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

Adaptive Reconstruction Methods for Low-Dose Computed Tomography Joseph Shtok Ph.D. supervisors: Prof. Michael Elad, Dr. Michael Zibulevsky. Technion IIT, Computer Science dept. Israel, 2011 Computer Science dept. Ph.D. Talk, Apr. 2012

Contents of this talk Intro to Computed • Scan model, Noise, Local reconstruction Tomography Framework of adaptive • General scheme of supervised learning reconstruction Adaptive FBP • Learned FBP filter for local reconstruction Sparsity-based sinogram • Adaptation of K-SVD to low-dose CT reconstruction restoration Learned shrinkage in a • Adaptation of the method to low-dose CT transform domain reconstruction Performance boosting of • Local fusion of multiple versions of the algorithm output existing algorithms Then you get tired of me. Computer Science dept. Ph.D. Talk, Apr. 2012

Short Intro to Computed Tomography And an aardvark has scheduled a head scan for today. Assume you have a shiny new CT scanner… Computer Science dept. Ph.D. Talk, Apr. 2012

Short Intro to Computed Tomography λ X-ray 0 source photons Sum contributions to x from all incident rays ( ) ∫ = ⋅ θ d θ R * g x g ( x ) ( x ) θ f θ Detectors l Line σ = σ σ F ( ) | | ( ) g g θ θ photon counts y l s Sinogram ∫ = = [R f ] f ( x ) dl g values l l y = − l l log( ) g λ l Radon (X-ray) θ 0 transform Computer Science dept. Ph.D. Talk, Apr. 2012

Noise in Low-Dose Reconstruction Accepted model for detector measurements (similar to one in CCD sensors) : λ + Ν(0, σ instance − y λ = λ [R ] f ~ ( ) ) Y Poisson - ideal count e l l 0 n l l l Poor photon statistics due Electronic noise in the hardware to low counts = + σ ≈ λ + σ 2 2 = λ + σ ( ) 2 Y Y Poisson instance y var( y ) l l n l n l l l n = instance = = + 1 z var( z ) ( ) Z Anscombe Y Y 3 8 l l l l l High Large High noise Streak integral Low count attenuation variance artifacts value ∫ − = = − = 1 g var( g ) ( ) y I e y g f x dl l 0 l l l l l Computer Science dept. Ph.D. Talk, Apr. 2012

Noise in Low-Dose Reconstruction + Projection and log- transform Ideal sinogram Stochastic data- dependent noise FBP FBP FBP+prep MAP estimate Adaptive Iterative Ideal inverse Smoothed Median Filtering statistically- Radon inverse Radon (Hsieh, 98) based PWLS Computer Science dept. Ph.D. Talk, Apr. 2012

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

Problem of local reconstruction FBP reconstruction from zero-padded truncated projections Basic sinogram completion: duplicate the margins. Non-linear sine-based sinogram completion Computer Science dept. Ph.D. Talk, Apr. 2012

Intro to Computed • Scan model, Noise, Local reconstruction Tomography Framework of adaptive • General scheme of supervised learning reconstruction Adaptive FBP • Learned FBP filter for local reconstruction Sparsity-based sinogram • Adaptation of K-SVD to low-dose CT reconstruction restoration Learned shrinkage in a • Adaptation of the method to low-dose CT transform domain reconstruction Performance boosting of • Local fusion of multiple versions of the algorithm output existing algorithms Computer Science dept. Ph.D. Talk, Apr. 2012

Error measure for CT reconstruction ~ − − f f reference image reconstructed image ( ) = ∑ ~ ~ ~ 2 2 ϕ − = − Basic error measure: Mean ( f ) f ( x ) f ( x ) f f 1 2 Square Error (MSE) x Problem: MSE can be reduced by blurring the image. ~ f Sharpness-promoting penalty: the gradient norm in should not fall f below the gradient norm in . ( ) + ~ ~ ~ ~ ~ 2 2 2 ϕ = − + µ − = ∇ = ∇ , ( ) f f f J J J f J f 2 x x 2 2 2 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. Computer Science dept. Ph.D. Talk, Apr. 2012

Supervised learning of adaptive processing tools High-quality Minimize the Degraded error measure reference CT photon counts w.r.t. images . parameters f Ψ Error measure Learnable parameters p,q,r y f ~ f Pre- Post- FBP recon processing processing F V U q p r Output image - ( ) + ~ ∑ a function of 2 Ψ = − + µ − (p, q, r) ( ) f U F V y J J p, q, r. r q p f 2 image f ~ ~ 2 2 = ∇ = ∇ , J f J f x x 2 2 Computer Science dept. Ph.D. Talk, Apr. 2012

Intro to Computed • Scan model, Noise, Local reconstruction Tomography Framework of adaptive • General scheme of supervised learning reconstruction Adaptive FBP • Learned FBP filter for local reconstruction Sparsity-based sinogram • Adaptation of K-SVD to low-dose CT reconstruction restoration Learned shrinkage in a • Adaptation of the method to low-dose CT transform domain reconstruction Performance boosting of • Local fusion of multiple versions of the algorithm output existing algorithms Computer Science dept. Ph.D. Talk, Apr. 2012

Learned FBP filter for ROI reconstruction κ = κ ∗ Train the convolution kernel to * T ( g ) R ( g ). FBP operator: κ pursuit reconstruction goals. g 2 2 -Truncated sinogram Training objective for Ψ κ = g f − Ψ κ = − Training objective: f ( ) T ( ) f ( ) T ( g ) f κ κ f after completion. 2 ROI reconstruction: 2 Q , Q - Binary mask, κ κ { ,..., } 1 5 restriction to ROI. Truncated sinogram 1. Use 2-D kernel. with completion κ Reference FBP reconstruction Reference AFBP reconstruction 1 . . . κ 2. Filter the sinogram 5 with radially-variant convolution kernel. Computer Science dept. Ph.D. Talk, Apr. 2012

ROI reconstruction Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP AFBP reconstruction reconstruction 22.9 dB 34.68 dB Computer Science dept. Ph.D. Talk, Apr. 2012

ROI reconstruction Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP AFBP reconstruction reconstruction 18.04 dB 29.63 dB Computer Science dept. Ph.D. Talk, Apr. 2012

ROI reconstruction Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP AFBP reconstruction reconstruction 19.48 dB 31.44 dB Computer Science dept. Ph.D. Talk, Apr. 2012

Intro to Computed • Scan model, Noise, Local reconstruction Tomography Framework of adaptive • General scheme of supervised learning reconstruction Adaptive FBP • Learned FBP filter for local reconstruction Sparsity-based sinogram • Adaptation of K-SVD to low-dose CT reconstruction restoration Learned shrinkage in a • Adaptation of the method to low-dose CT transform domain reconstruction Performance boosting of • Local fusion of multiple versions of the algorithm output existing algorithms Computer Science dept. Ph.D. Talk, Apr. 2012

Sparse-Land model for signals The concept: natural signals admit a faithful representation using only few columns (atoms) from a dedicated overcomplete dictionary. α ≤ Natural dictionaries: Wavelets, Haar k Number of non-zeros is small 0 functions, Discrete Cosines, Fourier. ν ≤ ε Residual is small 2 s ν + = α Dictionaries tailored to the specific family D of signals: obtained via a training process. ≈ = α E ( ) s f j D f Computer Science dept. Ph.D. Talk, Apr. 2012

Sparse-Land model for signals Denoising technique (Elad, Aharon, 2006): ~ ∑ ∑ 2 2 Φ α = δ − + µ α + α − (D, , ) D E f f f f j j j j 0 2 2 ~ - Noisy image patch j patch j f α Minimize w.r.t f Minimize w.r.t D, { } 1. ( K-SVD ) Train a dictionary D j along with sparse { } j α representations 2. Compute the image estimate (closed-form solution). 2 α = α α − ≤ ε arg min . . D E s t f Sparse coding: j j j j α 0 2 • State-of-the-art noise reduction. ∑ 2 Variable is the = α − d arg min D E Adaptive to current image or training set. d f • Dictionary update: i-th column in . i j j D 2 d • Uniform noise assumption. patch j Computer Science dept. Ph.D. Talk, Apr. 2012

Application to CT reconstruction Previous work (Liao, Sapiro, 2007):   { } ∑ ∑ 2 ~ 2 α = δ − + µ α + α − * * *   D , , arg min R D E f f g f j j j j 2 α  0 2  D, , f patch j patch j f Patch-wise sparse coding of CT image . • Online learning from noisy data. • Very nice results on geometric images under severe • angular subsampling. Drawbacks: Data fidelity term in the sinogram domain. • No reference to statistical model of the noise. • Sparse coding thresholds not treated. • Computer Science dept. Ph.D. Talk, Apr. 2012