Fast optimal linear approximation of the images of variably illuminated solid objects for recognition

A method to calculate an element R(x,y) of a viewpoint-independent spatial covariance matrix of a given object where there is no preferred direction in the illumination, the method comprising:

• calculating E(c(p(x),q(y))), according to the equation:
• E(c)=∫nεS2∩{C(n,c)>0}c2nx2−(1−c2)ny2
• where nεS2 is a particular direction on the unit sphere, S2, where C(n,c) is the joint-visibility condition, where n is the given direction, where p and q are vectors in which the x-axis is the bisector of the angle, 2θ, between them, where p=(c,+s,0) and q=(c,−s,0), where c=cos θ, s=sin θ, and (x,y) is the image point;
• solving for the viewpoint-independent spatial covariance matrix R(x,y) according to the equation:
• R(x,y)=α(x)α(y)E(c(p(x),q(y)))
• where α(x) is the intrinsic reflectance at the image point x and α(y) is the intrinsic reflectance at the image point y;
• and
• tabulating and storing the result of said viewpoint-independent spatial covariance matrix R(x,y).

A method for finding a set of Q pairs of reference values for the albedo and normals, {({circumflex over (α)}q,{circumflex over (q)})}qεQ, such that the perturbations are smallest for a given value of Q, the method comprising:

• using vector quantization algorithms to cluster a set of vectors {(α(x),p(x)))}xεV together in Q clusters;
• finding the centroids {({circumflex over (α)}q,{circumflex over (q)})}qεQ of said clusters such that the average distance from the vectors to the nearest respective cluster-centroid, q(x)≡qx, is minimal, where {circumflex over (α)}q is a reference albedo to an albedo (α(x), {circumflex over (q)} is a reference normal close to a vector normal to the surface, p(x), and x is a point on the surface of an object;
• and
• storing the results.

A method to generate a low-dimensional basis of the viewpoint-dependent illumination subspace from the pre-computed viewpoint-independent hierarchy of eigen-surfaces, the method comprising:

• generating and diagonalizing said object's vector-quantized viewpoint-independent covariance matrix RQ;
• determining the eigenbasis hierarchy defined on the surface of the object from the results of RQ, using the equation:
• R
• ⁡
• (
• u
• ,
• v
• )
• ≈
• ∑
• r
• =
• 1
• Q
• ⁢
• ψ
• r
• ⁡
• (
• u
• )
• ⁢
• σ
• r
• 2
• ⁢
• ψ
• r
• ⁡
• (
• v
• )
• where Q is the desired complexity, where σr2 is the non-increasing eigenspectrum of the spatial and the temporal covariance matrices, where ψr(u) and ψr(v) are the respective orthonormal eigenvectors of said spatial and temporal covariance matrices, and (u,v) is a point on the surface of the object;
• choosing a cutoff, N, such that the average residual signal power, trRN, is between the values of 0.1-10;
• keeping only the first N eigensurfaces and storing the results for subsequent use;
• warping said stored N eigensurfaces to a basis of the viewpoint-dependent illumination subspace at the viewpoint-dependent stage using the equation:
• {tilde over (ψ)}r(x)={tilde over (ψ)}r(x(u))
• when a query at a particular viewpoint needs to be matched to the objects in said database;
• and
• storing the results.