Adaptive Reconstruction Methods for Low-Dose Computed Tomography

Transcription

1 Adaptive Reconstruction Methods for Low-Dose Computed Tomography Joseph Shtok Ph.D. supervisors: Prof. Michael Elad, Dr. Michael Zibulevsky. Technion IIT, Israel, 011 Ph.D. Talk, Apr. 01

2 Contents of this talk Intro to Computed Tomography Scan model, Noise, Local reconstruction Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Then you get tired of me. Ph.D. Talk, Apr. 01

3 Short Intro to Computed Tomography And Assume an aardvark you have has scheduled a shiny new a head CT scanner for today. Ph.D. Talk, Apr. 01

4 Short Intro to Computed Tomography Sum contributions to from all incident rays x X-ray source λ 0 photons ( R ) * x = g ( x g θ ) dθ θ θ f (x) F g ( σ ) = σ θ g θ ( σ ) Detectors photon counts y l Line l gl = [R f ] = f ( x) dl l Radon (X-ray) transform l Sinogram values y g = log( l l λ 0 ) s θ Ph.D. Talk, Apr. 01

5 yl Noise in Low-Dose Reconstruction Accepted model for detector measurements (similar to one in CCD sensors): instance Y l Poisson( λ ) + Ν(0,σ ) ~ l n λ = λ e l 0 [R f ] l - ideal count Poor photon statistics due to low counts Electronic noise in the hardware Y l = Y l + σ Poisson( λ + σ n l n ) y var( ) = λ + σ instance l y l l n Z l = l l Anscombe( Y ) = Y instance z l var( z l ) =1 Large attenuation High integral value g l = f x) dl l y Low count ( l l High noise variance g 1 = I 0 e var(g l ) = y l Streak artifacts Ph.D. Talk, Apr. 01

6 Noise in Low-Dose Reconstruction Proection and logtransform + Ideal sinogram Stochastic datadependent noise FBP FBP FBP+prep MAP estimate Ideal inverse Radon Smoothed inverse Radon Adaptive Median Filtering (Hsieh, 98) Iterative statisticallybased PWLS Ph.D. Talk, Apr. 01

7 Problem of local reconstruction A point in the image draws a sine. Points outside the ROI contribute to its proections. ROI is not uniquely determined from the truncated data. Ph.D. Talk, Apr. 01

8 Problem of local reconstruction FBP reconstruction from zero-padded truncated proections Basic sinogram completion: duplicate the margins. Non-linear sine-based sinogram completion Ph.D. Talk, Apr. 01

9 Intro to Computed Tomography Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms Scan model, Noise, Local reconstruction General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Ph.D. Talk, Apr. 01

10 Error measure for CT reconstruction f reference image ~ f Basic error measure: Mean ( ) ϕ ( ~ f ) = f ( x) ~ f ( x) = f Square Error (MSE) x Problem: MSE can be reduced by blurring the image. reconstructed image Sharpness-promoting penalty: the gradient norm in below the gradient norm in. ~ ~ ~ ϕ ( f ) = f f + µ J f ( J ) + 1 f J = x f ~ f ~ should not fall ~ ~, J = f 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. 01

11 Supervised learning of adaptive processing tools y f Degraded photon counts V p Minimize the error measure w.r.t. parameters Learnable parameters p,q,r FBP recon High-quality reference CT images. Preprocessing Postprocessing ~ Ψ(p,q, r) = f U F V ( y ) + J image f F q r q p f µ J = x f U r ( J ) + f ~ ~, J = f x ~ f Error measure Output image - a function of p,q,r. Ψ Ph.D. Talk, Apr. 01

12 Intro to Computed Tomography Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms Scan model, Noise, Local reconstruction General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Ph.D. Talk, Apr. 01

13 Learned FBP filter for ROI reconstruction T ( g) = R ( κ g * FBP operator: ). κ Train the convolution kernel pursuit reconstruction goals. κ to Training obective: for ROI reconstruction: Truncated sinogram with completion Ψ ( κ ) = T ( g f ) f ( κ { 5 κ 1,..., κ } 1. Use -D kernel. g f, Q g f -Truncated sinogram after completion. Q - Binary mask, restriction to ROI. κ 1... κ 5 Reference. Filter the sinogram with radially-variant convolution kernel. AFBP reconstruction Ph.D. Talk, Apr. 01

14 ROI reconstruction Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP reconstruction.9 db AFBP reconstruction db Ph.D. Talk, Apr. 01

15 ROI reconstruction Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP reconstruction db AFBP reconstruction 9.63 db Ph.D. Talk, Apr. 01

16 ROI reconstruction Image size = 461 pixels. ROI radius = 34 pixels, Margin = 3 pixels. True ROI image FBP reconstruction db AFBP reconstruction db Ph.D. Talk, Apr. 01

17 Intro to Computed Tomography Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms Scan model, Noise, Local reconstruction General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Ph.D. Talk, Apr. 01

18 Sparse-Land model for signals The concept: natural signals admit a faithful representation using only few columns (atoms) from a dedicated overcomplete dictionary. Natural dictionaries: Wavelets, Haar functions, Discrete Cosines, Fourier. Dictionaries tailored to the specific family of signals: obtained via a training process. s = E( f) α ν k 0 ε Number of non-zeros is small Residual is small s + ν = Dα D α f Ph.D. Talk, Apr. 01

19 Sparse-Land model for signals Denoising technique (Elad, Aharon, 006): ~ Φ(D, f, α) = δ f f + µ α + 0 patch ~ f - Noisy image patch Dα E f 1. (K-SVD) Train a dictionary D Minimize Minimize w.r.t w.r.t along with sparse representations { α}. Compute the image estimate (closed-form solution). Sparse coding: Dictionary update: α = arg min α α f s. t. D, { α } Dα E State-of-the-art noise reduction. Adaptive d to current i = arg min image Dα or Etraining f set. d patch Uniform noise assumption. 0 f ε Variable d is the i-th column in D. Ph.D. Talk, Apr. 01

20 Application to CT reconstruction Previous work (Liao, Sapiro, 007): { * * * D, f, α } = arg min δ R f g~ + µ α + D, f, α 0 patch patch Dα E f Patch-wise sparse coding of CT image f. 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. Ph.D. Talk, Apr. 01

21 Application to CT reconstruction Our approach: 1. check data fidelity and perform sparse coding in the domain of noise-normalized raw data : { * * D,α, z} 1 z = arg min λ z ~ z + µ α + D,, z 0 1 α patch patch = arg min λ z ~ z + z patch ~ z = y + σ n ) + ( ~ * D α E ~ z T T = G + ~ D, D ( α) E E E Dα λz 1 patch = D 1 α E Solve for,α using K-SVD, but allow to use a different dictionary at restoration stage: D 1 ~ z D Ph.D. Talk, Apr. 01

22 Application to CT reconstruction * D D1,D D. Train a second dictionary optimized for image reconstruction using a designed error measure and pre-computed repersentations α : ~ D = arg min f T ΩG α + µ ( J J ) + Ω : z y g Reconstruction chain: Photon counts ~ y σ ( + n ) y~ Data CT image f z~ T { } E FBP reconstruction Small patches Sinogram g Sparse coding with D1 Representations Ω Improved data z~ α G, D 1 D Data restoration with D Ph.D. Talk, Apr. 01

23 Compared algorithms Adaptive Trimmed Mean (ATM) Filter Hsieh, 98. Extract M values from the neighborhood of a photon count. Remove α M extreme values and compute the average of the rest. are data-dependent; computed through M,α M( y βλ α m y l ) =, α( yl ) λ + = λ [ y δ ] l + Penalized Weighted Least Squares (PWLS) Elbakri, Fessler, 0. -nd order aproximation of a penalized log-likelihood expression for photon counts data: l. PWLS( y f ) = W ([R f ] g ) + λ ψ ( 1 l l l f p f ) k l p k N ( p) y l In our experience: depends highly on the parameters. Works quite well. Penalty weight. Controls Huber penalty variance-resolution tradeoff. (smoothed L1 norm). Ph.D. Talk, Apr. 01

24 Empirical results Thighs section FBP, 5.76 db ATM, 8.3 db PWLS, 8.90 db Sparse, 9.6 db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

25 Empirical results Thighs section FBP, 5.6 db ATM, 7.46 db PWLS, 8.6 db Sparse, 7.94 db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

26 Empirical results Same parameters, new anatomical region. Head section FBP, 9.84 db ATM, 9.83 db PWLS, 31.0 db Sparse, 3.36 db Recon. in [-170,50] HU Error images Ph.D. Talk, Apr. 01

27 Intro to Computed Tomography Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms Scan model, Noise, Local reconstruction General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Ph.D. Talk, Apr. 01

28 Learned shrinkage in a transform domain Convert to z-domain Analysis Shrinkage Synhtesis D + (pseudo-inverse) Scalar shrinkage functions Dictionary D Reconstruction Denoising by supression of small coefficients, which usually contain the noise. LUT D Examples of : Discrete Cosines, Wavelets, etc. Denosing by shrinkage of wavelet coeffs: Donoho & Johnston, The tool: Descriptive functions for descriptive dictionary. Denoising with learned shrinkage functions: Hel-Or and Shaked, 00. The tool: Learned functions for descriptive dictionary. Our goal: Solving non-linear inverse problems. The tool: Learned functions for learned dictionary in a look-ahead training. Ph.D. Talk, Apr. 01

29 Learned shrinkage in a transform domain Preparing the data Photon counts ~ ( y + σ n ) y~ Data z~ { } E Small patches Compare to Compare to referenve clean training T image data FBP reconstruction Obective 1: : best data image quality convert to sinogram p, Φ Ω train Φ Synthesis train S p LUT Pre-processing Φ p=, Φ = arg min gωφ ΩΦΨS E Ψ ~ p, arg min f T S ze ~ z+ µ ( J p, Φ P P J ~ ) + Ψ Analysis Ph.D. Talk, Apr. 01

30 Why not repeat the trick? Image shrinkage FBP recon Raw data shrinkage Post-processing with shrinkage functions, also trained by comparing to reference images. Difference made by the postprocessing : no image structure lost. ~ p, Φ = arg min f ΦS Ψ E fˆ + µ ( J J ) p, Φ P + Ph.D. Talk, Apr. 01

31 Empirical results Thighs section (male) FBP, 5.76 db ATM, 8.3 db PWLS, 8.90 db Shrinkage db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

32 Empirical results Thighs section FBP, 5.6 db ATM, 7.46 db PWLS, 8.6 db Shrinkage 8.94 db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

33 Empirical results Head section FBP, 9.84 db ATM, 9.83 db PWLS, 31.0 db Shrinkage 3.81dB Recon. in [-170,50] HU Error images Ph.D. Talk, Apr. 01

34 Alternative versions Optimizing for MSE introduces a blur into the image. Thighs section FBP, 7.70 db 5.76 db PWLS, db 8.90 db Shrinkage db db MSEoptimized versions Tuned by visual appearance Ph.D. Talk, Apr. 01

35 Effective dose reduction 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. ~ ~ ~ Error ( f, f ) = f f + µ ( J J ) + Normal X-ray dose range Low X-ray dose range Ph.D. Talk, Apr. 01

36 Contents of this talk Intro to Computed Tomography Scan model, Noise, Local reconstruction Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Ph.D. Talk, Apr. 01

37 Fusion over a smoothing parameter Scalar parameter Raw data Recon. algorithm Image 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 Artificial Neural Network (ANN) Decision rule: Use a regression to build one automatically, with a Neural network or Support Vector Regression. Ph.D. Talk, Apr. 01

38 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. Initial image Created with FBP versions from partial iterations Converged image Standard PWLS result Ph.D. Talk, Apr. 01

39 Empirical results Thighs section FBP 5.76 db FBP-ANN 30.6 db PWLS 8.90 db PWLS-ANN db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

40 Empirical results Thighs section FBP 5.6 db FBP-ANN 9.67 db PWLS 8.6 db PWLS-ANN db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

41 Empirical results Head section FBP 9.84 db FBP-ANN 33.41dB PWLS 31.0 db PWLS-ANN db Recon. in [-170,50] HU Error images Ph.D. Talk, Apr. 01

42 Intro to Computed Tomography Framework of adaptive reconstruction Adaptive FBP Sparsity-based sinogram restoration Learned shrinkage in a transform domain Performance boosting of existing algorithms Scan model, Noise, Local reconstruction General scheme of supervised learning Learned FBP filter for local reconstruction Adaptation of K-SVD to low-dose CT reconstruction Adaptation of the method to low-dose CT reconstruction Local fusion of multiple versions of the algorithm output Conclusions Ph.D. Talk, Apr. 01

43 Parade of proposed methods Thighs section Sparse 9.6 db Shrinkage db FBP-ANN 30.6 db PWLS-ANN db Recon. in [-0,350] HU Error images Ph.D. Talk, Apr. 01

44 Summary Adaptive methods can help improving CT reconstruction. FBP needs only a little help to allow truly local 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. Example-based training does not eopardize the image content (in the presented algorithms) and can be allowed for clinical use. Ph.D. Talk, Apr. 01

45 Thank you. Ph.D. Talk, Apr. 01