Riemann-Tensor Demos

Click on the underlined titles to watch QuickTime movies that show the Riemann-tensor components (in Cartesian coordinates) for different inputs. The input sequences are shown on the left and the six components of the curvature tensor in the right panel. These demos are related to a recent paper published in Optics Express and available at http://www.opticsexpress.org/oearchive/source/23045.htm
"Stationary" square Watch the stationary component (top left component). This component measures the spatial curvature and is the only active component in case of stationary images. Note, however, that the other components become active with transient patterns (short-time presentations, flicker, ...).

Square moves horizontally Watch the horizontal motion components (top right and bottom right).

Square moves in different directions Watch the horizontal-motion components (as above) and the vertical motion components (middle top and middle bottom). Note that the middle top component (the sectional curvature in x,t) is selective to discontinuously moving straight edges with vertical orientation and to continuous motion (of the corners) along that vertical direction. The top right component (the sectional curvature in y,t) is selective to horizontal edges and horizontal motions.

Endstopping Note the property of spatial endstopping (no response to straight patterns) for translating patterns.

Now we are looking at a more natural movie but only through some rather small windows that are placed according to different criteria. The number of windows is equal for the three cases we consider.
Random windows In this case the windows are placed at random positions in space and time and the underlying scene is hard to recognize.

Mean-curvature windows Now the the probability for a window to be placed is higher if the mean curvature of the movie is higher. The movie is still hard to see.

Curvature windows Finally, the probability for a window to be placed is higher, if the curvature (as measured by the Riemann tensor) is higher - the scene can now be recognize (at least by some people).

One large window ... for the original.

Curator: E. Barth

Last update: July 26, 2000

"Stationary" square	Watch the stationary component (top left component). This component measures the spatial curvature and is the only active component in case of stationary images. Note, however, that the other components become active with transient patterns (short-time presentations, flicker, ...).
Square moves horizontally	Watch the horizontal motion components (top right and bottom right).
Square moves in different directions	Watch the horizontal-motion components (as above) and the vertical motion components (middle top and middle bottom). Note that the middle top component (the sectional curvature in x,t) is selective to discontinuously moving straight edges with vertical orientation and to continuous motion (of the corners) along that vertical direction. The top right component (the sectional curvature in y,t) is selective to horizontal edges and horizontal motions.
Endstopping	Note the property of spatial endstopping (no response to straight patterns) for translating patterns.

Random windows	In this case the windows are placed at random positions in space and time and the underlying scene is hard to recognize.
Mean-curvature windows	Now the the probability for a window to be placed is higher if the mean curvature of the movie is higher. The movie is still hard to see.
Curvature windows	Finally, the probability for a window to be placed is higher, if the curvature (as measured by the Riemann tensor) is higher - the scene can now be recognize (at least by some people).
One large window ...	for the original.