L12与图正则化子空间聚类方法：鲁棒图像分割策略

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"《基于l12范数和图正则化的子空间聚类方法：一种鲁棒图像分割策略》(An l12 and Graph Regularized Subspace Clustering Method for Robust Image Segmentation)是一篇探讨在复杂图像分析背景下提高图像分割性能的研究论文。作者Jobin Francis、M. Baburaj和Sudhish N. George来自印度卡利库特国家技术学院，他们针对图像分割问题提出了一个无监督的解决方案，该方法将图像分割视为特征向量的子空间聚类。首先，论文中采用了一种超像素划分技术，将原始图像划分为一系列具有相似视觉特征的区域，这有助于减少背景噪声和纹理混淆的影响。接着，从每个超像素中提取Local Spectral Histogram特征，这是一种用于捕捉局部图像结构的重要特征表示方式。关键创新在于，他们构建了一个特征数据矩阵，然后应用子空间聚类算法。传统的l1范数在处理稀疏性和噪声方面表现出色，但在保持图像细节的同时可能会丢失某些信息。因此，论文引入了l12范数，结合了l1的稀疏性和l2的平滑性，这使得模型能够在保持图像分割精确性的同时，更有效地抑制噪声和异常值。为了进一步提升模型的稳健性，作者还引入了图正则化。图正则化利用图像邻域的信息，鼓励相邻区域在聚类结果上保持一致，这有助于在面对相似纹理和背景的情况下，增强分割的连贯性和一致性。整个方法通过单阶段优化模型实现，旨在通过l12范数和图正则化的协同作用，提供一个既能抵抗干扰又能捕捉图像内在结构的高效图像分割解决方案。这种无监督学习方法在图像分析领域具有广泛应用前景，如计算机视觉、物体识别和图像内容理解等领域。这篇毕业设计论文不仅阐述了理论框架，还可能包含实验结果和对比分析，展示了该方法在实际图像分割任务中的优越性能。"

An l

and Graph Regularized Subspace Clustering Method 53:5

Denition 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









i=1



j=1

. (3)

Denition 4 (l

2, 1

norm of a Matrix). Let A ∈ R

M×N

,itsl

2, 1

norm can be dened as

A

2, 1



i=1









j=1

. (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

, x

...x

] ∈ R

M×N

is created from the LSH features of the superpixels [6, 30]. Each column vector x

∈ 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 coecients 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

s.t. X = XZ, (5)

where .

is called l

norm and it counts the number of non-zero components of the sparse coe-

cient matrix Z. But, nding l

norm is NP-hard [20, 26] such that l

norm is replaced by its strong

convex surrogate l

norm [37, 42]. Thus, minimization using l

norm becomes a convex optimiza-

tion problem such that solution of Equation (5) can be eciently obtained [37]. Then, the convex

optimization problem can be formulated as [26, 37]

min

Z∈R

N ×N

Z

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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