Suppose that a sparse family of vectors {φp}p∈Λ{φp}p∈Λ has
been selected to approximate a signal ff.
An approximation can be recovered
as an orthogonal projection in
the space Vλ generated by these vectors.
We then face one of the following twoproblems.
- In a dual-synthesis problem, the orthogonal projection fλfλ
of ff in Vλ must be computed from dictionary coefficients,
{〈f,φp〉}p∈λ{〈f,φp〉}p∈λ, provided by an analysis operator.
This is the case when a signal transform {〈f,φp〉}p∈Γ{〈f,φp〉}p∈Γ
is calculated in some large dictionary and a subset of inner products
are selected. Such inner products may correspond to
coefficients above a threshold or local maxima values.
- In a dual-analysis problem, the decomposition
coefficients of fλfλ must be computed
on a family of selected vectors {φp}p∈Λ{φp}p∈Λ. This
problem appears when sparse representation algorithms
select vectors as opposed to inner products.
This is the case for
pursuit algorithms, which compute
approximation supports in highly redundant dictionaries.
The frame theory gives energy equivalence conditions
to solve both problems with stable operators.
A family {φp}p∈Λ{φp}p∈Λ is a frame of the space V it generates
if there exists B≥A>0B≥A>0 such that
∀
h
∈
V
,
A
∥
h
∥
2
≤
∑
m
∈
λ
|
〈
h
,
φ
p
〉
|
2
≤
B
∥
h
∥
2
.
∀
h
∈
V
,
A
∥
h
∥
2
≤
∑
m
∈
λ
|
〈
h
,
φ
p
〉
|
2
≤
B
∥
h
∥
2
.
(1)The representation is stable since
any perturbation of frame coefficients
implies a modification of similar magnitude
on h. Chapter 5
proves that the existence of a dual frame
{φ˜p}p∈Λ{φ˜p}p∈Λ that solves both the
dual-synthesis and dual-analysisproblems:
f
λ
=
∑
p
∈
λ
〈
f
,
φ
p
〉
φ
˜
p
=
∑
p
∈
λ
〈
f
,
φ
˜
p
〉
φ
p
.
f
λ
=
∑
p
∈
λ
〈
f
,
φ
p
〉
φ
˜
p
=
∑
p
∈
λ
〈
f
,
φ
˜
p
〉
φ
p
.
(2)
Algorithms are provided to calculate these decompositions.
The dual frame is also stable:
∀
f
∈
V
,
B
-
1
∥
f
∥
2
≤
∑
m
∈
γ
|
〈
f
,
φ
˜
p
〉
|
2
≤
B
-
1
∥
f
∥
2
.
∀
f
∈
V
,
B
-
1
∥
f
∥
2
≤
∑
m
∈
γ
|
〈
f
,
φ
˜
p
〉
|
2
≤
B
-
1
∥
f
∥
2
.
(3)The frame bounds A and B are redundancy factors.
If the vectors {φp}p∈Γ{φp}p∈Γ are normalized and
linearly independent, then A≤1≤BA≤1≤B. Such a dictionary
is called a Riesz basis of V and the dual frame is biorthogonal:
∀
(
p
,
p
'
)
∈
λ
2
,
〈
φ
p
,
φ
˜
p
'
〉
=
δ
[
p
-
p
'
]
.
∀
(
p
,
p
'
)
∈
λ
2
,
〈
φ
p
,
φ
˜
p
'
〉
=
δ
[
p
-
p
'
]
.
(4)When the basis is orthonormal, then both bases are equal.
Analysis and synthesis problems are then identical.
The frame theory is also used to construct redundant dictionaries
that define complete, stable, and redundant signal representations,
where V is then the whole signal space. The frame bounds
measure the redundancy of such dictionaries.
Chapter 5 studies the construction of
windowed Fourier and wavelet frame dictionaries by sampling
their time, frequency, and scaling parameters,
while controlling frame bounds.
In two dimensions, directional wavelet frames include wavelets
sensitive to directional image structures such as textures or edges.
To improve the sparsity of images having edges along regular geometric
curves, Candès and Donoho (CandesD:99)
introduced curvelet frames, with elongated waveforms having
different directions, positions, and scales.
Images with piecewise regular edges have representations that are
asymptotically more sparse by thresholding curvelet
coefficients than wavelet coefficients.
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