Demonstration of

The Windowed Image Second Moment Matrix

by Serge Belongie

This web page provides a simple exhibition of the use of the windowed-image second moment matrix. The windowed-image second moment matrix is an image processing tool which is useful in such tasks as locating junctions, determining the orientation of edges, describing textured regions and computing optical flow. A postscript file which describes the computation an interpretation of the following images in more detail may be obtained by clicking here.
Original image: 32 by 32 white square on black bacground.

Gradient: vectors point from dark to light in the direction of greatest change.

Detail of gradient vectors near corner.

Square of x-component of gradient, smoothed with 5 by 5 Gaussian.

Product of x- and y-components of gradient, smoothed with 5 by 5 Gaussian

Square of y-component of gradient, smoothed with 5 by 5 Gaussian.

Larger eigenvalue: stronger in singly-oriented regions (e.g. along the edges) and weaker in multiply-oriented regions (e.g. at the corners).

Smaller eigenvalue: negligible except in regions with multiple orientations.

Angle of 1st eigenvector, divided by two and weighted by larger eigenvalue: the argument of the principle eigenvector yields the dominant orientation. It is only meaningful in singly-oriented regions.

A more realistic image: 258 by 350 image of trees. (Matlab test image.)

Larger eigenvalue.

Smaller eigenvalue.