Laplacian pyramid-based change detection in multitemporal SAR images

Figures & data

Figure 1. Flowchart of Gaussian (G_l) and Laplacian pyramid (L_l)

Figure 2. Block diagram of LP-based change detection

Table

Algorithm 1 Laplacian pyramid-based change detection Algorithm.
Step 1	:	Use multitemporal SAR images $I_1$ and $I_2$ as input images.
Step 2	:	Preprocess the images by condensed anisotropic diffusion (CAD) method.
		Estimate the absolute gradient in $x,y$ direction $\left|\mathrm{\nabla }I\right|=\sqrt{I_x^2+I_y^2}$. Evaluate D, ${\lambda }_1$ and ${\lambda }_{2.}$ $D=\frac{1}{{\left|\mathrm{\nabla }I\right|}^2}{\left(\begin{array}{cc}{I}_x& I_y\\ -I_y& I_x\end{array}\right)}^T\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)\left(\begin{array}{cc}{I}_x& I_y\\ -I_y& I_x\end{array}\right)$ ${\lambda }_1=\left\{\begin{array}{ccc}\alpha \left(1-{\left(\frac{A}{S}\right)}^2\right)& ,& ifA=\left({\mu }_1-{\mu }_2\right)\le S^2\\ 0& ,& else\end{array}\right.;$ ${\lambda }_2=\alpha$ Calculate $I_{ij}^{n+1}$ from $I_{ij}^n$ using semi implicit scheme Equation (5)(5) $\frac{I_{i,j}^{n+1}-I_{i,j}^n}{\tau }=D_N\left({\mathrm{\nabla }}_N{I}_{i,j}^n\right)+D_S\left({\mathrm{\nabla }}_S{I}_{i,j}^n\right)+D_E\left({\mathrm{\nabla }}_E{I}_{i,j}^n\right)+D_W\left({\mathrm{\nabla }}_W{I}_{i,j}^n\right)+\beta \gamma {\left(I_{ci,j}^n\right)}^{\gamma }{\left(I_{i,j}^n\right)}^{\gamma -1}$(5) . Continue iterations until converge for time step $\tau .$
Step 3	:	Generate Gaussian and Laplacian pyramid images $G_l$, $L_l$ from despeckled images $I_1$ and $I_2$ using Equation (9)(9) $\begin{array}{rl}& G_0=I\\ & G_l=Reduce\left[G_{l-1}\right]\\ & L_l=G_l-Expand\left[G_{l+1}\right]\end{array}$(9) .
Step 4	:	Involving LR operator in each Laplacian pyramid decomposition layers using Equation (12)(12) $\begin{array}{rl}& \mathfrak{D}\tilde{\mathrm{I}}{ }_{decomp}=\left[log\left(L_{I_{2}l}\right)-log\left(L_{I_{1}l}\right)\right];\\ & \begin{array}{c}{I}_1,I_2-\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{s}\\ l=0,1,2,3-\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}\mathrm{s}\end{array}\end{array}$(12) .
Step 5	:	Reconstruct the degenerated images by reversing the construction steps to obtain the difference image using Equation (13)(13) $\mathfrak{D}\stackrel{\sim }{\mathrm{I}}=reconst(\mathfrak{D}{\stackrel{\sim }{\mathrm{I}}}_{decomp})$(13) .
Step 6	:	Apply automatic Otsu thresholding algorithm to get change map using Equations (14)(14) $P_i=\frac{n_i}{N}{P}_i\ge 0;\sum _{i=1}^L{P}_i=1$(14) –(17(17) ${\sigma }_B^2\left(t^{\ast }\right)={}_{0\le k\le L}^{max}{\sigma }_B^2\left(k\right)$(17) ).
Step 7	:	Analyze the change map image.
Step 8	:	Evaluate the performance metrics.

Table 1. Quality measures for speckle reduction assessment

Quality Measure	Definition	Description
SSIM	$\mathrm{S}\mathrm{S}\mathrm{I}\mathrm{M}\left(\mathrm{x},\mathrm{y}\right)=\frac{\left(2{\mu }_{\mathrm{x}}{\mu }_{\mathrm{y}}+{\mathrm{c}}_1\right)\left(2{\sigma }_{\mathrm{x}\mathrm{y}}+{\mathrm{c}}_2\right)}{\left({\mu }_{\mathrm{x}}^2+{\mu }_{\mathrm{y}}^2+{\mathrm{c}}_1\right)\left({\sigma }_{\mathrm{x}}^2+{\sigma }_{\mathrm{y}}^2+{\mathrm{c}}_2\right)}$, where ${\mathrm{c}}_1,{\mathrm{c}}_2$ are constant and chosen lesser than one.	Structural similarity index is used to characterize luminosity, disparity and structural variations, where mean intensity ${\mu }_x=\frac{1}{N}{\sum }_{i=1}^N{x}_i$  Standard deviation ${\sigma }_x={\left(\frac{1}{N-1}{\sum }_{i=1}^N{\left(x_i-{\mu }_x\right)}^2\right)}^{1/2}$  Covariance ${\sigma }_{xy}=\frac{1}{N}{\sum }_{i=1}^N\left(x_i-{\mu }_x\right)\left(y_i-{\mu }_y\right)$
ENL	$\mathrm{E}\mathrm{N}\mathrm{L}=\frac{\mathrm{E}{\left[{\mathrm{f}}_{\mathrm{I}}\right]}^2}{\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathrm{f}}_{\mathrm{I}}\right)}$	Equivalent number of looks. The amplitude of an image is calculated by the ratio of mean to variance. Maximum value corresponds to the better speckle reduction. Here, ${\mathrm{f}}_{\mathrm{I}}$ is filtered image.
SSI	$\mathrm{S}\mathrm{S}\mathrm{I}=\frac{\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathrm{f}}_{\mathrm{I}}\right)}{\mathrm{E}{\left[{\mathrm{f}}_{\mathrm{I}}\right]}^2}\cdot \frac{\mathrm{E}{\left[{\mathrm{s}}_{\mathrm{I}}\right]}^2}{\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathrm{s}}_{\mathrm{I}}\right)}$	Speckle Suppression Index. It is defined by the coefficient of variance of the filtered image and normalized by that of the speckle image. SSI provides better results for lesser than one. ${\mathrm{s}}_{\mathrm{I}}$ is a speckled image.

Table 2. Performance of various datasets using SSIM, ENL and SSI

Dataset	SSIM	ENL	SSI
Athabasca	0.8767	56.3572	0.2406
Alberta	0.8807	24.2520	0.1391
Aare River	0.9200	57.7557	0.2622

Figure 3. Experimental results of various dataset (a1) Athabasca image with marginal on 330th row, (a2) Image profile beside on 330th row of original image, (a3) Image profile beside on 330th row of despeckled image, (b1) Alberta image with marginal on 33rd row, (b2) Image profile beside on 33rd row of original image, (b3) Image profile beside on 33rd row of despeckled image, (c1) Aare image with marginal on 67th row, (c2) Image profile beside on 67th row of original image, (c3) Image profile beside on 67th row of despeckled image

Figure 4. Experimental results on LP-based multiscale decomposition combined with LR operator. Athabasca dataset (a₁) L₁₀−L₂₀, (a₂) L₁₁−L₂₁, (a₃) L₁₂−L₂₂, (a₄) L₁₃−L₂₃ and (a₅) Reconstructed image; Aare River dataset (b₁) L₁₀−L₂₀, (b₂) L₁₁−L₂₁, (b₃) L₁₂−L₂₂, (b₄) L₁₃−L₂₃ and (b₅) Reconstructed image; Alberta dataset (c₁) L₁₀−L₂₀, (c₂)) L₁₁−L₂₁, (c₃) L₁₂−L₂₂, (c₄) L₁₃−L₂₃ and (c₅) Reconstructed image

Figure 5. Experimental results of simulated images (a) I₁ (b) I₂ (c) Ground truth (d) MR (e) LR (f) MRKI (g) LRKI (h) MRO (i) LRO (j) FMLKI (k) FMLO (l) LPCD

Figure 6. Experimental results of Athabasca Dataset (a) I₁ (b) I₂ (c) Ground truth (d) MR (e) LR (f) MRKI (g) LRKI (h) MRO (i) LRO (j) FMLKI (k) FMLO (l) LPCD

Figure 7. Experimental results of Alberta Dataset (a) I₁ (b) I₂ (c) Ground truth (d) MR (e) LR (f) MRKI (g) LRKI (h) MRO (i) LRO (j) FMLKI (k) FMLO (l) LPCD

Figure 8. Experimental results of Aare River Dataset (a) I₁; (b) I₂; (c) Ground truth; (d) MR; (e) LR; (f) MRKI; (g) LRKI; (h) MRO; (i) LRO; (j) FMLKI; (k) FMLO; (l) LPCD

Table 3. Evaluation of change detection for simulated images with different methods

Methods	FP	FN	FAR	F1 Score	OE	ER	Kappa	MsR	DR	PCC (%)
MR	5889	457	0.1479	0.5448	6346	0.144	0.4743	0.1074	0.8926	85.60
LR	3372	315	0.0869	0.7284	3687	0.0836	0.6819	0.0599	0.9401	91.64
MRKI	5091	522	0.1271	0.5559	5613	0.1273	0.4927	0.1294	0.8706	87.27
LRKI	1897	948	0.0491	0.7595	2845	0.0645	0.7225	0.1743	0.8257	93.55
MRO	4217	605	0.1049	0.5758	4822	0.1094	0.5202	0.156	0.844	89.06
LRO	1450	961	0.0371	0.7703	2411	0.0547	0.7393	0.1921	0.8079	94.53
FMLKI	1188	289	0.0305	0.8678	1477	0.0335	0.8488	0.0562	0.9438	96.65
FMLO	1030	280	0.0264	0.8782	1310	0.0297	0.8614	0.056	0.944	97.03
LPCD	237	111	0.0028	0.9618	348	0.0079	0.9574	0.0513	0.9487	99.21

Table 4. Evaluation of change detection for Athabasca oil sand dataset with different methods

Methods	FP	FN	FAR	F1 Score	OE	ER	Kappa	MsR	DR	PCC (%)
MR	23,383	522	0.1992	0.2273	23,905	0.1969	0.1799	0.1293	0.8707	80.31
LR	16,142	192	0.1354	0.1991	16,334	0.1345	0.1721	0.0864	0.9136	86.55
MRKI	17,304	644	0.1474	0.2744	17,948	0.1478	0.2317	0.1595	0.8405	85.22
LRKI	11,283	192	0.0947	0.2613	11,475	0.0945	0.2374	0.0864	0.9136	90.55
MRO	12,753	778	0.1087	0.3252	13,531	0.1114	0.2873	0.1927	0.8073	88.86
LRO	6910	192	0.058	0.3637	7102	0.0585	0.3445	0.0864	0.9136	94.15
FMLKI	4033	106	0.0342	0.6209	4139	0.0341	0.6055	0.0303	0.9697	96.59
FMLO	2080	378	0.0178	0.7763	2458	0.0202	0.7659	0.0814	0.9186	97.98
LPCD	1123	64	0.0095	0.8148	1187	0.0098	0.8099	0.0239	0.9761	99.02

Table 5. Evaluation of change detection for Alberta flood dataset with different methods

Methods	FP	FN	FAR	F1 Score	OE	ER	Kappa	MsR	DR	PCC (%)
MR	7783	162	0.1453	0.305	7945	0.1432	0.2628	0.085	0.915	85.68
LR	2914	304	0.0554	0.6175	3218	0.058	0.5893	0.1048	0.8952	94.20
MRKI	4313	351	0.0803	0.3791	4664	0.0841	0.3472	0.1977	0.8023	91.59
LRKI	1964	273	0.038	0.7603	2237	0.0403	0.739	0.0715	0.9285	95.97
MRO	2559	410	0.0475	0.4391	2969	0.0535	0.4158	0.2608	0.7392	94.65
LRO	1008	124	0.0191	0.8227	1132	0.0204	0.8121	0.0451	0.9549	97.96
FMLKI	1016	165	0.0189	0.7203	1181	0.0213	0.7097	0.0979	0.9021	97.87
FMLO	867	199	0.0161	0.7361	1066	0.0192	0.7265	0.118	0.882	98.08
LPCD	456	73	0.0086	0.9068	529	0.0095	0.9018	0.0276	0.9724	99.05

Table 6. Evaluation of change detection for Aare River dataset with different methods

Methods	FP	FN	FAR	F1 Score	OE	ER	Kappa	MsR	DR	PCC (%)
MR	13,481	269	0.1602	0.1096	13,750	0.1613	0.0874	0.2413	0.7587	83.87
LR	8224	132	0.0985	0.2767	8356	0.098	0.2508	0.0763	0.9237	90.2
MRKI	10,469	281	0.1244	0.1343	10,750	0.1261	0.1132	0.252	0.748	87.39
LRKI	3889	155	0.048	0.6681	4044	0.0474	0.6451	0.0367	0.9633	95.26
MRO	9170	298	0.1089	0.1319	9468	0.1111	0.1127	0.293	0.707	88.89
LRO	2560	275	0.0314	0.701	2835	0.0333	0.6845	0.0764	0.9236	96.67
FMLKI	1922	161	0.0229	0.5372	2083	0.0244	0.5266	0.1175	0.8825	97.56
FMLO	1433	207	0.0171	0.5865	1640	0.0192	0.5776	0.1511	0.8489	98.08
LPCD	148	40	0.0018	0.9075	188	0.0022	0.9064	0.0416	0.9584	99.78

Figure 9. Comparison Result of PCC for: (a) Athabasca dataset, (b) Alberta dataset, (c) Aare River dataset; Comparison Result of Kappa Coefficient for: (d) Athabasca dataset, (e) Alberta dataset, (f) Aare River dataset

Table 7. Comparison of different multiscale methods

Multiscale methods	FAR	ER	DR	Kappa	PCC (%)	CT (sec)
SDA	0.4328	0.0188	0.9454	0.6999	98.12	2.3323
KL-MCP	0.6446	0.5806	0.6071	0.0251	41.94	1.8937
BA-MCD	0.0180	0.0203	0.9222	0.7671	97.97	3.1409
K matrix-MS	0.0244	0.0237	0.8321	0.6983	97.63	4.8321
Proposed method	0.0095	0.0098	0.9761	0.8099	99.02	0.9639

Figure 10. Experiment results of comparison with multiscale methods (a) SDA (b) KL-MCP (c) BA-MCD (d) K matrix-MS (e) Proposed LPCD method

Figure 11. Otsu thresholding results: ${\sigma }_{\mathrm{B}}^2$ Vs gray level (a) Athabasca image, (b) Alberta image, (c) Aare river image; Histogram with threshold values: (a) Athabasca image, (b) Alberta image, (c) Aare river image

Figure 12. Experiment results of Alberta dataset for Otsu thresholding (a) ${\sigma }_{\mathrm{B}}^2$ (b) ${\sigma }_{\mathrm{W}}^2$

Table 8. Experimental results for small percentage of changed pixels

Dataset	Image size	Ground Truth (changed pixels)	% of change	Class Separability	Change area pixels	OE	NU	AOM
Alberta Images	$175$ x $317$	$3033$	≤5%	${\sigma }_W^2$ (Within-class)	$2150$	$1385$	$0.0392$	$0.5783$
				${\sigma }_B^2$ (Between-class)	$3020$	$433$	$0.0551$	$0.8665$

Table 9. Area overlap measure (AOM) for Otsu thresholding for different between class variance value in values. Fig.13

	Otsu segmentation for different between-class variance values.
	${\sigma }_B^2=124.76$	${\sigma }_B^2=164.33$	${\sigma }_B^2=221.26$	${\sigma }_B^2=256.84$	${\sigma }_B^2=270.58$	${\sigma }_B^2=251.27$	${\sigma }_B^2=225.65$	${\sigma }_B^2=201.78$	${\sigma }_B^2=175.15$	${\sigma }_B^2=150.70$	${\sigma }_B^2=124.19$	${\sigma }_B^2=99.68$
AOM	0.0479	0.0750	0.2223	0.5829	0.8542	0.7386	0.5682	0.4579	0.3569	0.2832	0.2138	0.1307

Maximum value of between class variance${\sigma }_B^2=270.58$ yields accurate binary classification experimented by AOM.

Figure 13. Otsu segmentation result for Athabasca image (a) ${\sigma }_{\mathrm{B}}^2$ = 124.76; Threshold = 7, (b) ${\sigma }_{\mathrm{B}}^2=164.33$; Threshold = 14, (c) ${\sigma }_{\mathrm{B}}^2=221.26$; Threshold = 27, (d) ${\sigma }_{\mathrm{B}}^2=256.84$; Threshold = 39, (e) ${\sigma }_{\mathrm{B}}^2=270.57$; Threshold = 58, (f) ${\sigma }_{\mathrm{B}}^2=251.27$; Threshold = 88, (g) ${\sigma }_{\mathrm{B}}^2=225.65$; Threshold = 104, (h) ${\sigma }_{\mathrm{B}}^2=201.78$; Threshold = 115, (i) ${\sigma }_{\mathrm{B}}^2=175.15$; Threshold = 126, (j) ${\sigma }_{\mathrm{B}}^2=150.70$; Threshold = 135, (k) ${\sigma }_{\mathrm{B}}^2=124.19$; Threshold = 145, (l) ${\sigma }_{\mathrm{B}}^2=99.68$; Threshold = 154

Table 10. Performance measures (region nonuniformity)

Dataset	MRKI	MRO	LRKI	LRO	FMRKI	FMLO	LPCD
Athabasca	0.1705	0.1097	0.1319	0.0736	0.0611	0.0523	0.0296
Alberta	0.1326	0.0934	0.116	0.069	0.0367	0.0304	0.0126
Aare River	0.1034	0.0993	0.0671	0.0655	0.0557	0.0524	0.0446

Table 11. Performance measures (area overlap measure)

Dataset	MRKI	MRO	LRKI	LRO	FMRKI	FMLO	LPCD
Athabasca	0.159	0.1503	0.1942	0.2223	0.4503	0.6343	0.8542
Alberta	0.2307	0.5016	0.3316	0.5397	0.3673	0.4149	0.8306
Aare River	0.2339	0.6132	0.2813	0.6988	0.5629	0.5825	0.8296

Figure 14. Comparison results of (a) area overlap measure (AOM), (b) region nonuniformity (NU)