Sobel edge magnitude

Background

Classic Sobel filters approximate image derivatives. The vertical-edge ($G_x$) and horizontal-edge ($G_y$) kernels are

$$G_x = \begin{bmatrix}-1&0&1\\-2&0&2\\-1&0&1\end{bmatrix}, \qquad G_y = \begin{bmatrix}-1&-2&-1\\0&0&0\\1&2&1\end{bmatrix}.$$

Convolve the image with each (valid, stride 1), then combine into an edge magnitude:

$$M_{ij} = \sqrt{g_{x,ij}^2 + g_{y,ij}^2}.$$

Problem statement

Implement sobel_magnitude(img) returning the Sobel gradient-magnitude map of a grayscale image.

Input

• img — array-like of shape (H, W), H, W ≥ 3.

Output

Returns an np.ndarray of shape (H - 2, W - 2): $\sqrt{g_x^2 + g_y^2}$ at each valid position.

Examples

Example 1 — a flat image has no edges

Input:  img = 4×4 of all 7s
Output: 2×2 of all 0.0

Example 2 — a vertical edge lights up

Input:  img = [[0,0,9,9],[0,0,9,9],[0,0,9,9]]
Output: [[36. 36.]]

Explanation: $G_x$ responds strongly to the 0→9 column transition; $G_y$ is 0; magnitude $=\sqrt{36^2+0^2}=36$.

Constraints

• Use valid (no-padding) 3×3 cross-correlation for both kernels.
• Output shape is (H-2, W-2).
• Combine as sqrt(gx**2 + gy**2).

Notes

• Sobel is a hand-designed edge detector; a CNN's first layer learns a whole bank of such filters from data instead of using these fixed ones.