# Meta-Pixel-Driven Embeddable Discriminative Target and Background Dictionary Pair Learning for Hyperspectral Target Detection

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## Abstract

**:**

## 1. Introduction

- (1)
- The meta-pixel set for HSI is defined by inheriting the merits of the homogeneous superpixel spectral property and the local manifold affinity structure, which can significantly reduce the influence of the spectral variability and find the most informative and prototype spectral signatures of HSI.
- (2)
- A Meta-pixel-driven Embeddable Discriminative background and target Dictionary Pair (MEDDP) learning model is established to efficiently learn a discriminative and compact background dictionary from the constructed meta-pixel set by introducing the discriminative structural incoherence. In addition, an adaptive low-dimensional embeddable subspace is jointly derived to reduce spectral redundancy and extract meaningful features, which can lead to more accurate characterizations of the target and background.
- (3)
- An efficient optimization algorithm is designed to solve the MEDDP model. The key variables, i.e., the background dictionary and orthogonal embeddable projection matrix are optimized iteratively to find the satisfied solutions. Furthermore, a novel meta-pixel-level target detection is performed based on the MEDDP model and some representation learning strategies. Experiments on several benchmark HSI datasets verify the effectiveness of the proposed method in comparison with several state-of-the-art HSI target detectors.

## 2. Related Works

#### 2.1. Low-Rank Modeling

#### 2.2. Sparse Representation Theory-Based HSI Target Detection

_{0}-norm based sparsity regularization.

_{0}-norm ${\Vert \xb7\Vert}_{0}$ counts the number of the nonzero elements of a vector. The solution can be obtained via some sparse optimization algorithms, e.g., orthogonal matching pursuit (OMP) [32]. However, the problem of searching for the sparest solution of an underdetermined linear equation is NP-hard [33]. Recent advances in the sparse representation and compressed sensing theories reveal that if the solution $c$ is sufficiently sparse, one can use the following l

_{1}-minimization problem as a surrogate for the l

_{0}-minimization problem (4).

_{0}-norm minimization problems.

## 3. Metal-Pixel-Driven Embeddable Discriminative Target and Background Dictionary Pair Learning for HSI Target Detection

#### 3.1. Meta-Pixel Set Construction

#### 3.2. MEDDP Model Formulation

**,**as formulated below,

#### 3.3. MEDDP Model Optimization

- (1)
- Update$P$by solving the following problem with the other variables fixed.

- (2)
- Update$J$with the other variables fixed and solve the following problem.

- (3)
- Update${C}_{t}$with the other variables fixed by solving the following problem.

- (4)
- Update${D}_{b}$with the other variables fixed by solving the following problem.

- (5)
- Update$L$with the other variables fixed by solving the following problem.

- (6)
- Update the Lagrange multipliers and penalty parameter:

Algorithm 1. Solving problem (18) using Inexact ALM. |

Input: Meta-pixel set ${M}_{X}$, target prior spectra ${D}_{t},\text{}{\mu}_{max}={10}^{8},\text{}\rho =1.1,\text{}\mathsf{\u03f5}={10}^{-6},\text{}\mu ={10}^{-5},\text{}\alpha ,\text{}\beta ,\text{}\mathrm{and}\text{}\gamma $. Reduced dimension d. |

Initialization: Initialize $P$ by PCA, ${D}_{b}={C}_{t}={Y}_{1}={Y}_{2}=0$. |

Whilenot convergencedo |

1. Update $\mathbf{P},\text{}\mathbf{J},\text{}{\mathbf{C}}_{t},\text{}{\mathbf{D}}_{b},\text{}\mathbf{L}$ by successively solving the sub-problems in (23), (25), (27), (30) and (33). |

2. Update the Lagrange multipliers and penalty parameter as in (19). |

3. Examine the convergence conditions: $\Vert {\mathbf{D}}_{b}-\mathbf{J}{\Vert}_{F}^{2}/\Vert {\mathbf{M}}_{\mathbf{X}}{\Vert}_{F}^{2}<\mathsf{\u03f5}\text{}\mathrm{and}\text{}\Vert {\mathbf{C}}_{t}-\mathbf{L}{\Vert}_{F}^{2}/\Vert {\mathbf{M}}_{\mathbf{X}}{\Vert}_{F}^{2}\mathsf{\u03f5}$ |

End while |

Output:${D}_{b}\text{}\mathrm{and}\text{}\mathbf{P}$. |

#### 3.4. Meta-Pixel-Level Target Detection Based on MEDDP Model

_{0}-norm minimization problems.

Algorithm 2. MEDDP and meta-pixel-based target detection. |

Input: HSI dataset $\mathrm{X}$, target prior spectra ${\mathbf{D}}_{\mathit{t}}$, tradeoff parameters
$,\text{}\alpha ,\text{}\beta ,\text{}\mathrm{and}\text{}\gamma $. Reduced dimension d. |

1. Construct a training meta-pixel set of $\mathrm{X}$ based on ERS and local manifold preservation. |

2. Obtain the optimal target and background dictionary pair by solving Algorithm 1. |

3. Construct testing meta-pixel set, and use the obtained dictionary pair for me-ta-pixel-level target detection via different representation-based target detection strat-egies, such as the SRBBH presented in (35)–(37). |

Output: Detection map |

## 4. Experimental Verifications

#### 4.1. Benchmark HSI Datasets

#### 4.2. Comparison Methods and Performance Evaluation Metrics

- (1)
- ACE: ACE is a background unstructured detector by assuming that the background has the same covariance structure but different variances under the two hypotheses [9].
- (2)
- CEM: CEM detects target by designing a finite impulse response filter (FIRF) using the known target spectrum and minimizing the energy of the interference signal. However, CEM fails to consider the assumption of data distribution, which will restrict its performance [37].
- (3)
- SMF: Different from CEM, the SMF detector estimates the background covariance matrix and then employs the generalized likelihood ratio test for detection with only a single target spectrum, which cannot fully model the diversity of target spectra [10].
- (4)
- SRD: SRD represents a test pixel using the target and background combined dictionary, and then determines the label of test pixel (background or target) by examining which sub-dictionary yields a smaller representation residual for the test pixel [18].
- (5)
- SRBBH: SRBBH combines the idea of binary hypothesis and sparse representation, in which the test pixel is respectively represented by the background dictionary, and the background and target combined dictionary under the two hypotheses that the target is present or absent. The derived representation errors under the two hypotheses are used for the final detection decision [19].
- (6)
- BCRD: In BCRD, both pixels in the background and pixels in the target can be collaboratively represented by some pixels of the image. The detection result is achieved by estimating the residual difference of two collaborative representations [20].
- (7)
- SLRMDD: SLRMDD is based on sparse and low-rank matrix decomposition and regards the given HSI as a composition of the sum of low-rank background HSI and a sparse target HSI containing targets via a target dictionary constructed from some online spectral libraries. Strategy one is used for target detection by combining the separated background dictionary with the SRBBH detector. The ratio between the two key parameters is set as 5/2 [4].

#### 4.3. Qualitative and Quantitative Results

#### 4.4. Parameters Analysis and Convergence Analysis

- (1)
- Influence of the Reduced Dimensionality d.

- (2)
- Influence of Number of Training Meta-Pixel C.

- (3)
- Impact of the Number of Testing Meta-Pixel V.

- (4)
- Influence of the Balancing Parameters α, β, and γ.

^{−4}, 10

^{−3}, 10

^{−2}, 10

^{−1}, 10

^{0}, 10

^{1}, 10

^{2}, 10

^{3}, 10

^{4}}. The performance variation of the detector regarding the log-scaled β and γ were reported in Figure 12. From the MEDDP model formulation, one can see that a larger α will help enhance the discrimination between the target and background dictionaries by minimizing their correction, which can be observed from the experimental curves in Figure 12. However, an extreme large might lead to over-fitting, thus the optimal α can be selected from [10

^{−1}, 10

^{2}]. In addition, the ROC performance variations for our proposed detectors w.r.t different settings of the balance parameters β and γ with the optimal α fixed on different data sets are shown in Figure 13. The detection performances show higher sensitivity to the settings of β and γ. By comprehensively analyzing the performance variations rule w.r.t the different combinations of β and γ, the suggested setting ranges for the two parameters are β ∈ [10

^{1}, 10

^{4}], γ ∈ [10

^{−1}, 10

^{2}]. There is a higher probability that stable and satisfying performance can be yielded with these suggested parameters.

- (5)
- Convergence Analysis of the Optimization Algorithm

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Overview of the proposed HSI target detection method. In the training stage, the observed HSI data is segmented by entropy rate superpixel segmentation method, and then the training meta-pixel set is constructed, which is further decomposed to get a discriminative target and background dictionary pair with the guidance of target spectra and the structurally incoherent regularization in an adaptive lower-dimensional embeddable subspace. In the testing stage, the HSI data is segmented with a finer scale to construct the testing meta-pixel set as in the training stage. The discriminative target and background dictionary pair obtained in the training stage are then combined with some representative representation learning-based methods, such as the SRD, SRBBH, and BCRD, for meta-pixel level target detection.

**Figure 2.**Illustration for the center pixel and meta-pixel. As shown in (

**a**), the center pixel equally merges the pixels in the superpixel. Differently, contributions of the pixels in superpixel are weighted by considering the local manifold affinity structure between different pixels in the superpixel and finding the key typical spectral signature in each superpixel, i.e., meta-pixel, as in (

**b**).

**Figure 3.**The HSI dataset and the corresponding ground-truth used in the experiments. (

**a**) AVIRIS I dataset, (

**b**) AVIRIS II dataset, (

**c**) The Indian Pines dataset, (

**d**) The HYDICE dataset.

**Figure 4.**Visual comparisons between the detection maps of the proposed method and other comparative methods on the AVIRIS I dataset.

**Figure 5.**Visual comparisons between the Detection maps of the proposed method and other comparative methods on the AVIRIS II dataset.

**Figure 6.**Visual comparisons between the Detection maps of the proposed method and other comparing methods on the Indian Pines dataset.

**Figure 7.**Visual comparisons between the Detection maps of the proposed method and other comparative methods on the HYDICE dataset.

**Figure 8.**ROC performance for all the comparative detectors on different data sets. (

**a**) AVIRIS I dataset, (

**b**) AVIRIS II dataset, (

**c**) Indian Pines dataset, (

**d**) HYDICE dataset.

**Figure 9.**The ROC performance variations for our proposed detectors with different reduced dimensionality d on different data sets. (

**a**) MEDDP + SRD, (

**b**) MEDDP + SRBBH, (

**c**) MEDDP + BCRD.

**Figure 10.**The ROC performance variations for our proposed detectors with different number of training meta-pixel C on different data sets. (

**a**) MEDDP + SRD, (

**b**) MEDDP + SRBBH, (

**c**) MEDDP + BCRD.

**Figure 11.**The ROC performance variations for our proposed detectors with different number of testing meta-pixel V on different data sets. (

**a**) MEDDP + SRD, (

**b**) MEDDP + SRBBH, (

**c**) MEDDP + BCRD.

**Figure 12.**The ROC performance variations for our proposed detectors with different settings of the balance parameter α on different data sets. (

**a**) MEDDP + SRD, (

**b**) MEDDP + SRBBH, (

**c**) MEDDP + BCRD.

**Figure 13.**The ROC performance variations for our proposed detectors with different settings of the balance parameters β and γ with α fixed on different data sets. (

**a**) MEDDP + SRD on the AVIRIS I data set with α = 100; (

**b**) MEDDP + BCRD on the AVIRIS II data set with α = 10; (

**c**) MEDDP + SRBBH on the Indian Pines data set with α = 0.01; (

**d**) MEDDP + BCRD on the HYDICE data set with α = 0.1.

**Figure 14.**The convergence curves of Algorithm 1 for solving the proposed MEDDP model on different data sets. (

**a**) AVIRIS I, (

**b**) AVIRIS II, (

**c**) Indian Pines, (

**d**) HYDICE.

Notation | Meaning | Notation | Meaning |
---|---|---|---|

X | Observed HSI dataset | D_{t}, D_{b} | Target and background samples (dictionaries) |

L | Low-rank component of X | ${S}_{X}=\left\{{S}_{c}\right\}$, c = 1,2,…C | Superpixel set of X |

E | Sparse noise component of X | ${M}_{X}=\left\{{m}_{c}\right\}$, c = 1,2,…C | Meta-pixel set of X |

P | Embeddable projection matrix | ${\mathsf{\delta}}_{c}$ | Center pixel of the cth superpixel |

α, β, γ,λ | Tradeoff parameters | ${C}_{t}$ | Sparse target representation matrix |

Detectors | Datasets | |||
---|---|---|---|---|

AVIRIS I | AVIRIS II | Indian Pines | HYDICE | |

ACE | 0.7843 | 0.4716 | 0.4067 | 0.8903 |

CEM | 0.7069 | 0.7011 | 0.4794 | 0.9121 |

SMF | 0.7933 | 0.7299 | 0.6328 | 0.9147 |

SRD | 0.9629 | 0.9564 | 0.2155 | 0.9654 |

BCRD | 0.9774 | 0.9934 | 0.9969 | 0.9495 |

SRBBH | 0.9055 | 0.7675 | 0.5669 | 0.9214 |

SLRMDD | 0.6603 | 0.6211 | 0.5719 | 0.7982 |

MEDDP + SRD | 0.9826 | 0.9927 | 0.9988 | 0.9927 |

MEDDP + SRBBH | 0.9735 | 0.9921 | 0.9979 | 0.9879 |

MEDDP + BCRD | 0.9825 | 0.9943 | 0.9969 | 0.9895 |

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## Share and Cite

**MDPI and ACS Style**

Guo, T.; Luo, F.; Fang, L.; Zhang, B.
Meta-Pixel-Driven Embeddable Discriminative Target and Background Dictionary Pair Learning for Hyperspectral Target Detection. *Remote Sens.* **2022**, *14*, 481.
https://doi.org/10.3390/rs14030481

**AMA Style**

Guo T, Luo F, Fang L, Zhang B.
Meta-Pixel-Driven Embeddable Discriminative Target and Background Dictionary Pair Learning for Hyperspectral Target Detection. *Remote Sensing*. 2022; 14(3):481.
https://doi.org/10.3390/rs14030481

**Chicago/Turabian Style**

Guo, Tan, Fulin Luo, Leyuan Fang, and Bob Zhang.
2022. "Meta-Pixel-Driven Embeddable Discriminative Target and Background Dictionary Pair Learning for Hyperspectral Target Detection" *Remote Sensing* 14, no. 3: 481.
https://doi.org/10.3390/rs14030481