What does image gradient mean in the following paper?

What does image gradient mean in the following paper?

I am working in digital image processing using java, recently I am
implementing a paper in java, One portion of this paper is this:
What I have understood from that is, G(i,j) would be intensity of image at
location (i, j) after applying soble operator on it, does it mean so or
anything else,
I have used the following code to compute wG,
public void weightedGCalc() {
BufferedImage sobelIm = this.getSobelImage();
int width = sobelIm.getWidth();
int height = sobelIm.getHeight();
weightedG = new double[width][height];
for (int row = 0; row < width; row++) {
for (int col = 0; col < height; col++) {
int imgPix = new Color(sobelIm.getRGB(row, col)).getRed();
float val = -(float) (Math.pow(imgPix, 2) / (2 *
Math.pow(SIGMA_G[5], 2)));
weightedG[row][col] = (float) Math.exp(val);
}
}
}
Here this.getSobelImage(); will give me sobel Image of a given image. I am
working with gray level images hence i am considering only one plane
(RED). Here SIGMA_G[5] contains value of sigmaG as suggested by Author.