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# Computer examples - PowerPoint PPT Presentation

Computer examples. Tenenbaum, de Silva, Langford “A Global Geometric Framework for Nonlinear Dimensionality Reduction”. Statue face database. 698 64x64 grayscale images 2 mins, 12 secs on a ~600 (?) MHz PIII. The computed manifold. The computed manifold.

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### Computer examples

Tenenbaum, de Silva, Langford

“A Global Geometric Framework for Nonlinear Dimensionality Reduction”

• 698 64x64 grayscale images

• 2 mins, 12 secs on a ~600 (?) MHz PIII

• One test you could do:

• Sort all faces according to first manifold coordinate (“left-right”)

• View them in order

• See if the face makes a monotonic progression from left to right

Right  Left

Up  Down

Cleaner, since light variation is strictly azimuthal (consistent chin shadow)

Lit on left  Lit on right

Semantic consistency of a dimension value deteriorates between points that are far away on the manifold.

4 consecutive frames from right  left movie:

Well-lit faces are turning to the left with respect to each other

Dimly-lit faces also don’t turn right w.r.t each other

Semantic consistency of a dimension value deteriorates between points that are far away on the manifold.

Explanations:

Geodesic distance on the manifold is approximated by shortest-path distance in a neighbor graph.

 Sparsity in neighbor graphs result in distance error for points far away on the graph.

Geodesic distance approximator can’t be perfect in the face of sparse data

The test expected this face:

…to be a bit more left-facing than this face:

• Collapsing the manifold to one dimension isn’t the way to use it.

• Try tracing one dimension while keeping the other dimensions from jumping around too much.

Algorithm used:

Sort images by “left-right” coord as before

Draw a smooth line through the manifold

Only add images that are within a certain manifold distance D from this line.

D = 20

(Half the range of the “up-down” dimension)

(D = 30)

D = 40 (using 80% of the faces)

D = 50 (using 98% of the faces)

Run both algorithms on 100 of the statue faces (64 x 64 pixels)

Isomap

LLE

Running time for 100 64x64 images:

LLE: 5 secs

Isomap: 1.39 secs

The collapsing-to-primary-dimension-test:

Uh… the collapsing-to-second-dimension-test

The horizontal manifold traversal test (7 frames)

• LLE: once manifold is computed, meaningful paths through it need to be searched for.

• Images with a common background and a single translating object will have a rough time with pixel differences.

• Uniform translation, no overlap

Input images:

Output images:

• Uniform translation, 1-column overlap

Input images:

Output images:

• Uniform translation, 1-column overlap

• Uniform translation, with a skip

• Isomap with k = 1 (like before)

(Original)

(Reconstruction)

• Isomap with k = 2

(Original)

(Reconstruction)

• Isomap with k = 2