Low-Rank and Sparse Representation with Adaptive Neighborhood Regularization for Hyperspectral Image Classification

doi:10.11947/j.JGGS.2022.0108

Abstract

Abstract:

Low-Rank and Sparse Representation (LRSR) method has gained popularity in Hyperspectral Image (HSI) processing. However, existing LRSR models rarely exploited spectral-spatial classification of HSI. In this paper, we proposed a novel Low-Rank and Sparse Representation with Adaptive Neighborhood Regularization (LRSR-ANR) method for HSI classification. In the proposed method, we first represent the hyperspectral data via LRSR since it combines both sparsity and low-rankness to maintain global and local data structures simultaneously. The LRSR is optimized by using a mixed Gauss-Seidel and Jacobian Alternating Direction Method of Multipliers (M-ADMM), which converges faster than ADMM. Then to incorporate the spatial information, an ANR scheme is designed by combining Euclidean and Cosine distance metrics to reduce the mixed pixels within a neighborhood. Lastly, the predicted labels are determined by jointly considering the homogeneous pixels in the classification rule of the minimum reconstruction error. Experimental results based on three popular hyperspectral images demonstrate that the proposed method outperforms other related methods in terms of classification accuracy and generalization performance.

Key words: Hyperspectral Image (HSI); spectral-spatial classification; Low-Rank and Sparse Representation (LRSR); Adaptive Neighborhood Regularization (ANR)

Zhaohui XUE,Xiangyu NIE. Low-Rank and Sparse Representation with Adaptive Neighborhood Regularization for Hyperspectral Image Classification[J]. Journal of Geodesy and Geoinformation Science, 2022, 5(1): 73-90.

Figures/Tables 24

Fig.1

Algorithm 1: M-ADMM for Solving LRSR.

Input: Data set A, dictionary B, regularization parameters α and β
Output:Coefficient matrix X and noise matrix E
Initialize: X⁰=Z⁰=J⁰=0, E⁰=0,

Q 10

Q 20

Q 30

=0, μ⁰=10^-4, μ_max=10⁶, λ=1.1 (λ is a growth rate of μ for each iteration), k=0
While not converged do
Update the variable Z according to Eq. (6)
Update the variable J according to Eq. (7)
Update the variable E according to Eq. (8)
Update the variable X according to Eq. (10)
Update the Lagrange multipliers Q₁, Q₂, and Q₃

Q 1 k + 1

Q 1 k

+μ^k(A-BX^k⁺¹-E^k^-1)

Q 2 k + 1

Q 2 k

+μ^k

X k + 1 - Z k + 1

Q 3 k + 1

Q 3 k

+μ^k

X k + 1 - J k + 1

Update the parameter μ
μ^k⁺¹=min(λμ^k,μ_max)
Update k: k=k+1
End while

Fig.2

Fig.3

Tab.1

Fig.4

Tab.2

Fig.5

Tab.3

Tab.4

Fig.6

Fig.7

Tab.5

Classification accuracy of different methods for the Indian Pines data set (10% for training)(%)"

Class	SVM	SVMCK	SR	JSR	LRR	LRSR	JLRSR	FRPLRaS	LRSR-ANR
1	11.46±12.44	82.62±10.10	42.44±11.16	87.80±12.44	70.24±15.70	65.61±16.52	99.76±0.73	89.76±11.33	98.54±2.06
2	76.93±1,87	90.17±2.31	53.81±2.35	88.41±2.15	54.60±2.77	54.84±2.48	90.85±1.60	94.76±1.81	95.23±0.67
3	64.70±2.63	95.15±1.54	51.27±2.43	84.47±3.82	39.80±2.87	40.48±3.30	92.99±2.39	96.31±1.43	93.19±2.21
4	58.87±8.19	83.66±4.34	37.09±5.13	79.67±6.06	48.22±4.87	48.78±4.28	98.87±2.04	92.68±4.14	97.93±1.72
5	88.62±1.59	93.57±2.57	82.67±2.15	94.51±2.96	75.12±3.53	75.97±3.10	92.17±3.26	96.90±1.47	94.98±1.87
6	94.89±2.03	99.23±0.38	92.91±2.18	98.63±0.89	85.66±3.37	86.15±3.22	97.79±0.55	99.15±0.76	99.04±0.23
7	6.40±11.81	49.20±29.21	73.20±5.98	96.00±3.77	88.40±3.50	89.20±3.79	100.00±0.00	29.20±34.18	94.40±8.26
8	99.12±0.92	99.75±0.25	96.35±2.22	99.40±0.79	92.23±2.78	91.70±2.83	99.60±0.79	99.91±0.15	100.00±0.00
9	0.00±0.00	55.56±9.63	22.78±15.27	81.11±13.15	42.22±10.86	44.44±8.28	90.56±15.33	33.33±32.77	94.44±6.93
10	68.19±3.47	90.28±1.87	66.31±1.94	93.86±1.67	57.64±1.99	57.88±2.32	94.45±1.51	93.38±1.65	94.05±1.45
11	86.01±1.97	96.43±1.25	71.22±1.97	95.55±0.80	71.65±1.59	71.32±2.28	96.17±1.80	97.06±0.90	97.18±1.09
12	78.67±3.38	89.65±3.53	42.94±1.92	84.89±3.37	43.21±3.38	43.00±4.80	89.14±4.20	95.43±1.94	91.14±1.90
13	95.11±3.10	98.61±1.16	92.66±2.99	99.29±1.06	94.84±2.18	94.51±1.84	98.70±1.44	98.10±1.40	97.61±1.80
14	96.92±0.75	99.16±0.33	90.44±1.65	99.02±0.44	94.46±1.04	93.99±0.91	99.89±0.08	99.22±0.69	99.79±0.33
15	56.71±4.70	96.41±2.50	34.38±3.26	78.65±5.07	37.61±3.30	37.38±3.25	93.92±2.04	94.09±3.67	92.62±2.60
16	85.71±5.50	85.41±6.16	88.57±2.98	98.21±2.53	87.11±5.24	88.43±5.63	96.51±2.44	92.74±9.40	98.43±1.28
OA	81.35±0.71	94.41±0.50	68.90±0.67	92.69±0.45	67.40±0.65	67.43±0.69	95.19±0.46	96.20±0.46	96.29±0.30
AA	66.77±1.33	87.80±3.19	64.94±1.60	91.22±1.43	67.43±1.12	67.73±1.31	95.71±0.88	87.63±2.74	96.29±1.84
κ×100	78.58±0.81	93.63±0.57	64.48±0.76	91.65±0.52	62.62±0.73	62.66±0.78	94.52±0.52	95.67±0.52	95.77±0.66
Time/s	32	32	27	65	143	164	235	175	195

Tab.5

Fig.8

Tab.6

Fig.9

Tab.7

Classification accuracy of different methods for the Salinas data set (2% for training)(%)"

Class	SVM	SVMCK	SR	JSR	LRR	LRSR	JLRSR	FRPLRaS	LRSR-ANR
1	98.20± 0.91	99.08± 0.34	98.99± 0.36	99.99± 0.01	98.41± 0.38	97.48± 0.73	99.71±0.16	99.67±0.12	100.00± 0.00
2	99.46± 0.23	98.67± 0.52	98.19± 0.52	99.98± 0.02	97.23± 0.61	97.50± 0.57	99.95±0.06	99.27±0.22	99.95± 0.15
3	96.00± 2.95	98.78± 0.69	76.90± 3.31	96.59± 0.93	67.01± 3.88	67.92± 1.70	96.23±2.62	99.83±0.39	99.10± 2.09
4	98.37± 1.17	98.27± 1.94	92.88± 2.18	99.68± 0.23	98.55± 0.28	99.36± 0.06	72.03±4.88	99.37±0.81	81.23± 4.29
5	98.01± 0.52	97.80± 0.50	95.98± 0.87	95.01± 0.69	96.52± 1.22	97.67± 0.51	94.24±1.87	99.29±0.21	96.52± 1.04
6	99.52± 0.37	98.46± 1.56	98.80± 0.43	100.00± 0.00	99.44± 0.11	99.50± 0.07	96.4±0.76	99.82±0.13	99.93± 0.05
7	99.55± 0.25	98.97± 0.39	97.73± 0.52	100.00± 0.00	97.96± 0.28	97.42± 0.67	97.88±0.86	99.34±0.32	99.77± 0.11
8	88.87± 1.18	95.87± 0.74	84.88± 1.61	98.20± 0.50	64.12± 3.25	64.87± 2.29	94.55±1.96	91.01±0.70	98.22± 0.41
9	99.43± 0.38	99.22± 0.36	98.45± 0.51	99.99± 0.01	96.82± 0.23	97.05± 0.23	100.0±0.00	99.64±0.31	99.95± 0.03
10	90.65± 2.36	94.86± 2.76	91.31± 1.00	98.26± 0.53	76.21± 1.46	71.90± 1.19	92.91±1.50	95.52±1.03	97.69± 0.98
11	93.67± 2.78	94.59± 4.41	91.88± 4.39	99.87± 0.19	93.59± 1.11	93.71± 0.33	93.41±2.09	95.46±2.68	96.31± 1.78
12	99.52± 0.43	99.84± 0.09	83.04± 5.93	89.43± 7.05	91.06± 1.83	86.57± 3.90	88.87±2.58	99.57±0.28	92.86± 1.60
13	97.51± 1.96	94.44± 5.14	80.62± 5.11	87.82± 4.24	97.88± 0.26	97.41± 0.84	88.72±5.19	97.94±1.42	97.16± 1.79
14	90.94± 6.61	90.13± 4.30	83.34± 2.41	92.62± 3.53	87.14± 1.21	87.00± 0.97	85.22±6.30	93.33±4.38	92.66± 4.10
15	63.79± 3.29	91.51± 1.78	37.89± 3.76	56.07± 6.47	53.86± 4.33	53.08± 1.05	87.87±3.25	76.33±1.83	97.34± 0.97
16	96.05± 1.44	97.17± 2.22	94.85± 1.03	99.73± 0.27	88.71± 4.74	89.01± 3.28	99.01±1.69	97.47±0.69	99.21± 0.88
OA	91.24± 0.50	96.61± 0.35	84.59± 0.30	92.50± 0.86	81.58± 0.21	81.30± 0.48	94.21±0.54	94.10±0.28	97.91± 0.13
AA	94.35± 0.66	96.43± 0.70	87.86± 0.52	94.58± 0.53	87.78± 0.49	87.34± 0.32	92.94±0.45	96.43±0.29	96.74± 0.37
κ× 100	90.22± 0.56	96.22± 0.39	82.77± 0.33	91.61± 0.97	79.52± 0.24	79.21± 0.53	93.55±0.61	93.43±0.32	97.63± 0.21
Time/s	277	280	262	1274	768	889	2217	580	1690

Tab.7

Fig.10

Fig.11

Tab.8

Tab.9

Tab.10

Tab.11

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Class	Name
1	Alfalfa	5	41	46
2	Corn-notill	143	1285	1428
3	Corn-mintill	83	747	830
4	Corn	24	213	237
5	Grass-pasture	49	434	483
6	Grass-trees	73	657	730
7	Grass-pasture-mowed	3	25	28
8	Hay-windrowed	48	430	478
9	Oats	2	18	20
10	Soybean-notill	98	874	972
11	Soybean-mintill	246	2209	2455
12	Soybean-clean	60	533	593
13	Wheat	21	184	205
14	Woods	127	1138	1265
15	Buildings-grass-trees-drives	39	347	386
16	Stone-steel-towers	10	83	93
Total		1031	9218	10249

Class	Name	Training	Test	Total
1	Asphalt	133	6498	6631
2	Meadows	373	18276	18649
3	Gravel	42	2057	2099
4	Trees	62	3002	3064
5	Metal sheets	27	1318	1345
6	Bare soil	101	4928	5029
7	Bitumen	27	1303	1330
8	Bricks	74	3608	3682
9	Shadows	19	928	947
Total		858	41918	42776

Class	Name	Training	Test	Total
1	Brocoli_green_weeds_1	41	1968	2009
2	Brocoli_green_weeds_2	75	3651	3726
3	Fallow	40	1936	1976
4	Fallow_rough_plow	28	1366	1394
5	Fallow_smooth	54	2624	2678
6	Stubble	80	3879	3959
7	Celery	72	3507	3579
8	Grapes_untrained	226	11045	11271
9	Soil_vinyard_develop	125	6078	6203
10	Corn_senesced_green_weeds	66	3212	3278
11	Lettuce_romaine_4wk	22	1046	1068
12	Lettuce_romaine_5wk	39	1888	1927
13	Lettuce_romaine_6wk	19	897	916
14	Lettuce_romaine_7wk	22	1048	1070
15	Vinyard_untrained	146	7122	7268
16	Stone-steel-towers	37	1770	1807
Total		1092	53037	54129

Methods	Indian Pines	Pavia University	Salinas
SR	T=5	T=10	T=10
JSR	{T=20, W_S=5×5}	{T=25, W_S=11×11}	{T=20, W_S=11×11}
LRR	β=20	β=0.2	β =20
LRSR	{α=1, β=20}	{α=1, β=0.2}	{α=1, β=20}
JLRSR	{α=1, β=20, W_S=7×7}	{α=1, β=0.2, W_S=15×15}	{α=1, β=20, W_S=13×13}
FRPLRaS	{Rank=30, T=20, W_S=5×5}	{Rank=30, T=30, W_S=11×11}	{Rank=30, T=30, W_S=9×9}
Ours	{α=1, β=20, W_S=7×7, S=0.9}	{α=1, β=0.2, W_S=19×19, S=0.725}	{α=1, β=20, W_S=13×13, S=0.85}

Class	SVM	SVMCK	SR	JSR	LRR	LRSR	JLRSR	FRPLRaS	LRSR-ANR
1	90.97±2.60	95.87±0.60	83.70±5.00	88.03±4.94	62.32±1.29	59.63±3.62	91.63±2.69	96.07±0.55	97.14±1.03
2	97.18±0.79	99.32±0.41	89.17±1.66	97.82±1.09	83.11±1.88	81.51±1.10	99.35±0.54	97.73±0.57	99.84±0.15
3	69.65±5.27	96.12±2.39	50.28±9.24	69.23±13.53	58.70±4.58	59.45±3.54	87.21±4.35	79.37±4.23	96.22±2.07
4	88.91±3.25	99.36±0.16	82.42±1.62	87.28±1.61	89.59±0.78	88.67±1.77	89.83±3.46	93.41±2.76	93.20±2.86
5	98.36±0.76	73.76±9.21	94.23±3.49	100.00±0.00	99.36±0.09	99.35±0.04	99.67±0.76	99.34±0.92	99.92±0.04
6	81.88±2.87	95.81±1.25	47.33±4.89	75.12±7.36	43.20±2.48	43.04±1.72	85.77±3.61	88.18±1.95	93.42±4.16
7	72.59±9.26	96.66±1.16	30.42±7.14	46.83±16.28	59.69±4.52	58.79±5.57	95.33±3.62	94.32±1.52	98.97±1.01
8	88.65±2.39	97.84±0.90	24.63±3.27	56.70±13.16	58.24±1.61	57.07±1.96	90.64±3.84	92.33±2.46	96.40±1.32
9	99.52±0.34	80.02±13.59	73.02±3.64	81.76±3.19	84.98±0.93	82.46±1.45	86.25±0.93	99.89±0.16	88.45±2.06
OA	91.06±0.48	96.78±0.52	73.43±0.88	86.07±2.08	72.15±0.80	70.80±1.08	94.13±0.76	94.67±0.49	97.52±0.46
AA	87.52±1.41	92.75±1.97	63.91±1.71	78.09±3.88	71.02±0.70	70.00±1.18	91.74±1.17	93.41±0.77	95.88±0.67
κ×100	88.06±0.64	95.73±0.69	64.26±1.14	81.27±2.82	63.48±0.88	61.86±1.32	92.15±1.02	92.91±0.67	96.53±0.79
Time/s	100	104	125	455	409	468	977	347	726