Remote sensing change detection: a comparative study of spectral distances

Figures & data

Figure 1. Pléiades and WV-3 images of the Isfahan urban area for the years (a) 2012 and (b) 2016, respectively.

Table 1. Number of changed/unchanged pixels used as test/training data in each dataset.

Dataset	Label	Number of pixels
		Training	Test
Dataset 1 (Urban Area)	Changed	247565	61891
	Unchanged	474467	118616
Dataset 2 (Agricultural Area)	Changed	8838	2209
	Unchanged	12261	3065

Table 2. Implemented spectral distances.

Eq. Num.	Distance’s Name	Reference	Calculation (formula and description)
1	Euclidean Distance	Richards and Jia (1999)	$D_{\mathrm{Euclidean}}=\sqrt{{\sum }_{b=1}^B{\left(x_{i,j}^b\left(2\right)-x_{i,j}^b\left(1\right)\right)}^2}$
2	City-Block Distance	Duda et al. (2012)	$D_{\text{City Block}}=\sqrt{{\sum }_{b=1}^B\left|x_{i,j}^b\left(2\right)-x_{i,j}^b\left(1\right)\right|}$
3	Chebyshev Distance	Sharma and Batra (2014)	$D_{\mathrm{Chebchev}}=\underset{b}{\max }\left\{\left|x_{i,j}^b\right(2)-x_{i,j}^b(1\left)\right|\right\}$
4	Mahalanobis Distance	Richards and Jia (1999)	$D_{\mathrm{mahalanobis}}=\sqrt{{\left[x_{i,j}^b\left(2\right)-x_{i,j}^b\left(1\right)\right]}^T{K}^{-1}\mathrm{ }\left[x_{i,j}^b\left(2\right)-x_{i,j}^b\left(1\right)\right]}$ K: Covariance Matrix, a B × B Square matrix, includes the variance/ covariance values of the DN values of original images in different (B) spectral bands.
5	Pearson-Correlation Distance	Schreier et al. (2009)	$D_{\mathrm{Correlation}}=1-\frac{{\sum }_{b=1}^B\left(x_{i,j}^b\left(2\right)-\overline{x^b\left(2\right)}\right)\left(x_{i,j}^b\left(1\right)-\overline{x^b\left(1\right)}\right)}{\sqrt{{\sum }_{b=1}^B{\left(x_{i,j}^b\left(2\right)-\overline{x^b\left(2\right)}\right)}^2}\sqrt{{\sum }_{b=1}^B{\left(x_{i,j}^b\left(1\right)-\overline{x^b\left(1\right)}\right)}^2}}$ $\overline{x^b\left(t\right)}=\sum _{i,j}{x}_{i,j}^b(t)/N$    (6)
7	Spectral Angle Distance	Kruse et al. (1993)	$D_{\text{Spectral Distance}}=1-\frac{{<x}_{i,j}\left(1\right).x_{i,j}\left(2\right)>}{{\Vert x_{i,j}\left(1\right)\Vert }_2{\Vert x_{i,j}\left(2\right)\Vert }_2}=1-{\cos }^{-1}\left(\frac{{\sum }_{b=1}^B{x}_{i,j}^b\left(1\right)x_{i,j}^b\left(2\right)}{\sqrt{{\sum }_{b=1}^B{x_{i,j}^b\left(1\right)}^2}\sqrt{{\sum }_{b=1}^B{x_{i,j}^b\left(2\right)}^2}}\right)$ In the equation above, ${\Vert x_{i,j}(1)\Vert }_2\text{ and }{\Vert x_{i,j}(2)\Vert }_2\text{ are the size of the pixel’s vector in }\text{the feature space of the }1\text{st and }2\text{nd images},\text{ and }{<x}_{i,j}\left(1\right).x_{i,j}\left(2\right)>$ is the value obtained by internal multiplication of two previous vectors.
8	Spearman-Correlation Distance	Van Dongen and Enright (2012)	$D_{\mathrm{Spearman}}=1-\frac{{\sum }_{b=1}^B\left(r_{i,j}^b\left(2\right)-\overline{r^b\left(2\right)}\right)\left(r_{i,j}^b\left(1\right)-\overline{r^b\left(1\right)}\right)}{\sqrt{{\sum }_{b=1}^B{\left(r_{i,j}^b\left(2\right)-\overline{r^b\left(2\right)}\right)}^2}\sqrt{{\sum }_{b=1}^B{\left(r_{i,j}^b\left(1\right)-\overline{r^b\left(1\right)}\right)}^2}}$ $\overline{r^b\left(t\right)}=\sum _{i,j}{r}_{i,j}^b(t)/N$    (9) The gray levels may differ for various reasons, even for pixels whose land cover has not changed in the two images. The Spearman-Correlation similarity criterion is proposed as an improvement to the Pearson-Correlation distance, which uses the greatness rank of the pixel’s gray level in each band (b) compared to other bands of the image. So, $r_{i,j}^b\left(t\right)$ is the greatness rank of the pixel’s value in each band compared to values of the desired pixel in other bands.
10	Spectral Gradient Distance	Chen et al. (2013)	$D_{\text{Spectral Gradient}}=\sum _{b=2}^B\left|g_{i,j}^b\left(2\right)-g_{i,j}^b\left(1\right)\right|$ $g_{i,j}^b\left(t\right)=\frac{{\Delta R}_{\left(b,b-1\right)}(t)}{{\Delta \lambda }_{(b,b-1)}(t)}=\frac{R_b-R_{b-1}}{{\lambda }_b-{\lambda }_{b-1}}$    (11) In Equation (11), $g_{i,j}^b\left(t\right)$ is the spectral gradient between band b and b-1. So, the gradient distance calculates by subtracting the gradient values between similar parts of the bands in two images.
12	Cosine Distance	Van Dongen and Enright (2012)	$D_{\mathrm{cosine}}=1-\frac{{\sum }_{b=1}^B{x}_{i,j}^b{\left(1\right)x}_{i,j}^b(2)}{\sqrt{{\sum }_{b=1}^B{x_{i,j}^b(1)}^2}\sqrt{{\sum }_{b=1}^B{x_{i,j}^b(2)}^2}}$
If: $X=〈x(1){x(1)}^T〉\mathrm{ }\&\mathrm{ }Y=〈x(2){x(2)}^T〉\mathrm{ }\&\mathrm{ }C=〈x(1){x(2)}^T〉$ Then, three last distances are described below:
13	Covariance Equalization (CE)	Schaum and Stocker (1995)	$D_{\text{Covariance Equalization }}=x\left(1\right)-X^{1/2}{Y}^{-1/2}x\hspace{1em}(2)$
14	Chrono-chrome (CC)	Schaum and Stocker (2003)	$D_{\text{Chronochrome }}=x\left(1\right)-C{Y}^{-1}x(2)$
15	Hyperbolic Anomalous Change Detection (HACD)	Theiler (2008)	$D_{\mathrm{HACD}}=\left[\begin{array}{cc}{x(1)}^T& {x(2)}^T\end{array}\right]\left({\left[\begin{array}{cc}X& C^T\\ C& Y\end{array}\right]}^{-1}-{\left[\begin{array}{cc}X& 0\\ 0& Y\end{array}\right]}^{-1}\right)\left[\begin{array}{c}x(1)\\ x(2)\end{array}\right]$

Table 3. Accuracy assessment results of the applied CD methods in the urban (dataset 1) and agricultural (dataset 2) study areas.

Category	Method	Algorithm/distance	(Urban Region) Dataset 1 (OA%)	(Agricultural Region) Dataset 2 (OA%)
Unsupervised	Thresholding the spectral distances	Euclidean	66.68	72.52
		City-Block	61.91	76.29
		Chebyshev	68.16	69.90
		Mahalanobis	73.54	60.68
		Pearson-Correlation	85.78	78.79
		Spectral	79.89	78.45
		Spearman-Correlation	84.72	82.05
		Gradient	74.65	79.43
		Cosine	76.03	80.37
		Covariance Equalization	62.53	61.24
		ChronChrome	80.02	56.40
		HACD	67.82	50.02
	Majority Voting (On all 12 change maps)		82.67	81.23
	Linear Combination of the spectral distances		83.35	80.62
Supervised Post-classification comparison	Classification of the distance images	Maximum Likelihood	88.39	86.89
		Artificial Neural Network	92.83	87.21
		Support Vector Machine	95.21	88.73
	Classification of the stacked original bands	Support Vector Machine	95.31	89.35

Figure 2. Landsat 7 & 8 Satellite images of Zayandeh-Rud agricultural areas for (a) 2001 and (b) 2016.

Figure 3. Spectral distance images (between Pléiades and WV-3 images of 2012 and 2016).

Figure 4. Binary change maps (changed: black and unchanged: white) produced by applying SCD and fusion methods using the Pléiades and WV-3 images.

Figure 5. Change maps produced by applying SCD and fusion methods using the Landsat 7 & 8 images.

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