Sobel Edge Magnitude

Detect edges in a grayscale image with the Sobel operator.

An edge is a place where brightness changes quickly. The Sobel operator estimates the horizontal and vertical brightness gradients with two 3 x 3 kernels, each a smoothed finite difference:

Gx = [[-1, 0, 1],        Gy = [[-1, -2, -1],
      [-2, 0, 2],              [ 0,  0,  0],
      [-1, 0, 1]]              [ 1,  2,  1]]

Slide each kernel over the image without padding (valid mode). At each position, multiply the kernel with the 3 x 3 patch under it and sum. This is cross-correlation, the same operation deep learning calls convolution, so do not flip the kernels. The edge strength is sqrt(gx^2 + gy^2).

The first layers of a trained CNN often learn filters that look much like these.

Function signature

def sobel_magnitude(img):

Example

Input

img = np.array([[0, 0, 1, 1],
                [0, 0, 1, 1],
                [0, 0, 1, 1],
                [0, 0, 1, 1]], dtype=float)
sobel_magnitude(img)

Output

array([[4., 4.],
       [4., 4.]])

Reasoning

There is a vertical edge between columns 1 and 2. Every 3 x 3 patch has a left column darker than its right column by 1, so gx = 1 + 2 + 1 = 4. The rows are all the same, so gy = 0. The magnitude is 4 everywhere.

Constraints and notes

• img has shape (H, W) with H, W >= 3.
• The output has shape (H - 2, W - 2).