entries of Z has fewest non-zero
components [20, 37]. Then, the problem can be formulated as
min
Z∈R
N ×N
Z
0
s.t. X = XZ, (5)
where .
0
is called l
0
norm and it counts the number of non-zero components of the sparse coe-
cient matrix Z. But, nding l
0
norm is NP-hard [20, 26] such that l
0
norm is replaced by its strong
convex surrogate l
1
norm [37, 42]. Thus, minimization using l
1
norm becomes a convex optimiza-
tion problem such that solution of Equation (5) can be eciently obtained [37]. Then, the convex
optimization problem can be formulated as [26, 37]
min
Z∈R
N ×N
Z
1
s.t. X = XZ. (6)
ACM Trans. Multimedia Comput. Commun. Appl., Vol. 18, No. 2, Article 53. Publication date: February 2022.
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