Loosen Up! An introduction to frames.faculty.nps.edu/rgera/organized conferences/JointMeetings... · 2011-01-07 · Frames Vectors and Vector Spaces Orthonormal Bases Frames 4 (Interrelated)

Frames

Vectors and VectorSpaces

OrthonormalBases

Frames

4 (Interrelated)Research AreasApplied Math

Linear Algebra

Geometry

Operator Theory

Frames forUndergraduates

Loosen Up! An introduction to frames.

Keri A. Kornelson

University of Oklahoma - [email protected]

Joint AMS/MAA MeetingsPanel: This could be YOUR graduate research!

New Orleans, LAJanuary 7, 2011

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Quick review of vector spaces, Rn:

Vectors in Rn are sometimes represented as columns:

x =

x(1)x(2)

...x(n)

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Quick review of vector spaces, Rn:

Vectors in Rn are sometimes represented as columns:

x =

x(1)x(2)

...x(n)

and sometimes as arrows:

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Norms and Dot Products2 vectors

x =

x(1)x(2)

...x(n)

y =

y(1)y(2)

...y(n)

DefinitionThe norm or length of x is

‖x‖ =

(

n∑

i=1

x(i)2

)12

.

A vector with norm 1 is called a unit vector .

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Norms and Dot Products2 vectors

x =

x(1)x(2)

...x(n)

y =

y(1)y(2)

...y(n)

DefinitionThe norm or length of x is

‖x‖ =

(

n∑

i=1

x(i)2

)12

.

A vector with norm 1 is called a unit vector .

DefinitionThe dot product or inner product of x and y is

〈x , y〉 =n∑

i=1

x(i)y(i).

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Orthogonal Vectors

DefinitionTwo vectors are orthogonal if their inner (dot) product is zero,i.e. if their “arrows” are perpendicular.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Orthogonal Vectors

DefinitionTwo vectors are orthogonal if their inner (dot) product is zero,i.e. if their “arrows” are perpendicular.

DefinitionTwo vectors are orthonormal if they are orthogonal and areboth unit vectors.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Orthonormal Bases

DefinitionA collection of vectors {bi}

ni=1 is an orthonormal basis (ONB)

for Rn if the vectors are pairwise orthonormal and form a basis.

Some handy facts about ONBs:

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Orthonormal Bases





I The unique expansion coefficients are found by the dotproduct.

x =

n∑

i=1

cibi =

n∑

i=1

〈x , bi〉bi

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Orthonormal Bases





I The unique expansion coefficients are found by the dotproduct.

x =

n∑

i=1

cibi =

n∑

i=1

〈x , bi〉bi

I Parseval’s Identity:

‖x‖2 =

n∑

i=1

|〈x , bi〉|2

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


A Silly Example

b1 =

100

b2 =

010

b3 =

001

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


A Silly Example

b1 =

100

b2 =

010

b3 =

001

x =

456

= 4b1 + 5b2 + 6b3

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


A Silly Example

b1 =

100

b2 =

010

b3 =

001

x =

456

= 4b1 + 5b2 + 6b3

〈x , b1〉 = 4, 〈x , b2〉 = 5, 〈x , b3〉 = 6.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


A Silly Example

b1 =

100

b2 =

010

b3 =

001

x =

456

= 4b1 + 5b2 + 6b3

〈x , b1〉 = 4, 〈x , b2〉 = 5, 〈x , b3〉 = 6.

‖x‖2 =√

42 + 52 + 62

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Orthonormal Bases, cont.

ONB’s are pretty restrictive...they all look alike somehow.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I All vectors have norm 1.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory





I The number of vectors n equals the dimension of thespace: Rn.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory






I All the vectors are pairwise orthogonal.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory







Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory







I There’s not much flexibility to tailor an ONB to a particularapplication, and there is no resilience to losses or errors indata reconstruction.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Signal transmission

If we agree on an ONB {bi}ni=1, then I can just send you the

coefficients of x and you can find x .

In our silly example, I send you 4, 5, 6 and you can compute

x = 4b1 + 5b2 + 6b3 =

456

.

Same idea works for voice on a cell phone or pictures sentover the internet.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Signal transmission

If we agree on an ONB {bi}ni=1, then I can just send you the

coefficients of x and you can find x .

In our silly example, I send you 4, 5, 6 and you can compute

x = 4b1 + 5b2 + 6b3 =

456

.

Same idea works for voice on a cell phone or pictures sentover the internet.

If one data point gets lost using an ONB, there is noinformation about what it was.

4, ?, 6 −→ 4b1 + ?b2 + 6b3 =

4?

6

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Loosen up!The concept of a frame for a vector space allows for morewiggle room than ONBs. The vectors are allowed to

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory



Stretch out

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory



Stretch out

Move around a bit

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory



Stretch out

Move around a bit

Even invite a few friends over!

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Frames

Recall Parseval’s Identity for ONB {bi}: ‖x‖2 =n∑

i=1

|〈x , bi〉|2

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Frames


i=1

|〈x , bi〉|2

DefinitionA frame for Rn is a collection of vectors {fi}k

i=1 that satisfy alooser condition than Parseval’s identity. There are constantsA,B > 0 (called frame bounds ) such that

A‖x‖2 ≤

k∑

i=1

|〈x , fi〉|2 ≤ B‖x‖2.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Frames


i=1

|〈x , bi〉|2



A‖x‖2 ≤

k∑

i=1

|〈x , fi〉|2 ≤ B‖x‖2.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Frames


i=1

|〈x , bi〉|2



A‖x‖2 ≤

k∑

i=1

|〈x , fi〉|2 ≤ B‖x‖2.

A frame is tight if A = B and Parseval if A = B = 1.

‖x‖2 =

k∑

i=1

|〈x , fi〉|2 (look familiar?)

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Handy facts about frames:

I In finite-dimensional spaces, the frames are exactly thespanning sets for the space.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I Every ONB is a Parseval frame, but there are more!

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I Every ONB is a Parseval frame, but there are more!

I Parseval frames also satisfy the reconstruction property ofONBs:

x =

k∑

i=1

ci fi =k∑

i=1

〈x , fi〉fi

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


ProjectionsTheoremEvery frame is the projection of a basis for a larger space.Every Parseval frame is the projection of a orthonormal basisfor a larger space.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


ProjectionsTheoremEvery frame is the projection of a basis for a larger space.Every Parseval frame is the projection of a orthonormal basisfor a larger space.

ExampleR

3 orthonormal basis projected onto the plane.

yields the Parseval frame with 3 equal-norm vectors for R2.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


4 frame-related research areas

1. Applied Math

2. Linear Algebra

3. Geometry

4. Operator Theory

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Sample Applied Math Research Problems

I Which frames have the best resilience to 1, 2, or moreerasures?

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I Build tight frames which are tailored to a particularapplication.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory





I Build the sparsest possible tight frame of givensize/redundancy.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory






I Find an algorithm like Gram-Schmidt that generates tightframes from a given frame sequence.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory







I Find an algorithm that numerically converges to a tightframe under given constraints (same norms, for example).

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory







I Find an algorithm that numerically converges to a tightframe under given constraints (same norms, for example).

I Wavelet frames.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Synthesis operator, frame potential

Let {fi}ki=1 be a frame for Rn. We can create the n × k matrix S

which has the frame vectors as columns.

↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓

Theorem

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory





↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓

Theorem

I The frame {fi}ki=1 is Parseval if and only if the rows of S

are orthonormal.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory





↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓

Theorem

I The frame {fi}ki=1 is Parseval if and only if the rows of S

are orthonormal.

I {fi}ki=1 is Parseval iff SS∗ is the identity matrix on R

n.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Sample Linear Algebra Research Problems

I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I Find a tight frame with k vectors for Rn, where the vectorsare all lie on an ellipsoid.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory





I Find a tight frame with k vectors for Rn, where the vectorshave a given sequence of norms.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory






I These all relate to classical problems about writingoperators as sums of projections!

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory






I These all relate to classical problems about writingoperators as sums of projections!

I Frame potential - a real quantity that is minimized at tightframes, simulating electromagnetic potential.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


R2 frames

Another way to think about a vector in R2:[

a cos θa sin θ

]

Theorem

A frame{[

ai cos θi

ai sin θi

]}k

i=1is a tight frame for R2 if and only if

k∑

i=1

[

a2i cos 2θi

a2i sin 2θi

]

=

[

00

]

.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


R2 frames

Another way to think about a vector in R2:[

a cos θa sin θ

]

Theorem

A frame{[

ai cos θi

ai sin θi

]}k

i=1is a tight frame for R2 if and only if

k∑

i=1

[

a2i cos 2θi

a2i sin 2θi

]

=

[

00

]

.

Proof.Recall S is the 2 × k matrix with the frame vectors as columns,and the frame is tight iff SS∗ is a scalar multiple of the identity.Computing S and using some trigonometric identities gives theresult.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Sample Geometric Research Problems

I The theorem yields lots of facts about R2 tight frames —for example All 4-vector unit frames consist of 2 ONBs.

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I Is there a similar kind of characterization for tight framesin 3 or 4 dimensions?

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I Is there a similar kind of characterization for tight framesin 3 or 4 dimensions?

I Find/characterize equiangular equal-norm tight frames(related to packing problems).

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Kadison-Singer Problem (1951)

The Kadison-Singer problem in operator theory has been opensince 1951.

It has recently been shown equivalent to a variety of problemshaving to do with finite frames and finite matrices.

ExampleDoes there exist an ε > 0 and a natural number r such that forall equal-norm Parseval frames {fi}2n

i=1 for Rn , there is apartition {Aj}

rj=1 of {1, 2, . . . , 2n} such that {fi}i∈Aj has Bessel

bound ≤ 1 − ε for all j = 1, 2, . . . , r .

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


For further reading...

Frames for UndergraduatesDeguang Han, University of Central Florida, Orlando, FL, Keri Kornelson, Grinnell College, IA, David Larson, Texas A&M University, College Station, TX, and Eric Weber, Iowa State University, Ames, IA

Student Mathematical Library

2007; 295 pp; softcover

Volume: 40

ISBN: 978-0-8218-4212-6

List Price: US$49

Member Price: US$39

Order Code: STML/40

Frames are a generalization of bases.

�eir study has a powerful impact in both

abstract and applied settings. �is book

provides an undergraduate-level introduc-

tion to the theory of frames, primarily in

finite-dimensional Hilbert spaces.

Instructional Venues:

• A special topics course about

frames and bases.

• A second linear algebra course.

• A resource for an undergraduate

research activity.

AMERICAN MATHEMATICAL SOCIETY

For many more publications of interest,

visit the AMS Bookstore

www.ams.org/bookstore

1-800-321-4AMS (4267), in the U. S. and Canada, or 1-401-455-4000 (worldwide); fax:1-401-455-4046; email:

[email protected]. American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory


Where you might go to study frames.... . . in no particular order:

I University of Oklahoma

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory




I University of Maryland - also check out the NorbertWeiner Center

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory





I Texas A& M University

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory






I University of Missouri

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory







I University of Cincinnati

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory








I University of Colorado

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory









I University of Iowa

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory










I Iowa State University

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory











I University of Houston

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory












I University of Oregon

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory













I Georgia Institute of Technology

Frames


OrthonormalBases

Frames


Linear Algebra

Geometry

Operator Theory













I Georgia Institute of Technology

I Vanderbilt University

Documents

Loosen Up! An introduction to frames.faculty.nps.edu/rgera/organized conferences/JointMeetings... · 2011-01-07 · Frames Vectors and Vector Spaces Orthonormal Bases Frames 4 (Interrelated)