Estimation of multiple orientations in multi-dimensional signals

Analysis of Multiple Orientations

“…Mase [40], [16] (additive superposition model, gray value image sequences), followed by Shizawa and Iso [41] (additive superposition, gray value images, connection to steerable filters). More recent results can be found in [42], [43] (additive model, images; first theoretical steps towards higher multiplicity of signals beyond double orientations) and [13] (occluding model). An extensive discussion of these double-orientation approaches can be found in [17].…”

“…While the first space is the space of symmetric N × N matrices, the latter space is the space of N × N -matrices formed by u 1 ⊗ u 2 + u 2 ⊗ u 1 , i.e., the space of symmetric rank-2 matrices of size N × N . Methods to decompose such matrices into u 1 and u 2 can be found in, e.g., [40], and in [43], [48], and are outlined in the following. After estimating the MOP vectorũ, it is mapped to the space of fully symmetric tensors, which is now equivalent to the space of symmetric N × N -matrices, including the division by the permutation count according to (22).…”

“…This means that we have to qualify the above considerations: for images, and -apart from univariate signals where orientation cannot be defined -only for images, the MOP vector is in fact a minimal description of the sought parameters. For M = 2, decomposition techniques via roots of a polynomial can be found in [27], [47], [42], [43], [17]. For M > 2, the approach can be extended as follows: for instance, yields for the components of the MOP vector…”

Estimation of Multiple Orientations and Multiple Motions in Multi-Dimensional Signals

“…We model multiple orientation signals as an additive superposition of a set of one-dimensional oriented signals, but occluded orientations can be approached in the same manner [3,22,16,14]. As a major contribution, we then present a general solution for decomposing the MOP into the individual orientations, and thereby overcome previous limitations in either the dimension of the signal or the number of orientations [20,21,17,3,22,16,14]. This paper is organized as follows.…”

Nonlinear Analysis of Multi-Dimensional Signals: Local Adaptive Estimation of Complex Motion and Orientation Patterns

“…Based on (7.1), the intrinsic dimension [135] is defined by the dimension of the subspace E to which the signal is confined. More precisely, the intrinsic dimension of an n-dimensional signal s is n − d if s satisfies the constraint in (7.1) [74,81]. Therefore, when estimating subspaces, it is essential to know the intrinsic dimension of the signal.…”