Haag duality is established in conformal quantum field theory for observable fields on the compactified light ray S1 and Minkowski space S1×S1, respectively. This result provides the foundation for an algebraic approach to the classification of conformal theories. Haag duality can fail, however, for the restriction of conformal fields to the underlying non-compact spaces ℝ, respectively ℝ×ℝ. A prominent example is the stress energy tensor with central charge c>1.
This paper presents a general and model independent analysis of the problem of feature extraction in pattern recognition. We will derive two criteria which ensure the existence of a complete feature space. This is a space which contains exactly the information relevant for the classification process following feature extraction. We discuss several possibilities for the construction of such a complete feature space and present experimental results which indicate the potential of the proposed methods for practical applications.
The paper presents algorithms for the construction of features which are invariant with respect to a given transformation group. The methods are based on integral calculus and are applicable to parametric groups (Lie groups) and finite groups as well. To illustrate the concepts we discuss in detail how to construct invariant features for the general linear group which is of importance in computer vision applications. W e briefly mention how to combine the algorithms with Hilbert space techniques for the description of continuous signals. Finally we apply the presented methods to establish the existence of complete feature sets for arbitrary compact groups.
Invariant FeaturesIn this section we introduce the basic concepts and fix our notations. The signal space S is a subset of a finite dimensional complex vector space V. We call the elements of S patterns and denote them by vectors, e.g. GIG.. G is a group acting on V ; i.e. for every g E G exists an operator 7 ( g ) : V -, V and for these operators the following composition law is valid 7 ( 9 1 ) 1 ( 9 2 ) = 7 ( g l g z ) V g i , ~2 E G.(1) Note that g l g z is the group product in G whereas the left-hand side of (1) denotes the product of two operators. The group G is called transformation group. Furthermore we assume that the signal space S is stable under the action of the group G; i.e. 'T(g)V' E S V g E G ,~' E S. The action of G introduces an equivalence relationin S. Two patterns GI 2u' are called equivalent, v' N d, if a g E G exists with v' = 7 ( g ) G . The equivalence classes of this equivalence relation are called orbits of G in S. An orbit O(v') is a subset of S of the form O(3) = { 7 ( g ) v ' I g E G } .
The application of measuring techniques in practice tend to incorporate more and more complex sensors associated with the appropriate signal processing. Typical examples are problems in Visual inspection using a video camera as image sensor and applying pattern recognition methods. The recognition of patterns or objects independent of the viewing angle belongs to one of the fundamental tasks which are not yet sufficiently investigated. The paper gives some results for the construction, the existence and theproperties of invariants for the position-independent recognition of planar contour and gray-scale images. We consider modifications of the position caused by planar and spatial motion of rigid objects (Euclidean motion) with Orthographie projection into the camera plane which define corresponding equivalence classes. Dijferent Performance measures of the invariant features lilce completeness, robustness as well as the matter of computational complexity are discussed.
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