Image analysis problems such as feature tracking, edge detection, image enhancement, or texture analysis require thedetection of multi-oriented patterns which can appear at arbitrary orientations. Direct rotated matched filtering for feature detection is computationally expensive, but can be sped up with steerable filters. So far, steerable filter approaches were limited to only one direction. Many important low-level image features are, however, characterized by more than a single orientation. We therefore present here a framework for efficiently detecting specific multi-oriented patterns with arbitrary orientations in grayscale images. The core idea is to construct multisteerable filters by appropriate combinations of single-steerable filters. We exploit that steerable filters are closed under addition and multiplication. This allows to derive a design guide for multisteerable filters by means of multivariate polynomials. Furthermore, we describe an efficient implementation scheme and discuss the use of weighting functions to reduce angular oscillations. Applications in camera calibration, junction analysis of images from plant roots, and the discrimination of L, T, and X-junctions demonstrate the potential of this approach.

Estimation of local orientations in multivariate signals is an important problem in image processing and computer vision. This general problem formulation also covers optical flow estimation, which can be regarded as orientation estimation in space-time-volumes. Modelling a signal using only a single orientation, however, is often too restrictive, since occlusions and transparencies occur frequently, thus necessitating the modelling and analysis of multiple orientations. We, therefore, develop a unifying mathematical model for multiple orientations: Beyond describing an arbitrary number of orientations in scalar- and vector-valued image data such as color image sequences, it allows the unified treatment of additively and occludingly superimposed oriented structures as well as of combinations of these. Based on this model, we describe estimation schemes for an arbitrary number of additively or occludingly superimposed orientations in images. We confirm the performance of our framework on both synthetic and real image data.

The theory of steerable filters (introduced by Freeman and Adelson in 1991) offers a convenient and computationally efficient method to represent the rotation of a specific class of 2D filters as a linear combination of base filters. In this paper, we generalize this theory to filters which are modeled as point-by-point products of multiple (in our examples: two) base filters which each are rotated by individual rotation angles, thus introducing the term "multi-steerability". While standard steerable filters are useful for the detection of edges or lines (oriented in some unknown direction), our novel extended theory can be applied to the detection of features which are characterized by multiple orientations, for instance corners or line crossings, which greatly increases the applicability of the theory. We conclude our paper with an example for detecting the crossings in a checkerboard image using double-steerable filters.

Estimation of local orientation in images may be posed as the problem of finding the minimum gray-level variance axis in a local neighborhood. In bivariate images, the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2 x 2 tensor. For an ideal single orientation, the tensor is rank-deficient, i.e., the smaller eigenvalue vanishes. A large minimal eigenvalue signals the presence of more than one local orientation, what may be caused by non-opaque additive or opaque occluding objects, crossings, bifurcations, or corners. We describe a framework for estimating such superimposed orientations. Our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions. Unlike in the single-orientation case, the eigensystem analysis does not directly yield the orientations, rather, it provides so-called mixed-orientation parameters (MOPs). We, therefore, show how to decompose the MOPs into the individual orientations. We also show how to use tensor invariants to increase efficiency, and derive a new feature for describing local neighborhoods which is invariant to rigid transformations. Applications are, e.g., in texture analysis, directional filtering and interpolation, feature extraction for corners and crossings, tracking, and signal separation.

This paper outlines the ubiquitous presence of generalized orientation (or subspace) estimation problems in image analysis. We show the potential sources of bias in naive approaches to directional estimation problems, discuss countermeasures against this bias, and point out the direct relation to the total least squares problem. An improved method (using TLS and "equilibration") for a precise direct motion estimation of planar objects (8 parameter motion model, homography estimation) concludes this paper.