The discrete cosine transform is a valuable tool in analysis of data on undirected rectangular grids, like images. In this paper it is shown how one can define an analogue of the discrete cosine transform on triangles. This is done by combining algebraic signal processing theory with a specific kind of multivariate Chebyshev polynomials. Using a multivariate Christoffel-Darboux formula it is shown how to derive an orthogonal version of the transform.

In this paper, we study the problem of non-orthogonal joint diagonalisation of a set of real symmetric matrices via simultaneous conjugation. A family of block Jacobi-type methods are proposed to optimise two popular cost functions for the non-orthogonal joint diagonalisation, namely, the off-norm function and the log-likelihood function. By exploiting the appropriate underlying manifold, namely the so-called oblique manifold, rigorous analysis shows that, under the exact non-orthogonal joint diagonalisation setting, the proposed methods converge locally quadratically fast to a joint diagonaliser. Finally, performance of our methods is investigated by numerical experiments for both exact and approximate non-orthogonal joint diagonalisation.

The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the well-known sign-flipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the so-called QR FastICA algorithm, which employs the QR decomposition (Gram-Schmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA.

In this paper, a conjugate gradient method on the complex Grabmann manifold is proposed that computes the k-principal components of a Hermitian (n × n)-matrix. The algorithm is at most of order O(n2k) and yields locally good convergence results.

Extracting the motion parameters of a moving camera is an important issue in computer vision. This is due to the need of numerous emerging applications like telepresence and robot navigation. The key issue is to determine a robust estimate of the (3×3) essential matrix with its five degrees of freedom. In this work, a robust technique to compute the essential matrix is suggested under the assumption that the images are calibrated. The algorithm is a combination of the five-point relative pose problem using an optimization technique on a manifold, with the random sample consensus. The results show that the proposed method delivers faster and more accurate results than the standard techniques.

Independent subspace analysis (ISA) was developed as an extension of independent component analysis (ICA) when statistical independences are assumed to exist between groups of components rather than between individual components. Due to the superiority of FastICA against other linear ICA algorithms, an intuitive analogy, the so-called FastISA algorithm, has been proposed to solve the problem of ISA. Experimental evidences so far have shown the capability of FastISA, regardless of any independence criterion. Since standard FastICA can be viewed as a special case of an approximate Newton ICA method and moreover can be generalised as a scalar shifted fixed point algorithm, in this work, we propose two new classes of ISA algorithms, an approximate Newton-like ISA method and a matrix shifted fixed point ISA algorithm on the Grabmann manifold. As an aside, FastISA is a special case in the class of matrix shifted fixed point ISA algorithms. Performances of the proposed algorithms are investigated by numerical experiments.

In recent years, algorithms to perform Independent Component Analysis in blind identification, localisation of sources or more general in data analysis have been developed. Prominent example certainly is the socalled FastICA algorithms from the Finnish school. In this paper we will generalise the FastICA algorithm considered as a discrete dynamical system on the unit sphere to the case where all units converge simultaneously, i.e., we consider some kind of parallel FastICA algorithm living on the orthogonal group. In addition we present a local convergence analysis for the algorithms proposed in this paper building on earlier work. It turns out that one can treat these type of algorithms in a similar manner as the Rayleigh quotient iteration, well known in numerical linear algebra, i.e. considering the algorithm as a discrete dynamical system on a suitable manifold. The algorithms presented here are compared by several numerical experiments and simulations.

This paper investigates a novel procedure to calculate smooth interpolation curves of the rotation group SO3, which is commonly considered as the standard representation of rigid-body's orientations. The algorithm is a combination of rolling and wrapping with the pull back/push forward technique. One remarkable advantage of this approach is that interpolation curves are given in closed form, which brings convenience for implementations on real-time control systems. A numerical example along with some visualization results is presented as well

We present new algorithms for refining the estimates of the eigenvectors of a real symmetric matrix. Using techniques from calculus, we show that the algorithms converge locally cubically fast. By this we mean that locally all eigenvalue-eigenvector pairs simultaneously converge at a cubic rate. This is in contrast to well known shifted QR algorithms, which depending on the shifting strategy employed, have only one (or at most a small subset) of the eigenvalue-eigenvector pairs converging cubically at any one time. The algorithms are well suited to the situation where one needs to compute the eigenvectors of a perturbed matrix A + E based on a good estimate of the eigenvectors of a matrix A. Such a situation frequently appears in tracking applications.

There is a robotic balancing task, namely real-time dextrous-hand grasping, for which linearly constrained, positive definite programming gives a quite satisfactory solution from an engineering point of view. We here propose refinements of this approach to reduce the computational effort. The refinements include elimination of structural constraints in the positive definite matrices, orthogonalization of the grasp maps, and giving a precise Newton step size selection rule.

The problem of finding the generalized eigenvalues and eigenvectors of a pair of real symmetric matrices A and B, with B>0, can be viewed as a smooth optimization problem on a smooth manifold. We present a cost function approach to the generalized eigenvalue problem which is posed on the product of the n-sphere and Euclidian space R. The critical point set of this cost function is studied. An algorithm is presented based on constrained optimization. A proof of local quadratic convergence is given.

A VLSI-suited processor array is presented that performs a Jacobi-like parallel algorithm for computing the real Schur decomposition of real nonsymmetric (n/spl times/n) matrices. The array consists of n/sup 2//4 processor cells which only perform exchange operations and elementary (2/spl times/2) plane rotations. The underlying algorithm iterates on dense matrices, i.e., the matrices need not to be reduced to some canonical form in advance. The iteration algorithm is derived by a problem adapted discretization process of an equivalent dynamical system. The dynamical system belongs to the class of isospectral matrix flow equations, i.e. a matrix valued differential equation, whose time evolution converges for t/spl rarr//spl infin/ to upper quasi triangular form. The integration scheme is performed by computing some highly regular orthogonal matrix-matrix multiplications.>

In the past, there have been several attempts to relate Hamiltonian equations, which describe conservative systems, to electrical networks. In control theory the ideas of Hamiltonian systems were expanded to affine Hamiltonian systems with feedback that are input/output systems with Hamiltonian vectorfields. In this paper the principles of affine Hamiltonian systems are applied to a class of electrical networks. It is shown that voltage and current coordinates admit a Poisson structure and how input/output Poisson systems with feedback are related to electrical networks.>

The basic nonlinear operation in median filtering is a sorting of the input data vector. An analog structure for this filtering operation is presented. The sorting operation is embedded in a matrix-valued ordinary differential equation (ODE). The data to be sorted serve as initial values for the ODE. To induce the dynamics of the system a small perturbation is necessary. The circuit structure for the analog median filter consists of basic processing cells containing multipliers, integrators, and adders, arranged in an one-dimensional array. All connections are local only and no external control is necessary.>