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generalized_procrustes

PURPOSE ^

Rotationally align clouds of points (generalized Procrustes problem)

SYNOPSIS ^

function [A, R] = generalized_procrustes(A_measure)

DESCRIPTION ^

 Rotationally align clouds of points (generalized Procrustes problem)

 function X = generalized_procrustes(A_measure)

 The input is a 3D matrix A_measure of size nxmxN. Each of the N slices
 A_measure(:, :, i) is a cloud of m points in R^n. These clouds are
 assumed to be (noisy) rotated versions of a reference cloud Atrue.
 This algorithm tries to find the optimal rotations to apply to the
 individual clouds such that they will match each other as much as
 possible following a least-squares cost.

 The output A is an estimate of the cloud Atrue (up to rotation). The
 output R is a 3D matrix of size nxnxN containing the rotation matrices
 such that R(:, :, i) * A is approximately equal to A_measure(:, :, i).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SUBFUNCTIONS ^

SOURCE CODE ^

0001 function [A, R] = generalized_procrustes(A_measure)
0002 % Rotationally align clouds of points (generalized Procrustes problem)
0003 %
0004 % function X = generalized_procrustes(A_measure)
0005 %
0006 % The input is a 3D matrix A_measure of size nxmxN. Each of the N slices
0007 % A_measure(:, :, i) is a cloud of m points in R^n. These clouds are
0008 % assumed to be (noisy) rotated versions of a reference cloud Atrue.
0009 % This algorithm tries to find the optimal rotations to apply to the
0010 % individual clouds such that they will match each other as much as
0011 % possible following a least-squares cost.
0012 %
0013 % The output A is an estimate of the cloud Atrue (up to rotation). The
0014 % output R is a 3D matrix of size nxnxN containing the rotation matrices
0015 % such that R(:, :, i) * A is approximately equal to A_measure(:, :, i).
0016 
0017 % This file is part of Manopt and is copyrighted. See the license file.
0018 %
0019 % Main author: Nicolas Boumal, July 8, 2013
0020 % Contributors:
0021 %
0022 % Change log:
0023 %
0024     
0025     if ~exist('A_measure', 'var')
0026         % Generate random data to test the method.
0027         % There are N clouds of m points in R^n. Each of them is a noisy,
0028         % rotated version of a reference cloud A. Rotations are uniformly
0029         % random and noise on each rotated cloud is iid normal with
0030         % standard deviation sigma.
0031         n = 3;
0032         m = 10;
0033         N = 50;
0034         % The reference cloud
0035         Atrue = randn(n, m);
0036         % A 3D matrix containing the N measured clouds
0037         sigma = .3;
0038         A_measure = multiprod(randrot(n, N), Atrue) + sigma*randn(n, m, N);
0039     else
0040         [n, m, N] = size(A_measure);
0041     end
0042     
0043     % Construct a manifold structure representing the product of groups of
0044     % rotations with the Euclidean space for A. We optimize simultaneously
0045     % for the reference cloud and for the rotations that affect each of the
0046     % measured clouds. Notice that there is a group invariance because
0047     % there is no way of telling which orientation the reference cloud
0048     % should be in.
0049     tuple.R = rotationsfactory(n, N);
0050     tuple.A = euclideanfactory(n, m);
0051     M = productmanifold(tuple);
0052 
0053     % Define the cost function here. Points on the manifold M are
0054     % structures with fields X.A and X.R, containing matrices of sizes
0055     % respectively nxm and nxnxN. The store structure (the caching system)
0056     % is used to keep the residue matrix E in memory, as it is also used in
0057     % the computation of the gradient and of the Hessian. This way, we
0058     % prevent redundant computations.
0059     function [f, store] = cost(X, store)
0060         if ~isfield(store, 'E')
0061             R = X.R;
0062             A = X.A;
0063             store.E = multiprod(R, A) - A_measure;
0064         end
0065         E = store.E;
0066         f = (E(:)'*E(:))/(2*N);
0067     end
0068 
0069     % Riemannian gradient of the cost function.
0070     function [g, store] = grad(X, store)
0071         R = X.R;
0072         A = X.A;
0073         if ~isfield(store, 'E')
0074             [~, store] = cost(X, store);
0075         end
0076         E = store.E;
0077         % Compute the Euclidean gradient of the cost wrt the rotations R
0078         % and wrt the cloud A,
0079         egrad.R = multiprod(E, A'/N);
0080         egrad.A = A - mean(multiprod(multitransp(R), A_measure), 3);
0081         % then transform this Euclidean gradient into the Riemannian
0082         % gradient.
0083         g = M.egrad2rgrad(X, egrad);
0084         store.egrad = egrad;
0085     end
0086 
0087     % It is not necessary to define the Hessian of the cost. We do it
0088     % mostly to illustrate how to do it and to study the spectrum of the
0089     % Hessian at the solution (see further down).
0090     function [h, store] = hess(X, Xdot, store)
0091         R = X.R;
0092         A = X.A;
0093         % Careful: tangent vectors on the rotation group are represented as
0094         % skew symmetric matrices. To obtain the corresponding vectors in
0095         % the ambient space, we need a little transformation. This
0096         % transformation is typically not needed when we compute the
0097         % formulas for the gradient and the Hessian directly in Riemannian
0098         % form instead of resorting the egrad2rgrad and ehess2rhess. These
0099         % latter tools are convenient for prototyping but are not always
0100         % the most efficient form to execute the computations.
0101         Rdot = tuple.R.tangent2ambient(R, Xdot.R);
0102         Adot = Xdot.A;
0103         if ~isfield(store, 'egrad')
0104             [~, store] = grad(X, store);
0105         end
0106         E = store.E;
0107         egrad = store.egrad;
0108         
0109         ehess.R = multiprod(multiprod(Rdot, A) + multiprod(R, Adot), A') + ...
0110                   multiprod(E, Adot');
0111         ehess.R = ehess.R / N;
0112         ehess.A = Adot-mean(multiprod(multitransp(Rdot), A_measure), 3);
0113         
0114         h = M.ehess2rhess(X, egrad, ehess, Xdot);
0115     end
0116 
0117     % Setup the problem structure with manifold M and cost+grad functions.
0118     problem.M = M;
0119     problem.cost = @cost;
0120     problem.grad = @grad;
0121     problem.hess = @hess;
0122 
0123     % For debugging, it's always nice to check the gradient a few times.
0124     % checkgradient(problem);
0125     % pause;
0126     % checkhessian(problem);
0127     % pause;
0128     
0129     % Call a solver on our problem. This can probably be much improved if a
0130     % clever initial guess is used instead of a random one.
0131     X = trustregions(problem);
0132     A = X.A;
0133     R = X.R;
0134     
0135     % To evaluate the performance of the algorithm, see how well Atrue (the
0136     % reference cloud) matches A (the found cloud). Since the recovery is
0137     % up to rotation, apply Kabsch algorithm (or standard Procrustes),
0138     % i.e., compute the polar factorization to best align Atrue and A.
0139     if exist('Atrue', 'var')
0140         [U, ~, V] = svd(Atrue*A');
0141         Ahat = (U*V')*A;
0142         fprintf('Registration error: %g.\n', norm(Atrue-Ahat, 'fro'));
0143     end
0144     
0145     % Plot the spectrum of the Hessian at the solution found.
0146     % Notice that the invariance of f under a rotation yields dim SO(n),
0147     % that is, n*(n-1)/2 zero eigenvalues in the Hessian spectrum at the
0148     % solution. This indicates that critical points are not isolated and
0149     % can theoretically prevent quadratic convergence. One solution to
0150     % circumvent this would be to fix one rotation arbitrarily. Another
0151     % solution would be to work on a quotient manifold. Both can be
0152     % achieved in Manopt: they simply require a little more work on the
0153     % manifold description side.
0154     if M.dim() <= 512
0155         stairs(sort(hessianspectrum(problem, X)));
0156         title('Spectrum of the Hessian at the solution found.');
0157         xlabel('Eigenvalue number (sorted)');
0158         ylabel('Value of the eigenvalue');
0159     end
0160     
0161 end

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