The Sobel operator belongs to the family of first-derivative gradient operators, which detect edges by measuring the rate of intensity change.
First vs. Second Derivative
Gonzalez and Woods note that the first derivative is "zero on flat segments, nonzero at the onset of a step," producing thick edge responses. The second derivative (Laplacian) is "zero on flats and constant ramps, nonzero at step onset AND end" — more sensitive to fine detail but also much more sensitive to noise.
Sobel vs. Prewitt
The Prewitt operator uses [1 1 1] for smoothing instead of [1 2 1]. Solomon & Breckon explain the advantage: "The Sobel kernel implements differentiation in one direction and approximate Gaussian averaging in the other" — the center-weighted smoothing reduces noise sensitivity at the cost of slight blurring perpendicular to edges.
Sobel vs. Roberts Cross
Roberts Cross uses 2×2 kernels oriented at 45°. It is "fast but very sensitive to noise" (Solomon & Breckon) because it lacks any smoothing component. The Sobel's 3×3 size provides a critical smoothing row/column that makes it practical for real (noisy) images.
Isotropic Response
The Sobel operator is isotropic (direction-independent) at multiples of 45°. At intermediate angles, it slightly underestimates gradient magnitude — but the error is small and the operator is sufficiently rotation-invariant for most applications.
The Sobel filter is the workhorse gradient operator because it balances noise resistance with edge sensitivity. Simpler operators (Roberts) are too noise-sensitive. More complex operators (Scharr) offer minor improvements but at greater cost. The Sobel hits the sweet spot for practical tissue image analysis.
