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Linear Subspaces - Geometry

Linear Subspaces - Geometry. No Invariants, so Capture Variation. Each image = a pt. in a high-dimensional space. Image: Each pixel a dimension. Point set: Each coordinate of each pt. A dimension. Simplest rep. of variation is linear. Basis (eigen) images: x 1 …x k

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Linear Subspaces - Geometry

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  1. Linear Subspaces - Geometry

  2. No Invariants, so Capture Variation • Each image = a pt. in a high-dimensional space. • Image: Each pixel a dimension. • Point set: Each coordinate of each pt. A dimension. • Simplest rep. of variation is linear. • Basis (eigen) images: x1…xk • Each image, x = a1x1 + … + akxk • Useful if k << n.

  3. When is this accurate? • Approximately right when: • Variation approximately linear. Always true for small variation. • Some variations big, some small, can discard small. • Exactly right sometimes. • Point features with scaled-orthographic projection. • Convex, Lambertian objects and distant lights.

  4. Principal Components Analysis (PCA) • All-purpose linear approximation. • Given images (as vectors) • Finds low-dimensional linear subspace that best approximates them. • Eg., minimizes distance from images to subspace.

  5. Derivation on whiteboard • This is all taken from Duda, Hart and Stork Pattern Classification pp. 114-117. Excerpt in library.

  6. SVD • Scatter matrix can be big, so computation non-trivial. • Stack data into matrix X, each row an image. SVD gives X = UDVT • D is diagonal with non-increasing values. • U and V have orthonormal rows. • VT(:,1:k) gives first k principal components. • matlab

  7. Linear Combinations I S P Immediately apparent that u and v coordinates lie in a 4D linear subspace

  8. We Can Remove Translation (1) • This is trivial, because we can pick a simple origin. • World origin is arbitrary. • Example: We can assume first point is at origin. • Rotation then doesn’t effect that point. • All its motion is translation. • Better to pick center of mass as origin. • Average of all points. • This also averages all noise.

  9. Remove Translation (2) Notice this is just the first step of PCA.

  10. First Step: Remove Translation (3)

  11. Rank Theorem has rank 3. This means there are 3 vectors such that every row of is a linear combination of these vectors. These vectors are the rows of P. P S So, given any object, u and v coordinates of any image of it lie in a 3-dimensional linear subspace.

  12. Lower-Dimensional Subspace • (a4,b4) = affine invariant coordinates of point 4 relative to first three. • Represent image: (a4,b4, a5,b5,…an,bn) • This representation is complete. • The a or b coordinates of all images of an object occupy a 1D linear subspace.

  13. Representation is Complete • 3D-2D affine transformation is projection in some direction + 2D-2D affine transformation. • 2D-2D affine maps first three image points anywhere. So they’re irrelevant to a complete representation. • Once we use only affine coordinates, 2D-2D affine transformation no longer matters.

  14. 1D Subspace

  15. Summary • Projection can be linearized. • So images produced under projection can be linear and low-dimensional. • Are these results relevant to surfaces of real 3D objects projected to 2D? • Maybe to features; probably not to intensity images. • Why would images of a class of objects be low-dimensional?

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