Constructing invariant features by averaging techniques

Abstract

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 } .

Incorporating prior knowledge in support vector machines for classification: A review

“…Haar-integration has been introduced in [54] for the construction of invariant features. In a similar approach, Haar-integration has been used to generate invariant kernels known as Haar-integration kernels (HI-kernels) [24].…”

“…In a similar approach, Haar-integration has been used to generate invariant kernels known as Haar-integration kernels (HI-kernels) [24]. Consider a standard kernel k 0 and a transformation group T which contains the admissible transformations (see [54] for a complete definition). The idea is to compute the average of the kernel output k 0 (T x, T ′ z) over all pairwise combinations of the transformed samples (T x, T ′ z) , ∀T, T ′ ∈ T .…”

Adjustable invariant features by partial Haar-integration

“…We assume £ to be a group of transformations ¤ operating on patterns PSfrag replacements [10]. Using simple functions % , the resulting features can be computed efficiently.…”

Invariant kernel functions for pattern analysis and machine learning

“…(Nachbin 1965). Based on this, invariant feature representations of patterns are generated by integrating simple non-invariant functions h over the known transformation group (Schulz-Mirbach 1994). This results in Haar-integral invariants defined as…”

“…These are invariant functions, which can be derived by an averaging procedure (Schulz-Mirbach 1994. In this approach, the transformations T are assumed to be structured as a group G with invariant measure dg, the so called Haar-measure, which exists for locally compact topological groups and finite groups, cf.…”