sparse tomography for medical imaging...

Sparse Tomography For Medical ImagingReconstructions

Tatiana A. BubbaDepartment of Mathematics and Statistics, University of Helsinki

tatiana.bubba@helsinki.fi

FinTomo SeminarEspoo, May 17, 2018

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Finnish Centre of Excellence in

Inverse Modelling and Imaging 2018-20252018-2025

Finland

•HHHHHHH

����

•�������•�����

•`````` •�

�������

•HHHH

HHHH

The Mathematics Behind Tomography: the X-ray Transform

Xf(θ, τ)

θ

τ

f(θ, τ) θ

τ

∫R2

δ(τ − 〈x, ωθ〉)︸ ︷︷ ︸K(x,θ,τ)

f(x) dx = y(θ, τ) = (Xf)(θ, τ)

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

The Mathematics Behind Tomography: Discretization

θ

Object: target f Data: sinogram y

∫R2

δ(τ − 〈x, ωθ〉)︸ ︷︷ ︸K(x,θ,τ)

f(x) dx = y(θ, τ) = (Xf)(θ, τ)

K f = y

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Direct and Inverse Problem

Object f Data yDirect problem: K

Inverse problem

Direct problem: given object f , determine data y

Inverse problem: given (noisy) data y, recover object f

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Direct and Inverse Problem

Object f Data yDirect problem: K

Inverse problem

Direct problem: given object f , determine data y

Inverse problem: given (noisy) data y, recover object f

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

How to Solve the Inverse Problem?

Solving the inverse problems means reconstructing the object from the measureddata:

object data

@�

K

@�

?

K−1?K−1?�

��

� @@@

@

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

How to Solve the Inverse Problem?

Solving the inverse problems means reconstructing the object from the measureddata:

object data

@�

K

@�

?

K−1?

K−1?�

��

� @@@

@

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

How to Solve the Inverse Problem?

Solving the inverse problems means reconstructing the object from the measureddata:

object data

@�

K

@�

?

K−1?

K−1?�

��

� @@@

@

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

What Happens with the (Pseudo)inverse?

Naive reconstruction using the Moore-Penrose pseudoinverse:

Original phantom, values between0 (black) and 1 (white)

Naive reconstruction with minimum−14.9 and maximum 18.5(data has 0.1% relative noise)

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

The Problem is Ill-posedness

Hadamard (1903): a problem is well-posed if thefollowing conditions hold.

1. A solution exists,

2. The solution is unique,

3. The solution depends continuouslyon the input.

If one of these conditions fails, the problem is saidill-posed.

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Illustration of the Ill-posedness of Tomography

Difference 0.00254

@�

@�

K

K

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Illustration of the Ill-posedness of Tomography

Difference 0.00124

@�

@�

K

K

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Illustration of the Ill-posedness of Tomography

Difference 0.00004

@�

@�

K

K

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

How to Cure Ill-posedness?

Object space X Data space Y

D(K) K(D(K))

f=

Kf =

y

K

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Robust Solution: Regularization

Object space X Data space Y

D(K) K(D(K))

f

Kf

y

K

ε

ΓαΓα(y)

We need to define a family of continuous functions Γα : Y → X so that the reconstruc-tion error ‖Γα(ε)(y)− f‖X vanishes asymptotically at the zero-noise level ε→ 0.

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Classical Regularization Techniques

Truncated Singular Value Decomposition (TSVD)

Γk(y) =k∑i=0

〈y,ui〉σi

vi

Iterative regularization: the Landweber algorithm

Γn(y) = τ

n−1∑i=0

(1− τKTK)i KTy

f (n+1) = f (n) + τ KT (y − yf (n)), with f (0) = 0, 0 < τ <2

‖K‖2Tikhonov regularization

Γα(y) = argminf

{1

2‖Kf − y‖22 + α‖f‖22

}

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Naive Reconstruction (Moore-Penrose Pseudoinverse)

Original phantom ReconstructionRelative square norm error 100%

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Standard Tikhonov Regularization

Original phantom ReconstructionRelative square norm error 35%

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Non-negative Tikhonov Regularization

Original phantom ReconstructionRelative square norm error 13%

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Problem Solved?

Despite ill-posedness, CT is well understood when comprehensive projection dataare available:

Analytical techniques: FBP, FDK

Iterative techniques: ART-based methods, ML and LS approaches, MBIR

However, concrete practical issues:

lower the X-ray radiation dose

shorten the scanning time

take into account the non-stationarityof the target and the time-dependanceof the measurements

Limited Data tomography

Dymanic tomography

These are severely ill-posed problems and state-of-the-art techniques from clas-sical CT perform poorly.

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Problem Solved?

Despite ill-posedness, CT is well understood when comprehensive projection dataare available:

Analytical techniques: FBP, FDK

Iterative techniques: ART-based methods, ML and LS approaches, MBIR

However, concrete practical issues:

lower the X-ray radiation dose

shorten the scanning time

take into account the non-stationarityof the target and the time-dependanceof the measurements

Limited Data tomography

Dymanic tomography

These are severely ill-posed problems and state-of-the-art techniques from clas-sical CT perform poorly.

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Tikhonov-like Regularization

Variational problems of the form:

Γα(y) = argminf

{1

2‖Kf − y‖22 + α R(f)

}where R(f) incorporates prior information or assumption on the object f .

A non exhaustive list:

Tikhonov regularization: ‖f‖22Generalized Tikhonov regularization: ‖∇f‖22Total Variation regularization: ‖∇f‖1 or

∑ni=0 ‖[∇f ]i‖22

Regularization with higher-order derivatives

Sparsity: ‖f‖0 or ‖f‖1 or ‖Φf‖1, with Φ some sparsifying transform

Indicator functions of constraints sets: ιR+(f)

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

FBP with comprehensive data(1200 projections)

FBP with sparse data(20 projections)

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

Filtered back-projection Non-negative Tikhonov regularization

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

Filtered back-projection Non-negative TV regularization

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

Filtered back-projection Non-negative TGV regularization

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

Filtered back-projection Non-negative `1 regularization with Haarwavelets

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

Filtered back-projection Non-negative `1 regularization withDaubechies wavelets

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Sparse Tomography

Filtered back-projection Non-negative `1 regularization withshearlets

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

A Real Life Example: an Industrial Case Study

The VT device was developed in 2001–2012 by

Nuutti HyvonenSeppo JarvenpaaJari KaipioMartti KalkePetri KoistinenVille KolehmainenMatti LassasJan MobergKati NiinimakiJuha PirttilaMaaria RantalaEero SaksmanHenri SetalaSamuli SiltanenErkki SomersaloAntti VanneSimopekka VanskaRichard L. Webber

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Application: dental implant planning, where a missing tooth isreplaced with an implant

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Panoramic dental imaging shows all the teeth simultaneously

Panoramic imaging wasinvented by Yrjo VeliPaatero in the 1950’s.

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Nowadays, a digital panoramic imaging device is standard equipmentat dental clinics

A panoramic dental image offers a generaloverview showing all teeth and other struc-tures simultaneously.

Panoramic images are not suitable for den-tal implant planning because of unavoid-able geometric distortion.

•

X-ray source

Narrow detector

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

The panoramic X-ray device has been reprogrammed

Number of projection images: 11

Angle of view: 40 degrees

Image size: 1000×1000 pixels

The unknown vector f has 7 000 000 el-ements.

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

CBCT imaging gives 100 times more radiation than VTreconstruction

Navigation image VT slice CBCT slice

Images from the PhD thesis of Martti Kalke (2014).

[Kolehmainen, Vanne, Siltanen, Jarvenpaa, Kaipio, Lassas & Kalke 2006,Kolehmainen, Lassas & Siltanen 2008, Cederlund, Kalke & Welander 2009,Hyvonen, Kalke, Lassas, Setala & Siltanen 2010, U.S. patent 7269241]

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

Finnish Inverse Problems Society

University webpage:https://www.helsinki.fi/en/researchgroups/inverse-problems

Computational Blog: https://blog.fips.fi

Facebook: Finnish Inverse Problems Society

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar