ii About the cover The figure on the cover shows a visualization of a symmetric tensor in three dimensions, G = λ1ê1ê T 1 + λ2ê2ê T 2 + λ3ê3ê T 3 The object in the figure is the sum of a spear, a plate and a sphere. The spear describes the principal direction of the tensor λ1ê1ê T 1, where the length is proportional to the largest eigenvalue, λ1. The plate describes the plane spanned by the eigenvectors corresponding to the two largest eigenvalues, λ2(ê1ê T 1 + ê2ê T 2). The sphere, with a radius proportional to the smallest eigenvalue, shows how isotropic the tensor is, λ3(ê1ê T 1 + ê2ê T 2 + ê3ê T 3). The visualization is done using AVS [WWW94]. I am very grateful to Johan Wiklund for implementing the tensor viewer module used. This thesis deals with filtering of multidimensional signals. A large part of the thesis is devoted to a novel filtering method termed “Normalized convolution”. The method performs local expansion of a signal in a chosen filter basis which

In [10] we presented a new velocity estimation algorithm, using orientation tensors and parametric motion models to provide both fast and accurate results. One of the tradeoffs between accuracy and speed was that no attempts were made to obtain regions of coherent motion when estimating the parametric models. In this paper we show how this can be improved by doing a simultaneous segmentation of the motion field. The resulting algorithm is slower than the previous one, but more accurate. This is shown by evaluation on the well-known Yosemite sequence, where already the previous algorithm showed an accuracy which was substantially better than for earlier published methods. This result has now been improved further.

We present a robust image synthesis method to automatically infer missing information from a damaged 2D image by tensor voting. Our method translates image color and texture information into an adaptive ND tensor, followed by a voting process that infers non-iteratively the optimal color values in the ND texture space for each defective pixel. ND tensor voting can be applied to images consisting of roughly homogeneous and periodic textures (e.g. a brick wall), as well as difficult images of natural scenes which contain complex color and texture information. To effectively tackle the latter type of difficult images, a two-step method is proposed. First, we perform texture-based segmentation in the input image, and extrapolate partitioning curves to generate a complete segmentation for the image. Then, missing colors are synthesized using ND tensor voting. Automatic tensor scale analysis is used to adapt to different feature scales inherent in the input. We demonstrate the effectiveness of our approach using a difficult set of real images. 1

This thesis contains a presentation and an analysis of adaptive filtering strategies for multidimensional data. The size, shape and orientation of the filter are signal controlled and thus adapted locally to each neighbourhood according to a predefined model. The filter is constructed as a linear weighting of fixed oriented bandpass filters having the same shape but different orientations. The adaptive filtering methods have been tested on both real data and synthesized test data in 2D, e.g. still images, 3D, e.g. image sequences or volumes, with good results. In 4D, e.g. volume sequences, the algorithm is given in its mathematical form. The weighting coefficients are given by the inner products of a tensor representing the local structure of the data and the tensors representing the orientation of the filters. The procedure and filter design in estimating the representation tensor are described. In 2D, the tensor contains information about the local energy, the optimal orientation and a certainty of the orientation. In 3D, the information in the tensor is the energy, the normal to the best fitting local plane and the tangent to the best fitting line, and certainties of these orientations. In the case of time sequences, a quantitative comparison of the proposed method and other (optical flow) algorithms is presented. The estimation of control information is made in different scales. There are two main reasons for this. A single filter has a particular limited pass band which may or may not be tuned to the different sized objects to describe. Second, size or scale is a descriptive feature in its own right. All of this requires the integration of measurements from different scales. The increasing interest in wavelet theory supports the idea that a multiresolution approach is necessary. Hence the resulting adaptive filter will adapt also in size and to different orientations in different scales.

Motion estimation in image sequences is an important step in many computer vision and image processing applications. Several methods for solving this problem have been proposed, but very few manage to achieve a high level of accuracy without sacrificing processing speed. This paper presents a novel motion estimation algorithm, which gives excellent results on both counts. The algorithm starts by computing 3D orientation tensors from the image sequence. These are combined under the constraints of a parametric motion model to produce velocity estimates. Evaluated on the well-known Yosemite sequence, the algorithm shows an accuracy which is substantially better than for previously published methods. Computationally the algorithm is simple and can be implemented by means of separable convolutions, which also makes it fast. 1 Introduction Motion estimation algorithms always involve a trade-off between speed and accuracy. The method presented here is primarily intended to be accurate but ...

Retrieving images from very large collections using image content as a key is becoming an important problem. Classifying images into visual categories and finding objects in image databases are two major challenges in the field. This paper describes our approach toward the first of the two tasks, the generalization of which we believe will assist in the second task as well. We define a blobworld representation which provides a transition from the raw pixel data to a small set of localized coherent regions in color and texture space. Learning is then utilized to extract a probabilistic interpretation of the scene. Experimental results are presented for more than 1000 images from the Corel photo collection. 1. Introduction Very large collections of images are becoming common, and users have a clear preference for accessing images in these databases based on their content---be it the general image category (e.g., animal scenes, landscapes, urban scenes) or particular objects that are pre...

A theory for detecting general curve families by means of symmetry measurements in the coordinate transformed originals is presented. Symmetries are modeled by iso-gray curves of conjugate harmonic function pairs which also define the coordinate transformations. Harmonic function pair coordinates render the target curve patterns as parallel lines, which is defined here as linear symmetry. Detecting these lines, or generalized linear symmetry fitting as it will be called, corresponds to finding invariants of Lie groups of transformations. A technique based on least square error minimization for estimating the invariance parameters is presented. It uses the Lie infinitesimal operators to construct feature extraction methods that are efficient and simple to implement. The technique, which is shown to be an extension of the generalized Hough transform, enables detection by voting and accumulating evidence for the searched pattern. In this approach complex valued votes are permitted, where ...

Abstract—A robust synthesis method is proposed to automatically infer missing color and texture information from a damaged 2D image by ND tensor voting (N>3). The same approach is generalized to range and 3D data in the presence of occlusion, missing data and noise. Our method translates texture information into an adaptive ND tensor, followed by a voting process that infers noniteratively the optimal color values in the ND texture space. A two-step method is proposed. First, we perform segmentation based on insufficient geometry, color, and texture information in the input, and extrapolate partitioning boundaries by either 2D or 3D tensor voting to generate a complete segmentation for the input. Missing colors are synthesized using ND tensor voting in each segment. Different feature scales in the input are automatically adapted by our tensor scale analysis. Results on a variety of difficult inputs demonstrate the effectiveness of our tensor voting approach. Index Terms—Image restoration, segmentation, color, texture, tensor voting, applications. 1

Abstract. Perceptual experiments indicate that corners and curvature are very important features in the process of recognition. This paper presents a new method to efficiently detect rotational symmetries, which describe complex curvature such as corners, circles, star- and spiral patterns. The method is designed to give selective and sparse responses. It works in three steps; first extract local orientation from a gray-scale or color image, second correlate the orientation image with rotational symmetry filters and third let the filter responses inhibit each other in order to get more selective responses. The correlations can be made efficient by separating the 2D-filters into a small number of 1D-filters. These symmetries can serve as feature points at a high abstraction level for use in hierarchical matching structures for 3D-estimation, object recognition, etc. 1

This thesis presents a signal representation in terms of operators. The signal is assumed to be an element of a vector space and subject to transformations of operators. The operators form continuous groups, so-called Lie groups. The representation can be used for signals in general, in particular if spatial relations are undefined, and it does not require a basis of the signal space to be useful. Special attention is given to orthogonal operator groups which are generated by antiHermitian operators by means of the exponential mapping. It is shown that the eigensystem of the group generator is strongly related to properties of the corresponding operator group. For one-parameter orthogonal operator groups, a phase concept is introduced. This phase can for instance be used to distinguish between spatially even and odd signals and, therefore, corresponds to the usual phase for multi-dimensional signals. Given one operator group that represents the variation of the signal and one operator ...