Wavelet-based Bayesian inversion for tomographic problems with sparse data

Samuli Siltanen Department of Mathematics and Statistics University of Helsinki Workshop: Mathematics and Algorithms in Tomography Mathematisches Forschungsinstitut Oberwolfach, 15.4.2010

http://math.tkk.!/inverse-coe/

Finland

A series of projects started in 2001 aiming for a new type of low-dose 3D imaging

The goal was a mathematical algorithm with Input: small number of digital X-ray images taken with any X-ray device Output: three-dimensional reconstruction with high enough quality for the clinical task at hand

Relevant products of Instrumentarium Imaging in 2001:

Nuutti Hyvönen Seppo Järvenpää Jari Kaipio Martti Kalke Petri Koistinen Ville Kolehmainen Matti Lassas Jan Moberg Kati Niinimäki Juha Pirttilä Maaria Rantala Eero Saksman Henri Setälä Erkki Somersalo Antti Vanne Simopekka Vänskä

This work was done in 2001-2010 jointly with

1. The measurement model

5. Future work: dynamic X-ray tomography

4. Bayesian inversion with wavelets & Besov space priors

2. Bayesian inversion with total variation prior

3. Level set reconstruction

4 4 5

1 3 4

1 0 2

13 X-ray source

Detector

Every X-ray measures the sum of attenuation through tissue

4 4 5

1 3 4

1 0 2

11 7 6

13

8

3

9√2

8√2

1√2

1√2

5√2

Direct problem of tomography is to !nd the radiographs from given tissue

11 7 6

13

8

3

Inverse problem of tomography is to !nd the tissue from radiographs

9√2

8√2

1√2

1√2

5√2

m4

m5

m6

m2

m3

m1 x1 x4 x7

x2 x5 x8

x3 x6 x9

We write the reconstruction problem in matrix form and assume Gaussian noise

Our measurement is m=Ax+ε with Gaussian noise ε of standard deviation σ>0.

Construct system matrix A so that

1. The measurement model

5. Future work: dynamic X-ray tomography

4. Bayesian inversion with wavelets & Besov space priors

2. Bayesian inversion with total variation prior

3. Level set reconstruction

Bayes formula combines measured data and a priori information together

We reconstruct the most probable 3D tissue in light of 1.  Available radiographs and 2.  Physiological a priori information

Bayes formula gives the posterior distribution p(x|m):

We recover x as a point estimate from p(x|m)

Prior distribution, or tissue model Likelihood distribution,

or measurement model

~

Algorithms can be tailored to any measurement geometry.

Naturally modular software: measurement model (likelihood) and tissue model (prior) can be designed independently.

Estimating x leads to large-scale optimization or to integration in high-dimensional space:

Bayesian inversion algorithms are "exible and widely applicable

This is a brief history of Bayesian tomography 1983 Hanson and Wecksung 1987 Hanson 1994 Sauer, James and Klifa 1996 Bouman and Sauer 1997 Battle, Cunningham and Hanson 1997 Hanson, Cunningham and McKee 1997 Mohammad-Djafari and Sauer 1997 Nygrén, Markkanen, Lehtinen, Tereshchenko and Khudukon 1999 Mohammad-Djafari and Soussen 1998 Delaney and Bresler 1999 Sachs and Sauer 2001 Hsiao, Rangarajan and Gindi 2001 Persson, Bone and Elmqvist 2002 Yu and Fessler 2007 Chen, Ma, Feng, Luo, Shi and Chen 2009 Bodin, Sambridge and Gallagher

+ the articles between 2003-2010 described in this talk + a huge body of work in Bayesian PET and SPECT imaging

We build a prior distribution for dental tissue using total variation prior

Positivity constraint:

Approximate total variation penalty:

Computation of the MAP estimate

Large scale optimization problem:

We use the gradient method of Barzilai & Borwein, which is a modi!cation of Euler’s steepest descent method.

Step size differs from Euler’s:

Experimental setup for chairside 3D imaging models the clinical situation

Details of this limited angle experiment

Seven digital intraoral radiographs (664 x 872 pixels each)

Opening angle 60 degrees

There are 42 496 000 unknowns and 4 053 056 linear equations. Computation is divided into 400 approximately 2D problems.

S, Kolehmainen, Järvenpää, Kaipio, Koistinen, Lassas, Pirttilä and Somersalo 2003 Kolehmainen, S, Järvenpää, Kaipio, Koistinen, Lassas, Pirttilä and Somersalo 2003

Back- projection

Bayes-MAP

(singularities visible as analysed in Quinto 1993 and Ramm& Katsevich 1996)

1. The measurement model

5. Future work: dynamic X-ray tomography

4. Bayesian inversion with wavelets & Besov space priors

2. Bayesian inversion with total variation prior

3. Level set reconstruction

These results belong to the tradition of level set methods for inverse problems

1996 Santosa 1999 Sethian 2000 Dorn, Miller and Rappaport 2001 Osher and Santosa 2002 Suri, Liu, Singh, Laxminarayan, Zeng and Reden 2002 Vese and Chan 2003 Osher and Fedkiw 2003 Chan and Tai 2004 Tai and Chan 2005 Burger and Osher 2005 Chung, Chan and Tai 2006 Dorn and Lesselier 2006 Irishina, Moscoso and Dorn 2006 Villegas, Dorn, Moscoso, Kindelan and Mustieles 2008 Kolehmainen, Lassas and S 2008 Villegas, Dorn, Moscoso, Kindelan and Mustieles 2009 Irishina, Álvarez, Dorn and Moscoso 2009 Dorn and Lesselier (contains a more thorough review) 2010 Irishina, Álvarez, Dorn and Moscoso

We introduce a new level set reconstruction method

Note: in contrast to the classical level set method, the attenuation coefficient is represented inside the level set by the smooth level set function itself, not by a constant

Reconstruction is achieved at the long time limit of the solution

For proof of convergence, see Kolehmainen, Lassas & S 2008.

FBP levelset

Kolehmainen, Lassas & S 2008

C-arm X-ray device data from a knee phantom

FBP

Level set 60 projections 40 projections 20 projections 10 projections

truth BP level set

2D projection radiograph is not enough for dental implant planning

Panoramic X-ray device rotates around the head and produces a general picture

Panoramic imaging was invented by Yrjö Paatero in 1950’s.

Nowadays a panoramic device is standard equipment at every dental clinic around the world.

In our project, we reprogrammed the device so that it collects limited-angle data.

We consider the following limited angle experiment with the panoramic x-ray device:

11 projection images of the mandibular area

40 degrees angle of view

1000 x 1000 pixels per image, formed by a scanning movement

Kolehmainen, Vanne, S, Järvenpää, Kaipio, Lassas and Kalke (2006) Kolehmainen, Lassas and S (2008) Cederlund, Kalke and Welander (2009) Hyvönen, Kalke, Lassas, Setälä and S (2010)

Limited angle level set reconstruction can be used for locating the mandibular nerve

This is core technology for the PaloDEx Group’s VT product that has been in the market since 2007.

Remark that a software update transforms a 2D device into a 3D device.

1. The measurement model

5. Future work: dynamic X-ray tomography

4. Bayesian inversion: wavelets & Besov space priors

2. Bayesian inversion with total variation prior

3. Level set reconstruction

Why wavelets and Besov space priors?

In!nite-dimensional Wavelets and Bayesian inversion tomography 1970 Franklin 1984 Mandelbaum 1989 Lehtinen, Päivärinta & Somersalo 1991 Fitzpatrick 1995 Luschgy 2002 Lasanen 2004 Lassas and S 2005 Piiroinen 2008 Neubauer and Pikkarainen 2009 Helin and Lassas 2009 Lassas, Saksman and S

1994 Olson and DeStefano 1994 Sahiner and Yagle 1995 Delanay and Bresler 1996 Berenstein and Walnut 1996 Bhatia, Karl and Willsky 1997 Rashid-Farrokhi, Liu, Berenstein and Walnut 1997 Zhao, Welland and Wang 1999 Candès and Donoho 2000 Smith and Adhami 2002 Frese, Bouman and Sauer 2004 Soleski and Walter 2006 Soleski and Walter 2006 Rantala et al. 2007 Niinimäki, S and Kolehmainen 2009 Vänskä, Lassas and S

We introduce a renumbering of the wavelet basis functions using one index (from coarse to !ne)

Besov space norms can be written in terms of wavelet coefficients

Periodic boundary conditions: we work on d-dimensional torus.

Computation of the CM estimate reduces to sampling from well-known densities

Rantala, Vänskä, Järvenpää, Kalke, Lassas, Moberg and S 2006; United States patent 7215730

MAP estimate, Besov prior, p=1.5=q and s=0.5 Limited angle (30 degree) tomography

results for X-ray mammography

Local tomography results for dental X-ray imaging; data from dry mandible (jawbone)

Niinimäki, S and Kolehmainen (2007); Vänskä, Lassas and S (2009)

All 6 wavelet scales outside ROI Only coarsest wavelets outside ROI

Comparison of our local tomography results with Lambda-tomography

Lambda-tomography Besov prior, p=q=1.5 and s=0.5

1. The measurement model

5. Future work: dynamic X-ray tomography

4. Bayesian inversion with wavelets & Besov space priors

2. Bayesian inversion with total variation prior

3. Level set reconstruction

Combining several source-detector pairs enables 4-dimensional X-ray imaging

X-ray detectors are available with frame rates up to 200 Hz, providing dynamic data.

We can compute a 3D Bayesian estimate for each time; there are no moving parts.

Applications include -cardiac imaging -angiography -dental cone-beam imaging -veterinary medicine -non-destructive testing

Thank you!

Preprints available at www.siltanen-research.net

Newly purchased to the Helsinki Industrial Mathematics Lab:

Hamamatsu C7942CA #at panel X-ray sensor featuring • 2400x2400 pixels • Active area 120x120mm • Frame rate 2Hz or 9Hz (4x4 binning) • High dynamic range • Low noise

If you need tomographic data, please contact me and agree a data collection visit!