John Jasper
South Dakota State University
Pixels
Time
High resolution ⟺ Long scan time
Real world signals are sparse.
⇓
The conventional tradeoff is a lie!
We can solve "underdetermined"
Ax=b by L1 minimization provided:
Pixels
Time
We need an image with many pixels:
=
Image
Transformed Image
We need an image with many pixels:
=
Transformed Image
Column vectors need to be "spread out" in space
Compressed Sensing
Vectors that are "Spread out"
Equiangular tight frames
ETFs from groups
Z2×Z2×Z2×Z2
The design theory underneath
Real Flat ETFs
Some binary codes
Group divisible designs
ETFs and graphs
Definition. Given unit vectors Φ=(φi)i=1N, we define the coherence
μ(Φ)=i=jmax∣⟨φi,φj⟩∣.
μ(Φ)=cos(θ)
μ(Φ)=cos(θ)??
μ(Φ)=cos(θ)
Definition. Given unit vectors Φ=(φi)i=1N, we define the coherence
μ(Φ)=i=jmax∣⟨φi,φj⟩∣.
Example.
Definition. Given unit vectors Φ=(φi)i=1N, we define the coherence
μ(Φ)=i=jmax∣⟨φi,φj⟩∣.
Theorem (the Welch bound). Given a collection of unit vectors
Φ=(φi)i=1N in Cd, the coherence satisfies
μ(Φ)≥d(N−1)N−d.
Equality holds if and only if the following two conditions hold:
Welch bound equality ⟺ equiangular tight frame (ETF)
Useful matrix representation: Φ=∣φ1∣∣φ2∣⋯∣φN∣
Tightness: There is a constant A>0 such that i=1∑N∣⟨v,φi⟩∣2=A∥v∥2for all v.
( )
⇔ΦΦ∗=AI
⇔ the rows of Φ are orthogonal and equal norm
⟨v,ΦΦ∗v⟩=
⟨v,ΦΦ∗v⟩=
Example 2. Consider the (multiple of a) unitary matrix
Example 1. Consider the (multiple of a) unitary matrix
111−11−11−1−1−1−11
11111−11−111−1−11−1−11
1201−21231−21−23
20−2123−21−23
Example 3.
Z2×Z2×Z2×Z2
Some ETFs
arise from
groups...
a lot more ETFs
arise from
combinatorial designs!
Compressed Sensing
Vectors that are "Spread out"
Equiangular tight frames
ETFs from groups
Z2×Z2×Z2×Z2
The design theory underneath
Real Flat ETFs
Some binary codes
Group divisible designs
ETFs and graphs
Z7⎩⎨⎧012345611111111ωω2ω3ω4ω5ω61ω2ω4ω6ωω3ω51ω3ω6ω2ω5ωω41ω4ωω5ω2ω6ω31ω5ω3ωω6ω4ω21ω6ω5ω4ω3ω2ω
124111ωω2ω4ω2ω4ωω3ω6ω5ω4ωω2ω5ω3ω6ω6ω5ω3
(ω=e2πi/7)
Φ=111ωω2ω4ω2ω4ωω3ω6ω5ω4ωω2ω5ω3ω6ω6ω5ω3
Φ is tight, since it is rows out of a unitary.
Φ is equiangular, since D={1,2,4}⊂Z7 is a difference set.
That is, if we look at the difference table
− 124101326024450
every nonidentity group element shows up the same number of times
Φ∗Φ=circ(1^D)
∣Φ∗Φ∣2=circ(∣1^D∣2)
∣1^D∣2=1^D⊙1^D=F(1D∗1−D)
Want Φ= ETF, i.e., ∣1^D∣2=aδ0+b1G=spike + flat
F(spike + flat)=spike + flat
1D∗1−D=spike + flat⟺D is a difference set
Suppose: Φ=(rows of DFT indexed by D)
Z2×Z2×Z2×Z2
Some ETFs
arise from
McFarland Difference Sets...
a lot more ETFs
arise from
Steiner systems!
(0,0,0,1)(1,1,0,1)(0,0,1,0)(1,0,1,0)(0,0,1,1)(0,1,1,1,)(0,0,0,1)(0,0,0,0)(1,1,0,0)(0,0,1,1)(1,0,1,1)(0,0,1,0)(0,1,1,0)(1,1,0,1)(1,1,0,0)(0,0,0,0)(1,1,1,1)(0,1,1,1)(1,1,1,0)(1,0,1,0)(0,0,1,0)(0,0,1,1)(1,1,1,1)(0,0,0,0)(1,0,0,0)(0,0,0,1)(0,1,0,1)(1,0,1,0)(1,0,1,1)(0,1,1,1)(1,0,0,0)(0,0,0,0)(1,0,0,1)(1,1,0,1)(0,0,1,1)(0,0,1,0)(1,1,1,0)(0,0,0,1)(1,0,0,1) (0,0,0,0)(0,1,0,0)(0,1,1,1)(0,1,1,0)(1,0,1,0)(0,1,0,1)(1,1,0,1)(0,1,0,0)(0,0,0,0)
D={(0,0,0,1),(0,0,1,0),(0,0,1,1),(0,1,1,1),(1,0,1,0),(1,1,0,1)}
is a (McFarland) difference set in G=Z2×Z2×Z2×Z2
The subgroup H=Z2×Z2×0×0⩽G is disjoint from D.
Definition. A (2,k,v)-Steiner system is a {0,1}-matrix X such that:
Example. The matrix
X=
is a (2,2,4)-Steiner system.
=
Take a Steiner system with r ones per column
and an r×(r+1) ETF with unimodular entries
The Star product is a "Steiner" ETF
Compressed Sensing
Vectors that are "Spread out"
Equiangular tight frames
ETFs from groups
Z2×Z2×Z2×Z2
The design theory underneath
Real Flat ETFs
Some binary codes
Group divisible designs
ETFs and graphs
ETFs with all ±1 entries: Real Flat ETFs
Why real flat ETFs?
=
Example. A 276×576 real ETF
Theorem (J '13)
N×N Hadamard matrix ⟹ N(2N−1)×4N2 real flat ETF
Previously known real flat ETFs:
Theorem (Mixon, J, Fickus '13)
Real Flat
ETFs
Grey-Rankin
equality
binary codes
1-1 correspondence
We can also construct a real flat 317886556×1907416992 ETF.
Compressed Sensing
Vectors that are "Spread out"
Equiangular tight frames
ETFs from groups
Z2×Z2×Z2×Z2
The design theory underneath
Real Flat ETFs
Some binary codes
Group divisible designs
ETFs and graphs
Z2×Z2×Z3×Z3
Some ETFs
arise from
Spence Difference Sets...
a lot more ETFs
arise from
Group Divisible
Designs!
Ex:
G
Z2
×
Z2
×
Z3
×
Z3
=
⨂
≅
Unitary transformation
I3⊗(2×3 ETF)
3×4 ETF with unimodular entries
???
Definition. A K-GDD of type MU is a {0,1}-matrix X such that:
Example. The following is a 3-GDD of type 33:
X=
X⊤X=
Theorem (Fickus, J '19). Given a
d×n ETF
k-GDD of type MU
and
provided certain integrality conditions hold, there exists a D×N ETF with D>d, N>n and ND≈nd.
Φ=
Φ⊤Φ=
A. E. Brouwer maintains a table of known strongly regular graphs.
Our approach:
Real
ETFs
Combinatorial
designs
Strongly
regular graph
Construct
object
Certify
novelty
Theorem (J, '21). If there exists an
h×h Hadamard matrix with h≡1 or 2 mod 3,
then there exists a strongly regular graph with parameters:
v=h(2h+1),k=h2−1,λ=21(h2−4),μ=21h(h−1)
There exists a 20×20 Hadamard matrix, and hence an SRG(820,399,198,190), which is new!
From Brouwer's table online:
Overall: Five new infinite families!
Compressed Sensing
Vectors that are "Spread out"
Equiangular tight frames
ETFs from groups
Z2×Z2×Z2×Z2
The design theory underneath
Real Flat ETFs
Some binary codes
Group divisible designs
ETFs and graphs
L1 Minimization Solution
L2 Minimization Solution