Scale Space - Detector Examples

Detector Examples

Following the idea of expressing visual operation in terms of differential invariants computed at multiple scales using Gaussian derivative operators, we can express an edge detector from the set of points that satisfy the requirement that the gradient magnitude

should assume a local maximum in the gradient direction

By working out the differential geometry, it can be shown that this differential edge detector can equivalently be expressed from the zero-crossings of the second-order differential invariant

that satisfy the following sign condition on a third-order differential invariant:

Similarly, multi-scale blob detectors at any given fixed scale can be obtained from local maxima and local minima of either the Laplacian operator (also referred to as the Laplacian of Gaussian)

or the determinant of the Hessian matrix

In an analogous fashion, corner detectors and ridge and valley detectors can be expressed as local maxima, minima or zero-crossings of multi-scale differential invariants defined from Gaussian derivatives. The algebraic expressions for the corner and ridge detection operators are, however, somewhat more complex and the reader is referred to the articles on corner detection and ridge detection for further details.

Scale-space operations have also been frequently used for expressing coarse-to-fine methods, in particular for tasks such as image matching and for multi-scale image segmentation.