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K.: Frame potential and finite abelian groups
 Contemp. Math
, 2008
"... Abstract. This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of generators by elem ..."
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Abstract. This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of generators by elements of a fixed subgroup and it is shown, under certain norm conditions, that tight frames with this structure are characterized as local minimizers of the frame potential. Natural analogs to the downsampling and upsampling operators of finite cyclic groups are studied for arbitrary subgroups of finite abelian groups. Directions of further study are also proposed. 1.
Sums of Laplace eigenvalues: rotations and tight frames in higher dimensions
 J. Math. Phys
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GROUP RECONSTRUCTION SYSTEMS
 ELA
, 2011
"... We consider classes of reconstruction systems (RS’s) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS’s), that are associated with representations of finite groups G. These GRS’s generalize frames with high degree of symmetry, such as harmonic or geom ..."
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We consider classes of reconstruction systems (RS’s) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS’s), that are associated with representations of finite groups G. These GRS’s generalize frames with high degree of symmetry, such as harmonic or geometrically uniform ones. Their canonical dual and canonical Parseval are shown to be GRS’s. We establish simple conditions for oneerasure robustness. Projective GRS’s, that can be viewed as fusion frames, are also considered. We characterize the Gram matrix of a GRS in terms of block group matrices. Unitary equivalences and unitary symmetries of RS’s are studied. The relation between the irreducibility of the representation and the tightness of the GRS is established. Taking into account these results, we consider the construction of Parseval, projective and oneerasure robust GRS’s.
TIGHT FRAMES FOR EIGENSPACES OF THE LAPLACIAN ON DUAL POLAR GRAPHS
, 906
"... Abstract. We consider Γ = (X, E) a dual polar graph and we give a tight frame on each eigenspace of the Laplacian operator associated to Γ. We compute the constants associated to each tight frame and as an application we give a formula for the product in the Norton algebra attached to the eigenspace ..."
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Abstract. We consider Γ = (X, E) a dual polar graph and we give a tight frame on each eigenspace of the Laplacian operator associated to Γ. We compute the constants associated to each tight frame and as an application we give a formula for the product in the Norton algebra attached to the eigenspace corresponding to the second largest eigenvalue of the Laplacian. 1.
and Networking
"... Different signal processing transforms provide us with unique decomposition capabilities. Instead of using specific transformation for every type of signal, we propose in this paper a novel way of signal processing using a group of transformations within the limits of Group theory. For different typ ..."
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Different signal processing transforms provide us with unique decomposition capabilities. Instead of using specific transformation for every type of signal, we propose in this paper a novel way of signal processing using a group of transformations within the limits of Group theory. For different types of signal different transformation combinations can be chosen. It is found that it is possible to process a signal at multiresolution and extend it to perform edge detection, denoising, face recognition, etc by filtering the local features. For a finite signal there should be a natural existence of basis in it’s vector space. Without any approximation using Group theory it is seen that one can get close to this finite basis from different viewpoints. Dihedral groups have been demonstrated for this purpose. General Terms signal processing, algorithm, group theory.
GLets: Signal Processing using Transformation Groups
, 2013
"... We present an algorithm using transformation groups and their irreducible representations to generate an orthogonal basis for a signal in the vector space of the signal. It is shown that multiresolution analysis can be done with amplitudes using a transformation group. Glets is thus not a single t ..."
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We present an algorithm using transformation groups and their irreducible representations to generate an orthogonal basis for a signal in the vector space of the signal. It is shown that multiresolution analysis can be done with amplitudes using a transformation group. Glets is thus not a single transform, but a group of linear transformations related by group theory. The algorithm also specifies that a multiresolution and multiscale analysis for each resolution is possible in terms of frequencies. Separation of low and high frequency components of each amplitude resolution is facilitated by Glets. Using conjugacy classes of the transformation group, more than one set of basis may be generated, giving a different perspective of the signal through each basis. Applications for this algorithm include edge detection, feature extraction, denoising, face recognition, compression, and more. We analyze this algorithm using dihedral groups as an example. We demonstrate the results with an ECG signal and the standard ‘Lena ’ image.
FRAME MULTIPLICATION THEORY FOR VECTORVALUED HARMONIC ANALYSIS
, 2014
"... A tight frame Φ is a sequence in a separable Hilbert space H satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for Φ is a binary operat ..."
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A tight frame Φ is a sequence in a separable Hilbert space H satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for Φ is a binary operation on the frame elements that extends to a bilinear vector product on the entire Hilbert space. This is made possible, in part, by the reconstruction property of frames. The motivation for this work is the desire to define meaningful vectorvalued versions of the discrete Fourier transform and the discrete ambiguity function. We make these definitions and prove several familiar harmonic analysis results in this context. These definitions beget the questions we answer through developing frame multiplication theory. For certain binary operations, those with the Latin square property, we give a characterization of the frames, in terms of their Gramians, that have these frame multiplications. Combining finite dimensional representation theory and Naimark’s theorem, we show frames possessing a group frame multiplication are the projections of an orthonormal basis onto the isotypic components of the regular representations. In particular, for a finite group G, we prove there are only finitely many inequivalent frames possessing the group operation of G as a frame multiplication, and we give an explicit formula for the dimensions in which these frames exist. Finally, we connect our theory to a recently studied class of frames; we prove that frames possessing a group frame multiplication are the central Gframes, a class of frames generated by groups of operators on a Hilbert space.