Adaptive Reconstruction Methods for Low-Dose Computed Tomography - - PowerPoint PPT Presentation

Ph.D. Talk, Apr. 2012 Computer Science dept.

Adaptive Reconstruction Methods for Low-Dose Computed Tomography

Joseph Shtok

Technion IIT, Computer Science dept. Israel, 2011 Ph.D. supervisors: Prof. Michael Elad, Dr. Michael Zibulevsky.

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Contents of this talk

Then you get tired of me.

Ph.D. Talk, Apr. 2012 Computer Science dept.

Short Intro to Computed Tomography

Assume you have a shiny new CT scanner… And an aardvark has scheduled a head scan for today.

Ph.D. Talk, Apr. 2012 Computer Science dept.

X-ray source photon counts

Line

Radon (X-ray) transform

l

y

l

Detectors

) log( λ

l l

y g − =

λ

photons Sinogram values =

l

g

∫

=

l l

dl x f f ) ( ] [R Sum contributions to from all incident rays

Short Intro to Computed Tomography

) (x f x

θ

s

( )

∫

⋅ =

θ θ

θ θ d x g x g ) ( R*

) ( | | ) ( F σ σ σ

θ θ

g g =

Ph.D. Talk, Apr. 2012 Computer Science dept. Large attenuation High integral value Low count High noise variance Streak artifacts

Noise in Low-Dose Reconstruction

l

y

l

f l

e

] [R −

= λ λ Accepted model for detector measurements (similar to one in CCD sensors):

instance

• ideal count

Poor photon statistics due to low counts Electronic noise in the hardware

8 3

) ( + = =

l l l

Y Y Anscombe Z

instance

1 = ) var(

l l

z z

instance

2 n l l l

y y σ λ + = ) var(

∫

=

l l

dl x f g ) (

l

g l

e I y

−

=

1 −

=

l l

y g ) var(

) (

2 2 n l n l l

Poisson Y Y σ λ σ + ≈ + = ) ) ( ~

n

Ν(0,σ λ +

l l

Poisson Y

Ph.D. Talk, Apr. 2012 Computer Science dept.

Projection and log- transform

+

Ideal sinogram Stochastic data- dependent noise

Noise in Low-Dose Reconstruction

FBP

Ideal inverse Radon

FBP

Smoothed inverse Radon

FBP+prep

Adaptive Median Filtering (Hsieh, 98)

MAP estimate

Iterative statistically- based PWLS

Ph.D. Talk, Apr. 2012 Computer Science dept.

Problem of local reconstruction

A point in the image draws a sine. Points outside the ROI contribute to its projections. ROI is not uniquely determined from the truncated data.

Ph.D. Talk, Apr. 2012 Computer Science dept.

Problem of local reconstruction

Basic sinogram completion: duplicate the margins. FBP reconstruction from zero-padded truncated projections Non-linear sine-based sinogram completion

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Ph.D. Talk, Apr. 2012 Computer Science dept.

Error measure for CT reconstruction

Basic error measure: Mean Square Error (MSE)

− f − f ~

reference image reconstructed image

( )

2 2 2 1

f f x f x f f

x

~ ) ( ~ ) ( ) ~ ( − = − = ∑ ϕ

Problem: MSE can be reduced by blurring the image. Sharpness-promoting penalty: the gradient norm in should not fall below the gradient norm in .

f ~ f

( )+

− + − = J J f f f ~ ~ ) ~ (

2 2 2

µ ϕ

2 2 2 2

~ ~ , f J f J

x x

∇ = ∇ =

Nuances:

• The MSE component is restricted to regions of interest
• The gradient-based component is restricted to fine edges.
• The non-negativity function is smoothed for better optimization.

( )+

Ph.D. Talk, Apr. 2012 Computer Science dept.

Supervised learning of adaptive processing tools

High-quality reference CT images .

Pre- processing FBP recon Post- processing Learnable parameters p,q,r Minimize the error measure w.r.t. parameters

Degraded photon counts Error measure Output image - a function of

( )+

− + − = Ψ

∑

J J y V F U f

f image f

~ ) ( r) q, (p,

2 2 p q r

µ

r

U

q

F

p

V

f

y

f

f ~

2 2 2 2

~ ~ , f J f J

x x

∇ = ∇ =

Ψ

r. q, p,

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Ph.D. Talk, Apr. 2012 Computer Science dept.

Learned FBP filter for ROI reconstruction

2 2

f g f − = Ψ ) ( T ) (

κ

κ

FBP operator:

). ( R ) ( T

*

g g ∗ = κ

κ

Train the convolution kernel to pursuit reconstruction goals.

κ

Training objective:

2 2 Q f

f g

,

) ( T ) ( − = Ψ

κ

κ

Training objective for ROI reconstruction:

Q - Binary mask,

restriction to ROI.

5

κ

1

κ

• 2. Filter the sinogram

with radially-variant convolution kernel.

} ,..., {

5 1

κ κ

• 1. Use 2-D kernel.

Reference FBP reconstruction Reference AFBP reconstruction

Truncated sinogram with completion

. . .

• Truncated sinogram

after completion.

f

g

Ph.D. Talk, Apr. 2012 Computer Science dept.

ROI reconstruction

Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP reconstruction 22.9 dB AFBP reconstruction 34.68 dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

ROI reconstruction

Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP reconstruction 18.04 dB AFBP reconstruction 29.63 dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

ROI reconstruction

Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP reconstruction 19.48 dB AFBP reconstruction 31.44 dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Ph.D. Talk, Apr. 2012 Computer Science dept.

Sparse-Land model for signals

The concept: natural signals admit a faithful representation using only

few columns (atoms) from a dedicated overcomplete dictionary.

D s ν α + =

k α ≤

2

ν ε ≤

Number of non-zeros is small

α D

E ( )

j

s f =

f

Residual is small

Natural dictionaries: Wavelets, Haar functions, Discrete Cosines, Fourier. Dictionaries tailored to the specific family

• f signals: obtained via a training process.

≈

Ph.D. Talk, Apr. 2012 Computer Science dept.

Minimize w.r.t 1. (K-SVD) Train a dictionary D along with sparse representations

• 2. Compute the image estimate (closed-form solution).

Denoising technique (Elad, Aharon, 2006):

{ } j

α

Sparse-Land model for signals

∑ ∑

− + + − = Φ

j patch j j j j patch j

f f f f

2 2 2 2

E D ~ ) , (D, α α µ δ α

Sparse coding:

} { D,

j

α

Minimize w.r.t f

• State-of-the-art noise reduction.
• Adaptive to current image or training set.
• Uniform noise assumption.

∑

− =

j patch j j d i

f d

2 2

E D min arg α

Dictionary update:

j j j j

f t s ε α α α

α

≤ − =

2 2

E D . . min arg

Variable is the

i-th column in . D

d

f ~ - Noisy image

Ph.D. Talk, Apr. 2012 Computer Science dept.

Previous work (Liao, Sapiro, 2007):

Application to CT reconstruction

• Patch-wise sparse coding of CT image .
• Online learning from noisy data.
• Very nice results on geometric images under severe

angular subsampling.

f

Drawbacks:

• Data fidelity term in the sinogram domain.
• No reference to statistical model of the noise.
• Sparse coding thresholds not treated.

{ }

      − + + − =

∑ ∑

j patch j j j j patch j f

f g f f

2 2 2 2 , D, * * *

E D ~ R min arg , , D α α µ δ α

α

Ph.D. Talk, Apr. 2012 Computer Science dept.

Application to CT reconstruction

Our approach:

• 1. check data fidelity and perform sparse coding in the domain of

noise-normalized raw data :

=       − + − =

∑

j patch j j z

z z z z

2 2 2 2 2 *

~ E D ~ min arg α λ         +         ≡ =

∑ ∑

−

z

j j patch j j j j

~ D E E E ) ( , G

2 T 1 T D D

2 1

λ α α

{ }

      − + + − =

∑ ∑

j patch j j j patch j z

z z z z

2 2 1 2 2 , , D * * 1

~ E D ~ min arg , α , D

1

α α µ λ

α

8 3 2)

~ ( ~ + + =

n

y z σ

Solve for using K-SVD, but allow to use a different dictionary at restoration stage:

α , D1

2

D

Ph.D. Talk, Apr. 2012 Computer Science dept.

Application to CT reconstruction

+

− + Ω − = ) ~ ( G T min arg D

2 2 D , D D * 2

2 1 2

J J f µ α

2

D

α

• 2. Train a second dictionary optimized for image reconstruction

using a designed error measure and pre-computed repersentations :

z ~

Small patches

{ }

j

E

2 1 D

D ,

G

Represen- tations f Sparse coding with

1

D

Photon counts Improved data Data restoration with

2

D

T

FBP reconstruction CT image

8 3 2)

~ ( + +

n

y σ

Reconstruction chain:

g y z → → Ω :

Sinogram

Ω y ~

Data

z ~

j

α g

Ph.D. Talk, Apr. 2012 Computer Science dept.

Compared algorithms

Adaptive Trimmed Mean (ATM) Filter

Hsieh, ’98.

Penalized Weighted Least Squares (PWLS) Elbakri, Fessler, ’02.

• Extract M values from the neighborhood of a photon count .
• Remove extreme values and compute the average of the rest.
• are data-dependent; computed through

M 2α α M,

[ ]

. ) ( , 2 2 ) M( λ α α δ λ βλ

l m l l l

y y y y = − + =

+

2-nd order aproximation of a penalized log-likelihood expression for photon counts data:

∑ ∑ ∑

∈

− + − =

p p N k k p l l l l

f f g f W f y PWLS

) ( 2 2 1

) ( ) ] ([R ) | ( ψ λ

l

y

Huber penalty (smoothed norm). Penalty weight. Controls variance-resolution tradeoff.

1

L

In our experience: depends highly on the parameters. Works quite well.

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Thighs section

FBP, 25.76 dB ATM, 28.32 dB PWLS, 28.90 dB Sparse, 29.62 dB

Error images

• Recon. in

[-220,350] HU

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

FBP, 25.26 dB ATM, 27.46 dB PWLS, 28.26 dB Sparse, 27.94 dB

Error images

• Recon. in

[-220,350] HU

Thighs section

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Head section

FBP, 29.84 dB ATM, 29.83 dB PWLS, 31.02 dB Sparse, 32.36 dB

Error images

• Recon. in

[-170,250] HU Same parameters, new anatomical region.

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Ph.D. Talk, Apr. 2012 Computer Science dept.

Learned shrinkage in a transform domain

Scalar shrinkage functions

• Denosing by shrinkage of wavelet coeffs: Donoho & Johnston, 1994.

The tool: Descriptive functions for descriptive dictionary. LUT Examples of : Discrete Cosines, Wavelets, etc.

D+

D

Convert to z-domain Analysis Shrinkage Synhtesis Recon- struction Dictionary (pseudo-inverse) Denoising by supression of small coefficients, which usually contain the noise.

D

• Our goal: Solving non-linear inverse problems.

The tool: Learned functions for learned dictionary in a look-ahead training.

• Denoising with learned shrinkage functions: Hel-Or and Shaked, 2002.

The tool: Learned functions for descriptive dictionary.

Ph.D. Talk, Apr. 2012 Computer Science dept.

Learned shrinkage in a transform domain

Ψ Φ

LUT

Analysis Synthesis

p

S

train train convert to sinogram

Compare to clean training data Objective 1: best data quality T FBP reconstruction Objective 2: best image quality Compare to referenve image

z ~

{ }

j

E Photon counts

8 3 2)

~ ( + +

n

y σ

y ~

Data

Small patches

Ω

Preparing the data Pre-processing

2 2 ,

~ E min arg , z S g p

P p

Ψ ΩΦ − = Φ

Φ + Φ

− + Ψ ΩΦ − = Φ ) ~ ( ~ E T min arg ,

2 2 ,

J J z S f p

P p

µ

Ph.D. Talk, Apr. 2012 Computer Science dept.

Why not repeat the trick?

Raw data shrinkage FBP recon Image shrinkage

Post-processing with shrinkage functions, also trained by comparing to reference images. Difference made by the post- processing : no image structure lost.

+ Φ

− + Ψ Φ − = Φ ) ~ ( ˆ E min arg ,

2 2 ,

J J f S f p

P p

µ

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Thighs section (male) Error images

• Recon. in

[-220,350] HU

FBP, 25.76 dB ATM, 28.32 dB PWLS, 28.90 dB Shrinkage 30.05 dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Thighs section Error images

• Recon. in

[-220,350] HU

FBP, 25.26 dB ATM, 27.46 dB PWLS, 28.26 dB Shrinkage 28.94 dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Head section Error images

• Recon. in

[-170,250] HU

FBP, 29.84 dB ATM, 29.83 dB PWLS, 31.02 dB Shrinkage 32.81dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

Alternative versions

Thighs section

MSE-

• ptimized

versions Tuned by visual appearance

FBP, 27.70 dB 25.76 dB PWLS, 30.45 dB 28.90 dB Shrinkage 30.84 dB 30.05 dB

Optimizing for MSE introduces a blur into the image.

Ph.D. Talk, Apr. 2012 Computer Science dept.

Effective dose reduction

Normal X-ray dose range Low X-ray dose range

Estimating dose reduction factor:

• For each noise level, sweep over a range of FBP parameter and chose a

reconstruction with minimal error measure.

• Sweep over a range of the noise level and compare to learned shrinkage.

+

− + − = ) ~ ( ~ ) ~ , (

2 2

J J f f f f Error µ

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Contents of this talk

Ph.D. Talk, Apr. 2012 Computer Science dept.

Fusion over a smoothing parameter

Recon. algorithm

Scalar parameter Raw data Image

Decision rule The algorithm:

• Sweep the variance-resolution tradeoff.
• Extract pixel neighborhood (or other

features) from each version.

• Build a decision rule to perform the local

fusion. Decision rule: Use a regression to build one automatically, with a Neural network or Support Vector Regression.

Artificial Neural Network (ANN)

… …

Ph.D. Talk, Apr. 2012 Computer Science dept.

Fusion over a smoothing parameter

FBP algorithm: sweep the cut-off frequency of the low-pass sinogram filter. Collect few images with different resolution-variance trade-off. PWLS algorithm: perform the regular reconstruction while collecting versions along the iterations.

Created with FBP Standard PWLS result Initial image Converged image versions from partial iterations

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Thighs section

FBP 25.76 dB FBP-ANN 30.62 dB PWLS 28.90 dB PWLS-ANN 31.11 dB

Error images

• Recon. in

[-220,350] HU

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Thighs section Error images

• Recon. in

[-220,350] HU

FBP 25.26 dB FBP-ANN 29.67 dB PWLS 28.26 dB PWLS-ANN 30.03 dB

Ph.D. Talk, Apr. 2012 Computer Science dept.

Empirical results

Head section

FBP 29.84 dB FBP-ANN 33.41dB PWLS 31.02 dB PWLS-ANN 33.66 dB

Error images

• Recon. in

[-170,250] HU

Ph.D. Talk, Apr. 2012 Computer Science dept.

• Scan model, Noise, Local reconstruction

Intro to Computed Tomography

• General scheme of supervised learning

Framework of adaptive reconstruction

• Learned FBP filter for local reconstruction

Adaptive FBP

• Adaptation of K-SVD to low-dose CT reconstruction

Sparsity-based sinogram restoration

• Adaptation of the method to low-dose CT

reconstruction

Learned shrinkage in a transform domain

• Local fusion of multiple versions of the algorithm output

Performance boosting of existing algorithms

Conclusions

Ph.D. Talk, Apr. 2012 Computer Science dept.

Parade of proposed methods

Thighs section

Sparse 29.62 dB Shrinkage 30.05 dB FBP-ANN 30.62 dB PWLS-ANN 31.11 dB

Error images

• Recon. in

[-220,350] HU

Ph.D. Talk, Apr. 2012 Computer Science dept.

Summary

Adaptive methods can help improving CT reconstruction. Once the raw data is variance-normalized, the sparsity-based denoising mends most of the damage done by the low-dose scan. When the smootheness parameter is swept, reconstruction algorithms supply more information about the image; it is easily extracted by a regression function using only the intensity values. FBP needs only a little help to allow truly local reconstruction. Example-based training does not jeopardize the image content (in the presented algorithms) and can be allowed for clinical use.

Ph.D. Talk, Apr. 2012 Computer Science dept.

Thank you.