Frame (linear algebra)

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.Template:Sfn 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.Template:Sfn

History

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

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.Template:Sfn 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").Template:Sfn 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 $V$ over a field $F$ and we want to express an arbitrary element $𝐯\in V$ as a linear combination of the vectors $\{{𝐞}_k\}\in V$, that is, finding coefficients $\{c_k\}\in F$ such that

$𝐯=\sum _k{c}_k{𝐞}_k.$

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

Given that $\{{𝐞}_k\}$ spans $V$ 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 $V$ 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 $𝐯$. This means that we want to find the coefficients $c_k$ without removing elements in $\{{𝐞}_k\}$. The coefficients $c_k$ will no longer be uniquely determined by $𝐯$. Therefore, the vector $𝐯$ can be represented as a linear combination of $\{{𝐞}_k\}$ in more than one way.

Definition

Let $V$ be an inner product space and $\{{𝐞}_k{\}}_{k\in \mathbb{N}}$ be a set of vectors in $V$. The set $\{{𝐞}_k{\}}_{k\in \mathbb{N}}$ is a frame of $V$ if it satisfies the so called frame condition. That is, if there exist two constants $0<A\le B<\mathrm{\infty }$ such thatTemplate:Sfn

$A{\Vert 𝐯\Vert }^2\le \sum _{k\in \mathbb{N}}{|⟨𝐯,{𝐞}_k⟩|}^2\le B{\Vert 𝐯\Vert }^2,\mathrm{\forall }𝐯\in V.$

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) $A$ and $B$, respectively.Template:Sfn That is, a frame of $K\ge N$ normalized vectors $\Vert {𝐞}_k\Vert =1$ in an $N$-dimensional space $V$ has frame bounds which satisfiy

$0<A\le \frac{1}{N}\sum _{k=1}^K|⟨{𝐞}_k,{𝐞}_k⟩{|}^2=\frac{K}{N}\le B<\mathrm{\infty }.$

If the frame is a Riesz basis and is therefore linearly independent, then $A\le 1\le B$.

The frame bounds are not unique because numbers less than $A$ and greater than $B$ 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 asTemplate:Sfn

$𝐓:V\to {\ell }^2,𝐯\mapsto 𝐓𝐯=\{⟨𝐯,{𝐞}_{𝐤}⟩{\}}_{k\in \mathbb{N}},$

mapping $𝐯\in V$ to the sequence of frame coefficients $c_k=⟨𝐯,{𝐞}_{𝐤}⟩$, is called the analysis operator. Using this definition, the frame condition can be rewritten as

$A{\Vert 𝐯\Vert }^2\le {\Vert 𝐓𝐯\Vert }^2=\sum _k{|⟨𝐯,{𝐞}_k⟩|}^2\le B{\Vert 𝐯\Vert }^2.$

Synthesis operator

The adjoint of the analysis operator is called the synthesis operator of the frame and defined asTemplate:Sfn

${𝐓}^{*}:{\ell }^2\to V,\{c_k{\}}_{k\in \mathbb{N}}\mapsto \sum _k{c}_k{𝐞}_k.$

Frame operator

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

$𝐒:V\to V,𝐯\mapsto 𝐒𝐯={𝐓}^{*}𝐓𝐯=\sum _k⟨𝐯,{𝐞}_k⟩{𝐞}_k.$

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

$A{\Vert 𝐯\Vert }^2\le {\Vert 𝐓𝐯\Vert }^2=⟨𝐒𝐯,𝐯⟩\le B{\Vert 𝐯\Vert }^2.$

If the analysis operator exists, then so does the frame operator $𝐒$ as well as the inverse ${𝐒}^{-1}$. Both $𝐒$ and ${𝐒}^{-1}$ are positive definite, bounded self-adjoint operators, resulting in $A$ and $B$ being the infimum and supremum values of the spectrum of $𝐒$.Template:Sfn In finite dimensions, the frame operator is automatically trace-class, with $A$ and $B$ corresponding to the smallest and largest eigenvalues of $𝐒$ or, equivalently, the smallest and largest singular values of $𝐓$.Template:Sfn

Relation to bases

The frame condition is a generalization of Parseval's identity that maintains norm equivalence between a signal in $V$ and its sequence of coefficients in ${\ell }^2$.

If the set $\{{𝐞}_k\}$ is a frame of $V$, it spans $V$. Otherwise there would exist at least one non-zero $𝐯\in V$ which would be orthogonal to all ${𝐞}_k$ such that

$A{\Vert 𝐯\Vert }^2\le 0\le B{\Vert 𝐯\Vert }^2;$

either violating the frame condition or the assumption that $𝐯\ne 0$.

However, a spanning set of $V$ is not necessarily a frame. For example, consider $V={ℝ}^2$ with the dot product, and the infinite set $\{{𝐞}_k\}$ given by

$\{(1,0),(0,1),(0,\frac{1}{\sqrt{2}}),(0,\frac{1}{\sqrt{3}}),\dots \}.$

This set spans $V$ but since

$\sum _k{|⟨{𝐞}_k,(0,1)⟩|}^2=0+1+\frac{1}{2}+\frac{1}{3}+\cdots =\mathrm{\infty },$

we cannot choose a finite upper frame bound B. Consequently, the set $\{{𝐞}_k\}$ is not a frame.

Dual frames

Let $\{{𝐞}_k\}$ be a frame; satisfying the frame condition. Then the dual operator is defined as

$\widetilde{𝐓}𝐯=\sum _k⟨𝐯,{\widetilde{𝐞}}_k⟩,$

with

${\widetilde{𝐞}}_k=({𝐓}^{*}𝐓{)}^{-1}{𝐞}_k={𝐒}^{-1}{𝐞}_k,$

called the dual frame (or conjugate frame). It is the canonical dual of $\{{𝐞}_k\}$ (similar to a dual basis of a basis), with the property thatTemplate:Sfn

$𝐯=\sum _k⟨𝐯,{𝐞}_k⟩{\widetilde{𝐞}}_k=\sum _k⟨𝐯,{\widetilde{𝐞}}_k⟩{𝐞}_k,$

and subsequent frame condition

$\frac{1}{B}\Vert 𝐯{\Vert }^2\le \sum _k|⟨𝐯,{\widetilde{𝐞}}_k⟩{|}^2=⟨𝐓{𝐒}^{-1}𝐯,𝐓{𝐒}^{-1}𝐯⟩=⟨{𝐒}^{-1}𝐯,𝐯⟩\le \frac{1}{A}\Vert 𝐯{\Vert }^2,\mathrm{\forall }𝐯\in V.$

Canonical duality is a reciprocity relation, i.e. if the frame $\{{\widetilde{𝐞}}_k\}$ is the canonical dual of $\{{𝐞}_k\},$ then the frame $\{{𝐞}_k\}$ is the canonical dual of $\{{\widetilde{𝐞}}_k\}.$ To see that this makes sense, let $𝐯$ be an element of $V$ and let

$𝐮=\sum _k⟨𝐯,{𝐞}_k⟩{\widetilde{𝐞}}_k.$

Thus

$𝐮=\sum _k⟨𝐯,{𝐞}_k⟩({𝐒}^{-1}{𝐞}_k)={𝐒}^{-1}(\sum _k⟨𝐯,{𝐞}_k⟩{𝐞}_k)={𝐒}^{-1}𝐒𝐯=𝐯,$

proving that

$𝐯=\sum _k⟨𝐯,{𝐞}_k⟩{\widetilde{𝐞}}_k.$

Alternatively, let

$𝐮=\sum _k⟨𝐯,{\widetilde{𝐞}}_k⟩{𝐞}_k.$

Applying the properties of $𝐒$ and its inverse then shows that

$𝐮=\sum _k⟨𝐯,{𝐒}^{-1}{𝐞}_k⟩{𝐞}_k=\sum _k⟨{𝐒}^{-1}𝐯,{𝐞}_k⟩{𝐞}_k=𝐒({𝐒}^{-1}𝐯)=𝐯,$

and therefore

$𝐯=\sum _k⟨𝐯,{\widetilde{𝐞}}_k⟩{𝐞}_k.$

An overcomplete frame $\{{𝐞}_k\}$ allows us some freedom for the choice of coefficients $c_k\ne ⟨𝐯,{\widetilde{𝐞}}_k⟩$ such that $𝐯=\sum _k{c}_k{𝐞}_k$. That is, there exist dual frames $\{{𝐠}_k\}\ne \{{\widetilde{𝐞}}_k\}$ of $\{{𝐞}_k\}$ for which

$𝐯=\sum _k⟨𝐯,{𝐠}_k⟩{𝐞}_k,\mathrm{\forall }𝐯\in V.$

Dual frame synthesis and analysis

Suppose $V$ is a subspace of a Hilbert space $H$ and let $\{{𝐞}_k{\}}_{k\in \mathbb{N}}$ and $\{{\widetilde{𝐞}}_k{\}}_{k\in \mathbb{N}}$ be a frame and dual frame of $V$, respectively. If $\{{𝐞}_k\}$ does not depend on $f\in H$, the dual frame is computed as

${\widetilde{𝐞}}_k=({𝐓}^{*}{𝐓}_V{)}^{-1}{𝐞}_k,$

where ${𝐓}_V$ denotes the restriction of $𝐓$ to $V$ such that ${𝐓}^{*}{𝐓}_V$ is invertible on $V$. The best linear approximation of $f$ in $V$ is then given by the orthogonal projection of $f\in H$ onto $V$, defined as

$P_{V}f=\sum _k⟨f,{𝐞}_k⟩{\widetilde{𝐞}}_k=\sum _k⟨f,{\widetilde{𝐞}}_k⟩{𝐞}_k.$

The dual frame synthesis operator is defined as

$P_{V}f={\widetilde{𝐓}}^{*}𝐓f=({𝐓}^{*}{𝐓}_V{)}^{-1}{𝐓}^{*}𝐓f=\sum _k⟨f,{𝐞}_k⟩{\widetilde{𝐞}}_k,$

and the orthogonal projection is computed from the frame coefficients $⟨f,{𝐞}_k⟩$. In dual analysis, the orthogonal projection is computed from $\{{𝐞}_k\}$ as

$P_{V}f={𝐓}^{*}\widetilde{𝐓}f=\sum _k⟨f,{\widetilde{𝐞}}_k⟩{𝐞}_k$

with dual frame analysis operator $\{\widetilde{𝐓}f{\}}_k=⟨f,{\widetilde{𝐞}}_k⟩$.Template:Sfn

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.Template:Sfn 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

Template:Main From Harmonic analysis it is known that the complex trigonometric system $\{\frac{1}{\sqrt{2\pi }}{e}^{ikx}{\}}_{k\in \mathbb{Z}}$ form an orthonormal basis for $L^2(-\pi ,\pi )$. As such, $\{e^{ikx}{\}}_{k\in \mathbb{Z}}$ is a (tight) frame for $L^2(-\pi ,\pi )$ with bounds $A=B=2\pi$.Template:Sfn

The system remains stable under "sufficiently small" perturbations $\{{\lambda }_k-k\}$ and the frame $\{e^{i{\lambda }_{k}x}{\}}_{k\in \mathbb{Z}}$ will form a Riesz basis for $L^2(-\pi ,\pi )$. Accordingly, every function $f$ in $L^2(-\pi ,\pi )$ will have a unique non-harmonic Fourier series representation

$f(x)=\sum _{k\in \mathbb{Z}}{c}_k{e}^{i{\lambda }_{k}x},$

with $\sum |c_k{|}^2<\mathrm{\infty }$ and $\{e^{i{\lambda }_{k}x}{\}}_{k\in \mathbb{Z}}$ is called the Fourier frame (or frame of exponentials). What constitutes "sufficiently small" is described by the following theorem, named after Mikhail Kadets.Template:Sfn

Template:Math theorem The theorem can be easily extended to frames, replacing the integers by another sequence of real numbers $\{{\mu }_k{\}}_{k\in \mathbb{Z}}$ such thatTemplate:SfnTemplate:Sfn

$|{\lambda }_k-{\mu }_k|\le L<\frac{1}{4},\mathrm{\forall }k\in \mathbb{Z},\text{and}1-\cos (\pi L)+\sin (\pi L)<\sqrt{\frac{A}{B}},$

then $\{e^{i{\lambda }_{k}x}{\}}_{k\in \mathbb{Z}}$ is a frame for $L^2(-\pi ,\pi )$ with bounds

$A(1-\sqrt{\frac{B}{A}}(1-\cos (\pi L)+\sin (\pi L)){)}^2,B(2-\cos (\pi L)+\sin (\pi L){)}^2.$

Frame projector

Template:Distinguish Redundancy of a frame is useful in mitigating added noise from the frame coefficients. Let $𝐚\in {\ell }^2(\mathbb{N})$ denote a vector computed with noisy frame coefficients. The noise is then mitigated by projecting $𝐚$ onto the image of $𝐓$.

Template:Math theorem The ${\ell }^2$ sequence space and $\mathrm{im}(𝐓)$ (as $\mathrm{im}(𝐓)\subseteq {\ell }^2$) are reproducing kernel Hilbert spaces with a kernel given by the matrix $M_{k,p}=⟨{𝐒}^{-1}{𝐞}_p,{𝐞}_k⟩$.Template:Sfn As such, the above equation is also referred to as the reproducing kernel equation and expresses the redundancy of frame coefficients.Template:Sfn

Special cases Template:Anchor

Tight frames

A frame is a tight frame if $A=B$. A tight frame $\{{𝐞}_k{\}}_{k=1}^{\mathrm{\infty }}$ with frame bound $A$ has the property that

$𝐯=\frac{1}{A}\sum _k⟨𝐯,{𝐞}_k⟩{𝐞}_k,\mathrm{\forall }𝐯\in V.$

For example, the union of $k$ disjoint orthonormal bases of a vector space is an overcomplete tight frame with $A=B=k$. A tight frame is a Parseval frame if $A=B=1$.Template:Sfn Each orthonormal basis is a (complete) Parseval frame, but the converse is not necessarily true.Template:Sfn

Equal norm frame

A frame is an equal norm frame if there is a constant $c$ such that $\Vert {𝐞}_k\Vert =c$ for each $k$. An equal norm frame is a normalized frame (sometimes called a unit-norm frame) if $c=1$.Template:Sfn 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 $c$ such that $|⟨{𝐞}_i,{𝐞}_j⟩|=c$ for all $i\ne j$. In particular, every orthonormal basis is equiangular.Template:Sfn

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).

Generalizations

Semi-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.Template:Sfn The Bessel Sequence is an example of a set of vectors that satisfies only the upper frame inequality.

For any vector $𝐯\in V$ to be reconstructed from the coefficients $\{⟨𝐯,{𝐞}_k⟩{\}}_{k\in \mathbb{N}}$ it suffices if there exists a constant $A>0$ such that

$A\Vert x-y{\Vert }^2\le \Vert Tx-Ty{\Vert }^2,\mathrm{\forall }x,y\in V.$

By setting $𝐯=x-y$ and applying the linearity of the analysis operator, this condition is equivalent to:

$A\Vert 𝐯{\Vert }^2\le \Vert T𝐯{\Vert }^2,\mathrm{\forall }𝐯\in V,$

which is exactly the lower frame bound condition.

Fusion frame

Template:Main article A fusion frame is best understood as an extension of the dual frame synthesis and analysis operators where, instead of a single subspace $V\subseteq H$, a set of closed subspaces $\{W_i{\}}_{i\in \mathbb{N}}\subseteq H$ with positive scalar weights $\{w_i{\}}_{i\in \mathbb{N}}$ is considered. A fusion frame is a family $\{W_i,w_i{\}}_{i\in \mathbb{N}}$ that satisfies the frame condition

$A\Vert f{\Vert }^2\le \sum _i{w}_i^2\Vert P_{W_i}f{\Vert }^2\le B\Vert f{\Vert }^2,\mathrm{\forall }f\in H,$

where $P_{W_i}$ denotes the orthogonal projection onto the subspace $W_i$.Template:Sfn

Continuous frame

Suppose $H$ is a Hilbert space, $X$ a locally compact space, and $\mu$ is a locally finite Borel measure on $X$. Then a set of vectors in $H$, $\{f_x{\}}_{x\in X}$ with a measure $\mu$ is said to be a continuous frame if there exists constants, $0<A\le B$ such that

$A||f|{|}^2\le \underset{X}{\int }|⟨f,f_x⟩{|}^{2}d\mu (x)\le B||f|{|}^2,\mathrm{\forall }f\in H.$

To see that continuous frames are indeed the natural generalization of the frames mentioned above, consider a discrete set $\mathrm{\Lambda }\subset X$ and a measure $\mu ={\delta }_{\mathrm{\Lambda }}$ where ${\delta }_{\mathrm{\Lambda }}$ is the Dirac measure. Then the continuous frame condition reduces to

$A||f|{|}^2\le \sum _{\lambda \in \mathrm{\Lambda }}|⟨f,f_{\lambda }⟩{|}^2\le B||f|{|}^2,\mathrm{\forall }f\in H.$

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 $\{f_x{\}}_{x\in X}$ the continuous analysis operator is the operator mapping $f$ to a function on $X$ defined as follows:

$T:H\to L^2(X,\mu )$ by $f\mapsto ⟨f,f_x{⟩}_{x\in X}$.

Continuous synthesis operator

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

$T^{*}:L^2(X,\mu )\to H$ by $a_x\mapsto \underset{X}{\int }{a}_x{f}_{x}d\mu (x)$.

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 $\{f_x{\}}_{x\in X}$, it is defined as follows:

$S:H\to H$ by $Sf:=\underset{X}{\int }⟨f,f_x⟩f_{x}d\mu (x).$

In this case, the continuous frame projector $P:L^2(x,\mu )\to \mathrm{im}(T)$ is the orthogonal projection defined by

$P:=T{S}^{-1}{T}^{*}.$

The projector $P$ is an integral operator with reproducting kernel $K(x,y)=⟨S^{-1}{f}_x,f_y⟩$, thus $\mathrm{im}(T)$ is a reproducing kernel Hilbert space.Template:Sfn

Continuous dual frame

Given a continuous frame $\{f_x{\}}_{x\in X}$, and another continuous frame $\{g_x{\}}_{x\in X}$, then $\{g_x{\}}_{x\in X}$ is said to be a continuous dual frame of $\{f_x\}$ if it satisfies the following condition for all $f,h\in H$:

$⟨f,h⟩=\underset{X}{\int }⟨f,f_x⟩⟨g_x,h⟩d\mu (x).$

Framed positive operator-valued measure

Template:Main article 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 $(X,M)$ is a measurable space with $M$ a Borel σ-algebra on $X$ and let $F$ be a POVM from $M$ to the space of positive operators on $H$ with the additional property that

$0<AI\le F(M)\le BI<\mathrm{\infty },$

where $I$ is the identity operator. Then $F$ is called a framed POVM.Template:Sfn

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

$F(m)=\sum _{i\in m}{w}_i{P}_{W_i},m\in M.$

For the continuous frame operator, the framed POVM would beTemplate:Sfn

$⟨F(M)f_x,f_y⟩=\underset{M}{\int }⟨S{f}_x,f_y⟩d\mu (x).$

See also

• k-frame
• Biorthogonal wavelet
• Orthogonal wavelet
• Restricted isometry property
• Schauder basis
• Harmonic analysis
• Fourier analysis
• Functional analysis

References

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