In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent.
[1] Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics, computer science, and engineering.
[3] The Fourier transform has been used for over a century as a way of decomposing and expanding signals.
However, the Fourier transform masks key information regarding the moment of emission and the duration of a signal.
In 1946, Dennis Gabor was able to solve this using a technique that simultaneously reduced noise, provided resiliency, and created quantization while encapsulating important signal characteristics.
[1] This discovery marked the first concerted effort towards frame theory.
The frame condition was first described by Richard Duffin and Albert Charles Schaeffer in a 1952 article on nonharmonic Fourier series as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set (in their terminology, a "Hilbert space frame").
[4] In the 1980s, Stéphane Mallat, Ingrid Daubechies, and Yves Meyer used frames to analyze wavelets.
Today frames are associated with wavelets, signal and image processing, and data compression.
but is not linearly independent, the question of how to determine the coefficients becomes less apparent, in particular if
such that[5] A frame is called overcomplete (or redundant) if it is not a Riesz basis for the vector space.
If the frame condition is satisfied, then the linear operator defined as[7] mapping
are positive definite, bounded self-adjoint operators, resulting in
[9] In finite dimensions, the frame operator is automatically trace-class, with
[10] The frame condition is a generalization of Parseval's identity that maintains norm equivalence between a signal in
but since we cannot choose a finite upper frame bound B. Consequently, the set
(similar to a dual basis of a basis), with the property that[11] and subsequent frame condition Canonical duality is a reciprocity relation, i.e. if the frame
In dual analysis, the orthogonal projection is computed from
Representing a signal strictly with a set of linearly independent vectors may not always be the most compact form.
This facilitates fault tolerance and resilience to a loss of signal.
Finally, redundancy can be used to mitigate noise, which is relevant to the restoration, enhancement, and reconstruction of signals.
[14] The system remains stable under "sufficiently small" perturbations
What constitutes "sufficiently small" is described by the following theorem, named after Mikhail Kadets.
satisfies the Paley-Wiener criterion and thus forms a Riesz basis for
The theorem can be easily extended to frames, replacing the integers by another sequence of real numbers
denote a vector computed with noisy frame coefficients.
disjoint orthonormal bases of a vector space is an overcomplete tight frame with
[19] Each orthonormal basis is a (complete) Parseval frame, but the converse is not necessarily true.
[9] The Bessel Sequence is an example of a set of vectors that satisfies only the upper frame inequality.
: Just as a frame is a natural generalization of a basis to sets that may be linear dependent, a positive operator-valued measure (POVM) is a natural generalization of a projection-valued measure (PVM) in that elements of a POVM are not necessarily orthogonal projections.