Edges & Corners

Edges are 1D loci of intensity discontinuity; corners are 0D points where intensity changes in every direction. Both are derivative-based features built from the gradient operations of the previous chapter, and both feed downstream pipelines: edges into segmentation and contour analysis, corners into feature matching and SfM.

The Canny edge detector

A Computational Approach to Edge Detection (Canny, PAMI 1986) is the canonical edge detector and remains the reference algorithm for sharp, single-pixel-width edges. The pipeline is four stages:

1. Smooth the image with a Gaussian to suppress noise.
2. Compute the gradient $\mathrm{\nabla }I=(I_x,I_y)$ via convolution; the magnitude $|\mathrm{\nabla }I|=\sqrt{I_x^2+I_y^2}$ marks edge strength and the orientation $\arctan (I_y/I_x)$ marks the edge normal.
3. Non-maximum suppression along the gradient direction — keep only pixels that are local maxima of magnitude perpendicular to the edge.
4. Hysteresis thresholding — accept any pixel above $T_{\text{high}}$, reject any below $T_{\text{low}}$, accept the in-between band only if connected to a strong pixel.

Canny is the example of how a multi-stage classical pipeline (smoothing, derivatives, suppression, hysteresis) is needed to produce clean output from noisy inputs.

Harris corner detector

A corner is detected by looking at how the second-moment matrix behaves in a window:

$$M=\sum _{(x,y)\in W}w(x,y)\left[\begin{array}{cc}{I}_x^2& I_x{I}_y\\ I_x{I}_y& I_y^2\end{array}\right].$$

The eigenvalues of $M$ describe how intensity varies in the two principal directions. Two large eigenvalues = corner, one large = edge, two small = flat region. The Harris response avoids explicit eigendecomposition:

$$R=det(M)-\kappa (\mathrm{trace}M{)}^2,\kappa \in [0.04,0.06].$$

Local maxima of $R$ above a threshold are corners. Harris (1988) is rotation-invariant but not scale-invariant — that is what motivated the SIFT line of detectors.

FAST corners

FAST: Features from Accelerated Segment Test (Rosten & Drummond, ECCV 2006) is the algorithm of choice when corner-detection cost matters (real-time SLAM, embedded systems). For each candidate pixel $p$, examine the 16 pixels on a Bresenham circle of radius 3 around it. $p$ is a corner if a contiguous arc of $N\ge 9$ of those pixels are all brighter or all darker than $p$ by a threshold. A learned decision tree pre-filters the test, giving order-of-magnitude speedups over Harris.

Edges and corners in the deep era

Modern detectors and descriptors (SuperPoint, KeyNet, DISK) re-derive these signals as side outputs of CNNs trained for downstream tasks. SuperPoint, for example, predicts a corner heatmap and a per-pixel descriptor jointly, distilling years of hand-crafted detector design into a single forward pass. The classical edge/corner story matters because it explains what those networks learn and why the receptive-field, scale, and threshold considerations remain relevant.

What to read next

• Local Features (SIFT, SURF, ORB) — turning corners into matchable descriptors.
• Image Pyramids & Scale-Space — the multi-scale extension that makes detectors scale-invariant.
• Linear Filters & Convolution — the gradient operators these detectors are built on.