Sobel Edge Detection

The problem, from first principles

An edge is a place where brightness changes quickly, so edge detection is derivative estimation. The Sobel operator estimates the horizontal derivative Gx and the vertical derivative Gy by sliding two 3 x 3 kernels over the image. Each kernel differentiates in one direction and smooths in the other (the weights 1, 2, 1), which makes it less sensitive to noise than a plain difference. The gradient magnitude sqrt(Gx^2 + Gy^2) is large on edges, and the direction atan2(Gy, Gx) says which way the brightness changes. This is the hand-designed ancestor of the first convolutional layer of a CNN, which learns kernels like these.

From theory to code

Implement sobel_gradients and sobel_magnitude.

Constraints

• img is a 2-D array of shape (H, W) with H, W >= 3. Use Kx = [[-1, 0, 1], [-2, 0, 2], [-1, 0, 1]] and Ky = Kx.T.
• sobel_gradients(img) returns (gx, gy), each of shape (H - 2, W - 2) (valid positions only). Use cross-correlation (no kernel flip): g[i, j] = sum(img[i:i+3, j:j+3] * K).
• sobel_magnitude(img) returns sqrt(gx**2 + gy**2) with the same shape.

Hints