Jump to content

Frame (linear algebra)

From Wikipedia, the free encyclopedia

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. In the terminology of signal processing, a frame provides a redundant, stable way of representing a signal.[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.[2]



Because of the various mathematical components surrounding frames, frame theory has roots in harmonic and functional analysis, operator theory, linear algebra, and matrix theory.[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.

Definition and motivation


Motivating example: computing a basis from a linearly dependent set


Suppose we have a vector space over a field and we want to express an arbitrary element as a linear combination of the vectors , that is, finding coefficients such that

If the set does not span , then such coefficients do not exist for every such . If spans and also is linearly independent, this set forms a basis of , and the coefficients are uniquely determined by . If, however, spans but is not linearly independent, the question of how to determine the coefficients becomes less apparent, in particular if is of infinite dimension.

Given that spans and is linearly dependent, one strategy is to remove vectors from the set until it becomes linearly independent and forms a basis. There are some problems with this plan:

  1. Removing arbitrary vectors from the set may cause it to be unable to span before it becomes linearly independent.
  2. Even if it is possible to devise a specific way to remove vectors from the set until it becomes a basis, this approach may become unfeasible in practice if the set is large or infinite.
  3. In some applications, it may be an advantage to use more vectors than necessary to represent . This means that we want to find the coefficients without removing elements in . The coefficients will no longer be uniquely determined by . Therefore, the vector can be represented as a linear combination of in more than one way.



Let be an inner product space and be a set of vectors in . The set is a frame of if it satisfies the so called frame condition. That is, if there exist two constants such that[5]

A frame is called overcomplete (or redundant) if it is not a Riesz basis for the vector space. The redundancy of the frame is measured by the lower and upper frame bounds (or redundancy factors) and , respectively.[6] That is, a frame of normalized vectors in an -dimensional space has frame bounds which satisfiy

If the frame is a Riesz basis and is therefore linearly independent, then .

The frame bounds are not unique because numbers less than and greater than are also valid frame bounds. The optimal lower bound is the supremum of all lower bounds and the optimal upper bound is the infimum of all upper bounds.

Analysis operator


If the frame condition is satisfied, then the linear operator defined as[7]

mapping to the sequence of frame coefficients , is called the analysis operator. Using this definition, the frame condition can be rewritten as

Synthesis operator


The adjoint of the analysis operator is called the synthesis operator of the frame and defined as[8]

Frame operator


The composition of the analysis operator and the synthesis operator leads to the frame operator defined as

From this definition and linearity in the first argument of the inner product, the frame condition now yields

If the analysis operator exists, then so does the frame operator as well as the inverse . Both and are positive definite, bounded self-adjoint operators, resulting in and being the infimum and supremum values of the spectrum of .[9] In finite dimensions, the frame operator is automatically trace-class, with and corresponding to the smallest and largest eigenvalues of or, equivalently, the smallest and largest singular values of .[10]

Relation to bases


The frame condition is a generalization of Parseval's identity that maintains norm equivalence between a signal in and its sequence of coefficients in .

If the set is a frame of , it spans . Otherwise there would exist at least one non-zero which would be orthogonal to all such that

either violating the frame condition or the assumption that .

However, a spanning set of is not necessarily a frame. For example, consider with the dot product, and the infinite set given by

This set spans but since

we cannot choose a finite upper frame bound B. Consequently, the set is not a frame.

Dual frames


Let be a frame; satisfying the frame condition. Then the dual operator is defined as


called the dual frame (or conjugate frame). It is the canonical dual of (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 is the canonical dual of then the frame is the canonical dual of To see that this makes sense, let be an element of and let


proving that

Alternatively, let

Applying the properties of and its inverse then shows that

and therefore

An overcomplete frame allows us some freedom for the choice of coefficients such that . That is, there exist dual frames of for which

Dual frame synthesis and analysis


Suppose is a subspace of a Hilbert space and let and be a frame and dual frame of , respectively. If does not depend on , the dual frame is computed as

where denotes the restriction of to such that is invertible on . The best linear approximation of in is then given by the orthogonal projection of onto , defined as

The dual frame synthesis operator is defined as

and the orthogonal projection is computed from the frame coefficients . In dual analysis, the orthogonal projection is computed from as

with dual frame analysis operator .[12]

Applications and examples


In signal processing, it is common to represent signals as vectors in a Hilbert space. In this interpretation, a vector expressed as a linear combination of the frame vectors is a redundant signal. Representing a signal strictly with a set of linearly independent vectors may not always be the most compact form.[13] Using a frame, it is possible to create a simpler, more sparse representation of a signal as compared with a family of elementary signals. Frames, therefore, provide "robustness". Because they provide a way of producing the same vector within a space, signals can be encoded in various ways. 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.

Non-harmonic Fourier series


From Harmonic analysis it is known that the complex trigonometric system form an orthonormal basis for . As such, is a (tight) frame for with bounds .[14]

The system remains stable under "sufficiently small" perturbations and the frame will form a Riesz basis for . Accordingly, every function in will have a unique non-harmonic Fourier series representation

with and is called the Fourier frame (or frame of exponentials). What constitutes "sufficiently small" is described by the following theorem, named after Mikhail Kadets.[15]

Kadec's 14-theorem — Let be a sequence of real numbers such that

then 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 such that[16][17]

then is a frame for with bounds

Frame projector


Redundancy of a frame is useful in mitigating added noise from the frame coefficients. Let denote a vector computed with noisy frame coefficients. The noise is then mitigated by projecting onto the image of .

Theorem — Let be a frame of a Hilbert space of subspace thereof. The orthogonal projection is

The coefficients are frame coefficients in if and only if

The sequence space and (as ) are reproducing kernel Hilbert spaces with a kernel given by the matrix .[9] As such, the above equation is also referred to as the reproducing kernel equation and expresses the redundancy of frame coefficients.[18]

Special cases


Tight frames


A frame is a tight frame if . A tight frame with frame bound has the property that

For example, the union of disjoint orthonormal bases of a vector space is an overcomplete tight frame with . A tight frame is a Parseval frame if .[19] Each orthonormal basis is a (complete) Parseval frame, but the converse is not necessarily true.[20]

Equal norm frame


A frame is an equal norm frame if there is a constant such that for each . An equal norm frame is a normalized frame (sometimes called a unit-norm frame) if .[21] A unit-norm Parseval frame is an orthonormal basis; such a frame satisfies Parseval's identity.

Equiangular frames


A frame is an equiangular frame if there is a constant such that for all . In particular, every orthonormal basis is equiangular.[22]

Exact frames


A frame is an exact frame if no proper subset of the frame spans the inner product space. Each basis for an inner product space is an exact frame for the space (so a basis is a special case of a frame).





Sometimes it may not be possible to satisfy both frame bounds simultaneously. An upper (respectively lower) semi-frame is a set that only satisfies the upper (respectively lower) frame inequality.[9] The Bessel Sequence is an example of a set of vectors that satisfies only the upper frame inequality.

For any vector to be reconstructed from the coefficients it suffices if there exists a constant such that

By setting and applying the linearity of the analysis operator, this condition is equivalent to:

which is exactly the lower frame bound condition.

Fusion frame


A fusion frame is best understood as an extension of the dual frame synthesis and analysis operators where, instead of a single subspace , a set of closed subspaces with positive scalar weights is considered. A fusion frame is a family that satisfies the frame condition

where denotes the orthogonal projection onto the subspace .[23]

Continuous frame


Suppose is a Hilbert space, a locally compact space, and is a locally finite Borel measure on . Then a set of vectors in , with a measure is said to be a continuous frame if there exists constants, such that

To see that continuous frames are indeed the natural generalization of the frames mentioned above, consider a discrete set and a measure where is the Dirac measure. Then the continuous frame condition reduces to

Just like in the discrete case we can define the analysis, synthesis, and frame operators when dealing with continuous frames.

Continuous analysis operator


Given a continuous frame the continuous analysis operator is the operator mapping to a function on defined as follows:

by .

Continuous synthesis operator


The adjoint operator of the continuous analysis operator is the continuous synthesis operator, which is the map

by .

Continuous frame operator


The composition of the continuous analysis operator and the continuous synthesis operator is known as the continuous frame operator. For a continuous frame , it is defined as follows:


In this case, the continuous frame projector is the orthogonal projection defined by

The projector is an integral operator with reproducting kernel , thus is a reproducing kernel Hilbert space.[9]

Continuous dual frame


Given a continuous frame , and another continuous frame , then is said to be a continuous dual frame of if it satisfies the following condition for all :

Framed positive operator-valued measure


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.

Suppose is a measurable space with a Borel σ-algebra on and let be a POVM from to the space of positive operators on with the additional property that

where is the identity operator. Then is called a framed POVM.[23]

In case of the fusion frame condition, this allows for the substitution

For the continuous frame operator, the framed POVM would be[24]

See also





  • Antoine, J.-P.; Balazs, P. (2012). "Frames, Semi-Frames, and Hilbert Scales". Numerical Functional Analysis and Optimization. 33 (7–9): 736–769. arXiv:1203.0506. doi:10.1080/01630563.2012.682128. ISSN 0163-0563.
  • Casazza, Peter; Kutyniok, Gitta; Philipp, Friedrich (2013). "Introduction to Finite Frame Theory". Finite Frames: Theory and Applications. Berlin: Birkhäuser. pp. 1–53. ISBN 978-0-8176-8372-6.
  • Christensen, Ole (2016). "An Introduction to Frames and Riesz Bases". Applied and Numerical Harmonic Analysis. Cham: Springer International Publishing. doi:10.1007/978-3-319-25613-9. ISBN 978-3-319-25611-5. ISSN 2296-5009.
  • Duffin, Richard James; Schaeffer, Albert Charles (1952). "A class of nonharmonic Fourier series". Transactions of the American Mathematical Society. 72 (2): 341–366. doi:10.2307/1990760. JSTOR 1990760. MR 0047179.
  • Kovačević, Jelena; Chebira, Amina (2008). "An Introduction to Frames" (PDF). Foundations and Trends in Signal Processing. 2 (1): 1–94. doi:10.1561/2000000006.
  • Kovacevic, Jelena; Dragotti, Pier Luigi; Goyal, Vivek (2002). "Filter Bank Frame Expansions with Erasures" (PDF). IEEE Transactions on Information Theory. 48 (6): 1439–1450. CiteSeerX doi:10.1109/TIT.2002.1003832.
  • Mallat, Stéphane (2009). A wavelet tour of signal processing: the sparse way. Amsterdam Boston: Elsevier/Academic Press. ISBN 978-0-12-374370-1.
  • Moran, Bill; Howard, Stephen; Cochran, Doug (2013). "Positive-Operator-Valued Measures: A General Setting for Frames". Excursions in Harmonic Analysis, Volume 2. Boston: Birkhäuser Boston. doi:10.1007/978-0-8176-8379-5_4. ISBN 978-0-8176-8378-8.
  • Robinson, Benjamin; Moran, Bill; Cochran, Doug (2021). "Positive operator-valued measures and densely defined operator-valued frames". Rocky Mountain Journal of Mathematics. 51 (1). arXiv:2004.11729. doi:10.1216/rmj.2021.51.265. ISSN 0035-7596.
  • Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93. Academic Press. ISBN 978-0-12-772955-8.