FIT5221 Lecture 2 - Edge and Corner Detection

Vocabulary and mathematical formulations for edge detection systems and the Harris corner detector as presented in FIT5221.

Edge detection

The goal of identifying sudden changes or discontinuities in an image's intensity function to encode semantic and shape information.

Image gradient

A vector that points in the direction of most rapid change in intensity, where the magnitude indicates edge strength and the direction represents the orientation of the edge normal.

Sobel edge detector

An operator using separable filters to approximate the gradient of the image intensity. The magnitude is calculated as G(x, y) = \text{\sqrt{I_x(x, y)^2 + I_y(x, y)^2}}, and it uses specific kernels like $M_x = \begin{pmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \end{pmatrix}$.

Derivative theorem of convolution

A property used in image processing to combine smoothing and differentiation: $\frac{\partial}{\partial x} (h * f) = (\frac{\partial}{\partial x} h) * f$, which allows for the use of a derivative of Gaussian filter.

2D Gaussian

A function used for image smoothing defined as $G_{\sigma}(x, y) = \frac{1}{2\pi\sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}$.

Canny edge detector

A multi-step edge detection algorithm that involves filtering with a derivative of Gaussian, finding gradient magnitude and orientation, non-maximum suppression, and linking edges via hysteresis thresholding.

Non-maximum suppression

A technique used in edge detection to thin wide ridges down to a single pixel width by checking if a pixel's gradient magnitude is a local maximum along the gradient direction.

Hysteresis thresholding

A method that uses two thresholds (low and high) to define edges: the high threshold starts edge curves and the low threshold is used to continue them through iterative neighbor checks.

Repeatability

A characteristic of a good keypoint where the same keypoint can be found in several images despite geometric and photometric transformations.

Saliency

A property of keypoints requiring each keypoint to be distinctive within the image.

Second moment matrix

A 2x2 matrix $M = \sum_{x,y} w(x,y) \begin{pmatrix} I_x^2 & I_x I_y \\ I_x I_y & I_y^2 \end{pmatrix}$ computed from image derivatives used to identify corners.

Harris corner response function

A formula used to classify image regions: $R = \det(M) - \alpha \text{trace}(M)^2 = \lambda_1\lambda_2 - \alpha(\lambda_1 + \lambda_2)^2$, where $R > 0$ indicates a corner, $R < 0$ an edge, and $|R|$ small a flat region.

Invariance

A property where an image is transformed but the detected feature or corner locations do not change.

Covariance

A property where if two transformed versions of the same image exist, features are detected in corresponding locations (e.g., corner locations are covariant with respect to translation and rotation).