高光谱图像压缩：基于补丁的低秩张量分解方法

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"PLTD：高光谱图像的基于补丁的低秩张量分解" 这篇研究论文探讨了高光谱图像（HSI）处理中的一个关键问题，即数据量庞大和冗余信息的问题。高光谱图像由于其丰富的光谱信息，通常包含大量数据，这在很多应用中成为处理的瓶颈。为了缓解这一问题，论文提出了基于补丁的低秩张量分解（PLTD）的新方法来实现HSI的压缩和重构。传统的图像编码算法往往侧重于空间维度的压缩，而忽略了光谱维度的冗余。PLTD方法则打破了这一局限，它首先将HSI划分为一系列局部补丁，并将每个补丁表示为一个三维张量，这考虑到了HSI的空间和光谱特性。接着，通过聚类算法将相似的张量补丁分组，形成簇，每个簇对应一个四维张量。低秩张量分解是PLTD的核心。由于同一簇内的张量补丁具有相似性，组合成的四维张量在一定程度上可以被表示为低秩的形式，这意味着可以通过较少的基来近似整个张量。这种方法有效地利用了HSI的内在结构，降低了数据的表示复杂度，从而实现了有效的压缩。在压缩过程中，PLTD通过保持张量的低秩属性来保留HSI的关键信息。而在解压或重构阶段，低秩张量分解后的数据可以被恢复，重构出与原始HSI质量相近的图像。这一过程不仅减少了数据存储和传输的需求，还能在一定程度上降低噪声影响，提高图像质量和分析效率。此外，PLTD方法的创新性在于它结合了补丁处理、张量理论以及聚类算法，提供了一种新的HSI处理框架。这种方法对于HSI的分类、目标检测、异常检测等应用都具有潜在价值，可能推动HSI在环境监测、遥感、军事侦察等领域的发展。 "PLTD：高光谱图像的基于补丁的低秩张量分解"这篇论文提出了一种高效且有针对性的HSI压缩方法，它充分利用了HSI的多维特性，有望改善现有技术在处理大数据量HSI时的性能，对于理解和应用HSI的低秩特性有深远影响。

DU et al.: PLTD: FOR HSI 69

projection [54]. Although, at ﬁrst glance, these methods may

appear suitable for HSI compression and reconstruction, they

are basically designed for a set of tensor objects, and they can-

not perform feature reduction with only single tensor data as the

input.

To fully address the advantages of tensor representation for

HSI compression, we propose a novel patch-based low-rank

tensor decomposition (PLTD) method, which combines patch-

based tensor representation and tensor dictionary learning. In-

stead of compressing the HSI by spectral channel or by pixel,

we represent each local patch of the HSI as a third-order tensor,

which helps to preserve both the spatial and spectral correlations.

Matrix dictionary learning [55] aims to decompose the observed

data matrix to a dictionary and a sparse coefﬁcient, respectively,

which leads to a low-rank representation of the input matrix. In

order to share the nonlocal similarity over the spatial domain

[56], [57], the similar tensor patches are grouped by a cluster-

ing algorithm, and we construct a fourth-order tensor per cluster.

Since the grouped tensor is assumed to be redundant, by extend-

ing the matrix dictionary learning to tensor dictionary learning,

each cluster can be low-rank decomposed to a sparse coefﬁcient

tensor and three dictionary matrices, respectively. Since the co-

efﬁcient tensor is sparse, we proposed to extract all the nonzero

entries to form an intrinsic dense tensor whose dimensions along

the two spatial modes and an additional spectral mode are much

smaller than the original cluster, thus leading to simultaneous

compression of both the spatial and spectral domains. Finally,

the reconstructed HSI is recovered by the combination of the

reconstructed patches, which are obtained by the tensor product

of the coefﬁcient and dictionaries per cluster. In this way, the

proposed PLTD algorithm optimally removes the redundancy in

both the spatial and spectral domains in a uniﬁed framework.

The proposed PLTD algorithm can overcome the shortcom-

ings of the existing gray-level based image compression meth-

ods. For example, JPEG and SPIHT can only deal with the

spatial redundancy, while the high dimension of the spectral

channel is left unchanged; PCA treats each pixel vector as an

individual observation, and thus only the spectral dimension is

considered while the redundancy in the spatial dimensions is ig-

nored in the compression, which results in a poor compression

ratio in practice; MPCA is not suitable for single tensor data

as the input if it is used to address both the spatial and spec-

tral redundancies. In summary, these methods cannot take full

advantage of the characteristics of HSIs. In contrast, the pro-

posed PLTD algorithm is expected to be able to overcome these

limitations. Patch-based tensor representation helps to preserve

both the spatial and spectral correlations, and the tensor dictio-

nary learning and low-rank decomposition make it possible to

simultaneously remove the redundancy in both the spectral and

spatial domains.

The main contributions of the proposed PLTD algorithm are

summarized as follows:

1) A patch-based tensor representation for HSI data com-

pression is proposed, and the nonlocal similarity measure

of nonlocal patches in the spatial domain is employed. A

tensor can describe 3D data as an entirety, keeping the

spatial and spectral correlations at the same time. In the

PLTD algorithm, the HSI cube is split into many over-

lapping tensor patches in the spatial domain by a ﬁxed

local window, while the spectral signature is unchanged

in the patch-based tensor representation. In this way, the

proposed method can keep as much of the spatial and

spectral correlation as possible, which may be very im-

portant in the subsequent HSI analysis. Through the for-

mulated nonlocal similarity, similar patches are grouped

together. Patches in a certain cluster can be considered as a

fourth-order tensor, which helps to share the nonlocal sim-

ilarity among patches and leads to a better compression

performance.

2) The tensor dictionary learning is generalized based on the

matrix dictionary learning, and a new deﬁnition of tensor

sparsity is provided to support this generalization. By us-

ing the suggested tensor sparsity, we can simultaneously

enforce the sparse constraint of an HSI data patch on the

height, width, and spectral modes. Based on this sparsity,

the matrix dictionary learning is extended to the tensor-

based application, and the tensor cluster is decomposed to

the sparse coefﬁcient tensor and a few matrix dictionaries.

With the constraint that the coefﬁcient should be sparse

in both the spatial and spectral modes, the data size of the

intrinsic dense tensor is far less than the original tensor,

and thus both the spatial and spectral redundancies can be

compressed.

3) A low-rank tensor decomposition based optimization al-

gorithm is developed. In the proposed method, in order to

ﬁnd the optimal coefﬁcient tensor and the corresponding

dictionaries for each cluster, we extend the matrix-based

dictionary learning to a tensor version according to multi-

linear algebra, and we further develop an optimization ap-

proach based on low-rank tensor decomposition. Through

such tensor decomposition, an HSI is split into numerous

clusters, while each of the clusters is represented by a co-

efﬁcient tensor and its dictionaries, by which the data size

of the coefﬁcient tensor is far less than the original cluster

from the HSI.

The rest of this paper is organized as follows. Section II

introduces the notation and some basic multilinear algebra.

Section III presents the PLTD algorithm for HSI compression

and reconstruction. The experimental results and analysis are

provided in Section IV, followed by the summary and conclu-

sion in Section V.

II. N

OTATION AND BASIC MULTILINEAR ALGEBRA

In this section, we introduce the notation and multilinear

algebra used in the paper. Since a tensor is a multidimensional

array of numbers that transforms linearly under a coordinate

transformation, we distinguish the vectors (ﬁrst-order tensor),

matrices (second-order tensor), and tensors which are higher

than the second-order as follows. A vector is denoted as an

italic lowercase letter such as a, a matrix is denoted as a bold

italic uppercase letter such as A, and an Nth-order tensor is

denoted as a calligraphic uppercase letter A∈R

× I

× ···× I

The elements of an object (including a vector and matrix) are

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