Sparse Tomography For Medical Imaging Reconstructionsprojects.gtk.fi/export/sites/projects/FinTomo/liitteet/TBubba_FinTomo2018.pdf · The panoramic X-ray device has been reprogrammed

Sparse Tomography For Medical ImagingReconstructions

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

[email protected]

FinTomo SeminarEspoo, May 17, 2018

Tatiana Bubba Sparse tomography for medical imaging reconstructions FinTomo seminar

[email protected]

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)


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


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


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


How to Solve the Inverse Problem?

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

object data

@�

K

@�

?

K−1?K−1?�

��

� @@@

@




object data

@�

K

@�

?

K−1?

K−1?�

��

� @@@

@




object data

@�

K

@�

?

K−1?

K−1?�

��

� @@@

@


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)


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.


Illustration of the Ill-posedness of Tomography

Difference 0.00254

@�

@�

K

K



Difference 0.00124

@�

@�

K

K



Difference 0.00004

@�

@�

K

K


How to Cure Ill-posedness?

Object space X Data space Y

D(K) K(D(K))

f=

Kf =

y

K


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.


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

}


Naive Reconstruction (Moore-Penrose Pseudoinverse)

Original phantom ReconstructionRelative square norm error 100%


Standard Tikhonov Regularization



Non-negative Tikhonov Regularization



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.


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.


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)


Sparse Tomography

FBP with comprehensive data(1200 projections)

FBP with sparse data(20 projections)


Sparse Tomography

Filtered back-projection Non-negative Tikhonov regularization


Sparse Tomography

Filtered back-projection Non-negative TV regularization


Sparse Tomography

Filtered back-projection Non-negative TGV regularization


Sparse Tomography

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


Sparse Tomography

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


Sparse Tomography

Filtered back-projection Non-negative `1 regularization withshearlets


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


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


Panoramic dental imaging shows all the teeth simultaneously

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


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


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.


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]


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


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

https://blog.fips.fi

Documents

Sparse Tomography For Medical Imaging Reconstructionsprojects.gtk.fi/export/sites/projects/FinTomo/liitteet/TBubba_FinTomo2018.pdf · The panoramic X-ray device has been reprogrammed