Sparsity / Wavelet Transform (WT) / Signals and Natural images // Textured images are not strongly sparse in the wavelet domain. “Doppler” Signal

“Lena” image

“Fabric” texture / VisTeX

WT “Doppler”

WT “Lena’

WT “Fabric”

Sparsity = Ok.

Sparsity = Weak!

Weak sparsity???

Degenerated penalized shrinkage ⇒ Thresholding.

Smooth penalized shrinkage, large differential. Smooth penalized shrinkage, small differential.

Sparsity - Signal/Images

Sigmoid Shrinkage Regularization

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Sigmoid Shrinkage:

a single family of functions for performing thresholding and non-degenerated shrinkage

 Problem
:

Estimating a sparse (strong / weak) vector d = {di }16i6N from observations ξ = {ξi }16i6N : ξi = di + i , iid

i = 1, 2, . . . , N,

(1)

2

where i ∼ N (0, σ ) for i = 1, 2, . . . , N (nota: d can be sparse in the strong or weak sense).

Regularization framework [extending the standard `1 penalty qλ = Id function]: An estimate b d of d is derived as a penalized least square solution, associated with a penalty function qλ = qλ (·): N X 2 b qλ (|di |), (2) d = argd min ||d − ξ||` + 2 2

i=1

or, equivalently, use a shrinkage/thresholding function δλ to retrieve an estimate b d = δλ (ξ). [Strongly] Sparse vector d =⇒ Use Degenerated shrinkage functions. Examples: Hard, Soft Thresholding (these functions pertain to the sigmoid shrinkage class). Weakly sparse vector d =⇒ Use Non-degenerated sigmoid shrinkage functions. Example: Smooth Sigmoid Based Shrinkage (SSBS) with t = 0, λ 6= 0, θ 6= 0.

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Sigmoid shrinkage and regularization. Not provided
SURE, iterative sigmoid shrinkage. Sparsity - Signal/Images

Sigmoid Shrinkage Regularization

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Sigmoid Shrinkage and Regularization Sigmoid Shrinkage δt,τ,λ = product of soft thresholding and a “sigmoid” δt,τ,λ (x) =

sgn(x)(|x| − t)+ , 1 + e −τ (|x|−λ)

(3)

with sgn(x) = 1 (resp. -1) if x > 0 (resp. x < 0), and (x)+ = x (resp. 0) if x > 0 (resp. x < 0). Parameters t, λ act as a thresholds. A particular subclass of interest is that associated with t = 0 (vanishing attenuation at infinity). Denote by δτ,λ = δ0,τ,λ this subclass and re-paremeterize it to exhibit an attenuation degree θ, δτ,λ = δθ,λ and qτ,λ = qθ,λ , with Z x   x 10 sin2 θ + 2 sin θ cos θ 1 −τ (z−λ) δτ,λ (x) = , qτ,λ (x) = L τ ze dz, τ = . 1 + e −τ (|x|−λ) τ 0 λ 5 cos2 θ − 1

δθ,λ =⇒=⇒=⇒ θ = π6 (—) θ = π4 (· · · ) θ = π3 (−−) =⇒=⇒=⇒

Sparsity - Signal/Images

Sigmoid Shrinkage Regularization

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Sigmoid shrinkage regularization ERS

on synthetic aperture radar images

(European Remote-Sensing) satellite image

Arcachon Bay, France

Sparsity - Signal/Images

WT - δπ/6,λD Result λD is the detection threshold

Sigmoid Shrinkage Regularization

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