Gradient Covariance · Second-Moment Analysis

The structure
of an image.

A real-time pipeline that converts RGB to luminance, takes its partial derivatives, and computes the covariance of the gradient field. The resulting 2×2 matrix encodes local orientation, edges, and corners.

rec. 709
N × 2 → 2 × 2
eigen · decomp

Input

Synthetic Samples

Local Window (W) 7 px

Corner Threshold 35%

Input RGB i.

Source image · pixel space ℝ³

Luminance L(x,y) ii.

0.2126 R + 0.7152 G + 0.0722 B

‖∇L‖ — Gradient Magnitude iii.

√(Gₓ² + G_y²) · edge energy

Gₓ — ∂L/∂x iv.

Horizontal derivative · red(+) / blue(−)

G_y — ∂L/∂y v.

Vertical derivative · red(+) / blue(−)

Corner Response — λ₂ vi.

Shi-Tomasi · min eigenvalue of J_W

Gradient Distribution

Each point is one pixel's gradient vector (Gₓ, G_y). The ellipse is the 2σ iso-contour of the global covariance matrix C. Its axes are the eigenvectors; their lengths are √λ.

pixel gradient

2σ covariance ellipse

major axis · v₁ · √λ₁

minor axis · v₂ · √λ₂

Pipeline formulas

1 · Luminance (Rec. 709) $$L(x,y) = 0.2126\,R + 0.7152\,G + 0.0722\,B$$

2 · Spatial Gradients $$G_x = \frac{\partial L}{\partial x}, \quad G_y = \frac{\partial L}{\partial y}$$

3 · Flatten — N pixels into M $$\mathbf{M} = \begin{bmatrix} G_x(1) & G_y(1) \\ G_x(2) & G_y(2) \\ \vdots & \vdots \\ G_x(N) & G_y(N) \end{bmatrix} \in \mathbb{R}^{N\times 2}$$

4 · Second-Moment Matrix $$\mathbf{C} = \frac{1}{N}\,\mathbf{M}^\top \mathbf{M}$$

Live statistics

N (pixels)—

C · 2×2 second-moment

— —
— —

λ₁ · major eigenvalue—

λ₂ · minor eigenvalue—

anisotropy · 1 − λ₂/λ₁—

dominant angle · v₁—

trace · tr(C)—

det · det(C)—