Image Reconstruction Tutorial Part 1: Sparse optimization and ...Image Reconstruction Tutorial Part...

Image Reconstruction Tutorial

Part 1: Sparse optimization and learning approaches

Aleksandra Pizurica1 and Bart Goossens2

1Group for Artificial Intelligence and Sparse Modelling (GAIM)

2Image Processing and Interpretation (IPI) group

TELIN, Ghent University - imec

Yearly workshop FWO–WOG

Turning images into value through statistical parameter estimation

Ghent, Belgium, 20 September 2019

Outline

1 Model-based iterative reconstruction algorithms

Sparse optimization

Solution strategies: greedy methods vs. convex optimization

Optimization methods in sparse image reconstruction

2 Structured sparsity

Wavelet-tree sparsity

Markov Random Field (MRF) priors

3 Machine learning in image reconstruction

Main ideas and current trends

A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 2 / 45

A fairly general formulation

Reconstruct a signal (image) x ∈ X from data y ∈ Y where

y = T (x) + n

X and Y are Hilbert spaces, T : X 7→ Y is the forward operator and n is noise.

A common model-driven approach is to minimize the negative log-likelihood L:

minx∈XL(T (x), y)

Typically, ill-posed and leads to over-fitting. Variational regularization, also called

model-based iterative reconstruction seeks to minimize a regularized objective

function

minx∈XL(T (x), y) + τφ(x)

φ : X 7→ R ∪ −∞,∞ is a regularization functional. τ ≥ 0 governs the influence of

the a priori knowledge against the need to fit the data.


Linear inverse problems

Many image reconstruction problems can be formulated as a linear inverse problem.

A noisy indirect observation y, of the original image x is then

y = Ax + n

Matrix A is the forward operator. x ∈ Rn; y,n ∈ Rm (or x ∈ Cn; y,n ∈ Cm).

Here, image pixels are stacked into vectors (raster scanning). In general, m 6= n.

Some examples

CT: A is the system matrix modeling the X-ray transformation

MRI: A is (partially sampled) Fourier operator

OCT: A is the first Born approximation for the scattering

Compressed sensing: A is a measurement matrix (dense or sparse)


Linear inverse problems

For the linear inverse problem y = Ax + n, model-based reconstruction seeks to solve:

minx

1

2‖Ax− y‖2

2 + τφ(x) (Tikhonov formulation)

Alternatively,

minxφ(x) subject to ‖Ax− y‖2

2 ≤ ε (Morozov formulation)

minx‖Ax− y‖2

2 subject to φ(x) ≤ δ (Ivanov formulation)

Under mild conditions, these are all equivalent [Figueiredo and Wright, 2013], and

which one is more convenient is problem-dependent.


Outline


Sparse optimization









Sparse optimization

A common assumption: x is sparse in a well-chosen transform domain.

We refer to a wavelet representation meaning any wavelet-like multiscale

representation, including curvelets and shearlets..

x = Ψθ, θ ∈ Rd , Ψ ∈ Rn×d

The columns of Ψ are the

elements of a wavelet frame

(an orthogonal basis or an

overcomplete dictionary)

The main results hold for learned dictionaries, trained on a set of representative

examples to yield optimally sparse representation for a particular class of images.


Compressed sensing

Consider y = Ax + n, with y,n ∈ Rm, x = Ψθ, x ∈ Rn, θ ∈ Rd , m < n

θ = arg minθ

1

2‖AΨθ− y‖2

2 + τφ(θ), x = Ψθ

Commonly: minθ‖θ‖0 s.t. ‖AΨθ− y‖2

2 ≤ ε or minθ

12‖AΨθ− y‖2

2 + τ‖θ‖1

[Candes et al., 2006], [Donoho, 2006], [Lustig et al., 2007]A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 8 / 45

Compressed sensing: recovery guarantees

Consider y = Ax + n, with y,n ∈ Rm, x = Ψθ, x ∈ Rn, m < n

Matrix Φ = AΨ has K -restricted isometry property (K-RIP) with constant εK < 1 if

∀ K -sparse (having only K non-zero entries) θ:

(1− εK )‖θ‖22 ≤ ‖Φθ‖2

2 ≤ (1 + εK )‖θ‖22

Suppose matrix A ∈ Rm×n is formed by subsampling a given sampling operator

A ∈ Rn×n. The mutual coherence between A and Ψ:

µ(A,Ψ) = maxi ,j|a>i ψj |

If m > Cµ2(A,Ψ)Kn log(n), for some constant C > 0, then

minθ

1

2‖AΨθ− y‖2

2 + τ‖θ‖1

recovers x with high probability, given the K-RIP holds for Φ = AΨ.A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 9 / 45

Analysis vs. synthesis formulation

Consider y = Ax + n, with y ∈ Rm, x = Ψθ, x ∈ Rn, θ ∈ Rd

Synthesis approach:

minθ

1

2‖AΨθ− y‖2

2 + τφ(θ)

or in the constrained form:

minθφ(θ) subject to ‖AΨθ− y‖2

2 ≤ ε

Analysis approach:

minx

1

2‖Ax− y‖2

2 + τφ(x)

or in the constrained form:


2 ≤ ε




Synthesis approach:

minθ

1

2‖AΨθ− y‖2

2 + τφ(θ)

or in a constrained form:


2 ≤ ε

Analysis approach:

minx

1

2‖Ax− y‖2

2 + τφ(x)



2 ≤ ε




Synthesis approach:

minθ

1

2‖AΨθ− y‖2

2 + τφ(θ)



2 ≤ ε

Analysis approach that also applies to wavelet regularization

minx

1

2‖Ax− y‖2

2 + τφ(Px)


minxφ(Px) subject to ‖Ax− y‖2

2 ≤ ε




Synthesis approach:

minθ

1

2‖AΨθ− y‖2

2 + τφ(θ)



2 ≤ ε

Analysis approach that also applies to wavelet regularization

minx

1

2‖Ax− y‖2

2 + τφ(Px)


minxφ(Px) subject to ‖Ax− y‖2

2 ≤ ε

P: a wavelet transform operator or P = I (standard analysis)A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 13 / 45

Outline


Sparse optimization










Solution strategy is problem-dependent. For non-convex problems like

minx‖x‖0 subject to ‖Ax− y‖2

2 ≤ ε

Greedy algorithms, e.g.,

Matching Pursuit (MP) [Mallat and Zhang, 1993]

OMP[Tropp, 2004], CoSaMP [Needell and Tropp, 2009]

IHT [Blumensath and Davies, 2009]

or convex relaxation can be applied leading to:

minx‖x‖1 subject to ‖Ax− y‖2

2 ≤ ε

minx

1

2‖Ax− y‖2

2 + τφ(x)

known as LASSO [Tibshirani, 1996] or BPDN [Chen et al., 2001] problem.A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 15 / 45

Greedy methods: OMP

OMP algorithm for solving minx‖x‖0 subject to Ax = y

Require: k = 1, r(1) = y,Λ(0) = ∅1: repeat

2: λ(k) = arg maxj |Aj · r(k)|3: Λ(k) = Λ(k−1) ∪ λ(k)4: x(k) = arg minx‖AΛkx− y‖2

5: y(k) = AΛkx(k)

6: r(k+1) = r(k) − y(k)

7: k = k + 1

8: until stopping criterion satisfied

Aj is the j-th column of A, and AΛ a sub-matrix of A with columns indicated in Λ.


Outline


Sparse optimization









Proximal operator

Many state-of-the-art image reconstruction algorithms solve problems of the kind

minx

1

2‖Ax− y‖2

2 + τφ(Px)

making use of the proximity operator i.e., the Moreau proximal mapping

[Combettes and Wajs, 2005]

proxτφ(y) = argminx

1

2‖x− y‖2

2 + τφ(x)

For certain choices of φ(x), this operator has a closed-form, e.g.,

φ(x) = ‖x‖1 → proxτ`1(y) = soft(y, τ) component-wise soft thresholding

φ(x) = ‖x‖0 → proxτ`0(y) = hard(y,

√2τ) component-wise hard thresholding

Another common regularization function is total variation (TV):

φ(x) = ‖x‖TV → proxτTV (y) Chambolle’s algorithm [Chambolle, 2004]


Iterative shrinkage/thresholding (IST)

The standard algorithm for solving

minx

1

2‖Ax− y‖2

2 + τφ(x)

is iterative shrinkage/thresholding (IST) algorithm [Figueiredo and Nowak, 2003],

[Daubechies et al., 2004]:

xk+1 = proxτφ

(xk −

1

γAH(Axk − y)︸︷︷︸

gradient of the datafidelity term

)

Its key ingredient is the proximity operator proxτφ(y) = argminx

12‖x− y‖2

2 + τφ(x)

A general approach in [Daubechies et al., 2004]: φ(x) is a weighted `p norm of the

coefficients of x with respect to a wavelet basis.A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 19 / 45

Iterative shrinkage/thresholding (IST) and extensions

The standard algorithm for solving

minx

1

2‖Ax− y‖2

2 + τφ(x)

is iterative shrinkage/thresholding (IST) algorithm [Figueiredo and Nowak, 2003],

[Daubechies et al., 2004]:

xk+1 = proxτφ

(xk −

1

γAH(Axk − y)︸︷︷︸

gradient of the datafidelity term

)

Different accelerated versions:

TwIST [Bioucas-Dias and Figueiredo, 2007]

FISTA [Beck and Teboulle, 2009]

SpaRSA [Wright et al., 2009]


Variable splitting

A very old idea (back to at least [Courant, 1943]): Represent minx f1(x) + f2(Gx) as

minxf1(x) + f2(z) subject to Gx = z

The rationale: it may be easier to solve the constrained problem.

Variable splitting (VS) together with the augmented Lagrangian method (ALM) and

non linear block Gauss-Seidel (NLBGS) leads to a form of Alternating Direction

Method of Multipliers (ADMM). It is this interpretation:

(VS + ALM + NLBGS)→ ADMM

that we give in the next few slides, following [Afonso et al., 2010]


Variable splitting and Augmented Lagrangian Method



Lµ(x, z,λ) = f1(x) + f2(z) + λT (Gx− z)︸︷︷︸Lagrangian

+µ

2‖Gx− z‖2

2︸︷︷︸“augmentation”

Basic augmented Lagrangian method (ALM), a.k.a., method of multipliers (MM),:

(x(k), z(k)) = argmin(x,z)

Lµ(x, z,λ(k−1))

λ(k) = λ(k−1) + µ(Gx(k) − z(k))

Goes back to at least [Hestenes, 1969], [Powell, 1969]


Variable splitting and Augmented Lagrangian Method



Lµ(x, z,λ) = f1(x) + f2(z) + λT (Gx− z)︸︷︷︸Lagrangian

+µ

2‖Gx− z‖2

2︸︷︷︸“augmentation”

After simple “complete-the-squares” ALM/MM yields [Afonso et al., 2010]:


f1(x) + f2(z) +µ

2‖Gx− z− d(k−1)‖2

2

d(k) = d(k−1) − (Gx(k) − z(k))


ADMM as Variable splitting and ALM

Use variable splitting (VS) to represent minx f1(x) + f2(Gx) as


ALM/MM yields :


f1(x) + f2(z) +µ

2‖Gx− z− d(k−1)‖2

2 (P)

d(k) = d(k−1) − (Gx(k) − z(k))

Solve (P) with one step of NLBGS → “scaled” ADMM version [Boyd et al., 2011]:

x(k) = argminxf1(x) +

µ

2‖Gx− z(k−1) − d(k−1)‖2

2

z(k) = argminzf2(z) +

µ

2‖Gx(k−1) − z− d(k−1)‖2

2

d(k) = d(k−1) − (Gx(k) − z(k))


ADMM algorithm

ADMM algorithm for solving: minx f1(x) + f2(Gx)

Require: k = 0, µ > 0, z0,d0

1: repeat

2: x(k) = argminxf1(x) + µ

2 ‖Gx− z(k−1) − d(k−1)‖22

3: z(k) = argminzf2(z) + µ

2‖Gx(k−1) − z− d(k−1)‖22

4: d(k) = d(k−1) − (Gx(k) − z(k))5: k = k + 1

6: until stopping criterion is satisfied

Equivalent to split-Bregman method [Goldstein and Osher, 2009].

Connections with Douglas-Raschford splitting [Setzer, 2009].


ADMM algoritm for linear inverse problems

Instantiate ADMM to our linear inverse problem: minx‖Ax− y‖22 + τφ(Px)

Require: k = 0, µ > 0, z0,d0

1: repeat

2: x(k) = argminx‖Ax− y‖2

2 + µ2 ‖Px− z(k−1) − d(k−1)‖2

2

3: z(k) = argminz

τφ(z) + µ2 ‖Px(k−1) − z− d(k−1)‖2

2= proxτφ/µ(Px(k−1) − d(k−1))

4: d(k) = d(k−1) − (Px(k) − z(k))5: k = k + 1

6: until stopping criterion is satisfied


A variant of ADMM algorithm for more than two functions

Consider minx∈Rn∑Jj=1 gj(Hjx) and map it into the previous: minx f1(x) + f2( Gx︸︷︷︸

z

)

f1(x) = 0, f2(z) =J∑j=1

gj(zj), G =

H1...

HJ

∈ Rp×n, z(k) =

z(k)1...

z(k)J

, d(k) =

d(k)1...

d(k)J

,

x(k) =( J∑j=1

((Hj)>Hj

)−1( J∑j=1

(Hj)>(z

(k−1)j + dk−1

j ))

z(k)1 = proxg1µ(H1x(k−1) − d

(k−1)1 )

...

z(k)J = proxgJµ(HJx

(k−1) − d(k−1)J ) C-SALSA [Afonso et al., 2011]

d(k) = d(k−1) − (Gx(k) − z(k))A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 27 / 45

Example: MRI reconstruction with shearlet regularization

A transversal slice of a FLAIR sequence, resampled along a non-Cartesian trajectory

based on an Archimedean spiral (sampling rate 15%). [Aelterman et al., 2011].A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 28 / 45

Example: CT reconstruction with shearlet regularization

x = arg minx

1

2‖C−1(Ax− y)‖2

2 + τ‖Px‖1

Matrix C is a “prewhitener” for the acquisition system [Vandeghinste et al., 2013].A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 29 / 45

Example: CT reconstruction with shearlet regularization

Top left: reference; Top right: SIRT; Bottom left: ADMM with TV regularization;

Bottom right: ADMM with shearlet regularization [Vandeghinste et al., 2013]A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 30 / 45

Modelling structured sparsity

Two main approaches to modelling structured sparsity in image reconstruction

in the acquisition stage

in the reconstruction stage

In the following we only focus on the second approach.

For the the improved design of the sampling patterns/sampling trajectories making

use of the structured sparsity, see [Roman et al., 2015], [Adcock et al., 2017],

[Gozcu, 2018]


Outline


Sparse optimization









Wavelet tree sparsity

[Jacob et al., 2009], [He and Carin, 2009], [Rao et al., 2011].

Application to MRI [Chen and Huang, 2014].


Outline


Sparse optimization









Sparse reconstruction with Markov Random Field priors

Use Markov Random Field (MRF) as a statistical model for the spatial clustering of

important wavelet coefficients [Cevher et al., 2010], [Pizurica et al., 2011]A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 35 / 45

Sparse reconstruction with Markov Random Field priors

Consider (a little simpler): y = Ax + n, with y,n ∈ Rm, x ∈ Rn, si ∈ 0, 1

P(s) =1

Zexp

[−(∑i

αsi +∑〈i ,j〉

βsisj

)]P(s, x, y) = P(y|x)P(x|s)P(s)

[x, s] = arg maxx,s

∑〈i ,j〉

βsisj +∑i

[αsi + log(p(xi |si)]−1

2σ2‖y − Ax‖2

2


Sparse MRI reconstruction with MRF priors

Consider: y = Ax + n, with y,n ∈ Cm, x ∈ Cn, x = Ψθ, θ ∈ Cd , si ∈ 0, 1

Let Ωs = i ∈ N : si = 1. Define a model for θ that conforms to the support s:

Ms = θ ∈ CD : supp(θ) = Ωs

Our objective is:

minx∈CN

‖Ax− y‖22 subject to Px ∈Ms

or equivalently:

minx∈CN

‖Ax− y‖22 + ιΩs(supp(Px))

where

ιQ(q) =

0, q ∈ Q+∞, otherwise

LaSB [Pizurica et al., 2011], GreeLa [Panic et al., 2016], LaSAL [Panic et al., 2017]A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 37 / 45

Example: CS-MRI with LaSB - early motivating results for using MRFs

SB (split-Bregman) and LaSB implemented with the same shearlet transform.A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 38 / 45

Example: CS-MRI with MRF priors

20% measurements, with variable density random sampling

Left: zero fill (PSNR = 19.87 dB)

Middle: WaTMRI [Chen and Huang, 2014] (wavelet-tree; PSNR = 28.78 dB)

Right: LaSAL [Panic et al., 2017] (MRF-based; PSNR = 33.43 dB)A. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 39 / 45

Example: CS-MRI with MRF priors

Reconstructions from 20% measurements, with radial sampling

Left: reconstructions; Right: error images; Top: LaSAL, Bottom: WaTMRIA. Pizurica and B. Goossens (UGent) Image Reconstruction Tutorial: Part 1 FWO-WOG TIVSPE 2019 40 / 45

Outline


Sparse optimization









Machine learning in image reconstruction

Covered in many recent workshops, special sessions and special issues of journals:

Three main direction have been proposed

learned postprocessing or learned denoisers;

learn a regularizer and use it in a classical variational regularization scheme;

learning the full reconstruction operator


Learning fast approximations of sparse coding

Core idea: time-unfolded version of an iterative reconstruction algorithm, like IST,

truncated to a fixed number of iterations.

Representatives: LISTA [Gregor and LeCun, 2010],[Moreau and Bruna, 2017]


Deep CNN models in image reconstruction

A central question is whether one can combine elements of model and data driven

approaches for solving ill-posed inverse problems.

[McCann et al., 2017]


Summary

Sparse optimization is a fundamental concept in inverse problems like image

reconstruction.

This tutorial covered some basic components of sparse image recovery

algorithms, including ADMM-based methods.

The concept of structured sparsity was underlined with particular attention to

using Markov Random Field priors in sparse image recovery.

A new frontier: machine learning in image reconstruction. Great potential, a

huge variability of approaches.


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