Recipe

Classify a window with the Harris response

Use when: you have Sobel gradients $G_x,G_y$ over a small window and need to decide flat, edge, or corner without solving for eigenvalues

Goal: a single signed number $R$ that classifies the window

Taught in Corner detection: Harris, Shi-Tomasi, and FAST

Quick recipe

Compute $G_x,G_y$ at every pixel in the window
Build the structure tensor $M$
Compute $\det(M)$ and $\operatorname{trace}(M)$
Compute $R=\det(M)-k\operatorname{trace}(M)^2$
Classify by $R$ 's sign (and $\operatorname{trace}(M)$ 's size)

1
Run Chapter 0's Sobel kernels over the window to get one $(G_x,G_y)$ pair per pixel.

Why: everything downstream is built from these gradient readings, and Sobel is this course's own noise-resistant way to compute them.
2
Sum $G_x^2$ , $G_y^2$ , and $G_xG_y$ over every pixel in the window into the $2\times2$ matrix $M$ .

Why: one pixel's gradient could be a fluke; summing over the whole window confirms the pattern holds throughout the neighborhood, not just at one point.
3
Read these two numbers straight off $M$ 's own entries: no eigenvalue solve needed.

⚡ Shortcut

$\det(M)=Sxx\cdot Syy-Sxy^2$ and $\operatorname{trace}(M)=Sxx+Syy$ are both a couple of multiplications and one subtraction — the same $2\times2$ trace/determinant shortcut that skips straight to a matrix's eigenvalues without ever writing out $\det(A-\lambda I)$ by hand.

Why: $\det(M)=\lambda_1\lambda_2$ and $\operatorname{trace}(M)=\lambda_1+\lambda_2$ carry everything the classification below needs about the two eigenvalues, without solving for either one.
4
Use $k=0.04$ unless a specific value is given — the standard empirical range is $0.04$ – $0.06$ .

Why: this single number is large and positive exactly when both eigenvalues are large and comparable, and negative exactly when one dominates the other.
5
Read $R$ 's sign against $\operatorname{trace}(M)$ 's own size to reach the final verdict.
- $\operatorname{trace}(M)$ itself is small — flat, regardless of $R$ 's sign.
- $R$ is strongly positive — corner.
- $R$ is negative — edge.
Why: each case matches one of the three eigenvalue signatures this section already derived — flat (both eigenvalues small), edge (one large, one ≈0), corner (both large).

Common mistakes

Checking $R$ 's sign without first confirming $\operatorname{trace}(M)$ is meaningfully large — a truly flat region can also produce $R$ near zero, but for the opposite reason (nothing there at all, not a narrowly-missed corner).
Treating $\det(M)$ alone as the verdict. A window can have a sizeable $\det(M)$ and an even larger $\operatorname{trace}(M)$ , so the $k\operatorname{trace}(M)^2$ penalty still pushes $R$ negative — both terms have to be weighed together, which is the entire reason the penalty exists.
Assuming a bigger $k$ always finds more corners. A larger $k$ makes the penalty stronger, which makes the detector pickier — fewer borderline windows clear $R>0$ , not more.