Robust Principle Component Analysis Based 4D Computed Tomography Hongkai Zhao Department of Mathematics, UC Irvine Joint work with H. Gao, J. Cai and Z. Shen

Motivation A spatio-temporal formulation using matrix model for 4D CT that explores the maximum temporal coherence of spatial structure among different phases to Improve the image quality, e.g., denoising. Detect motion/changes Reduce the radiation dose by avoiding redundant measurements of the common background structure at different phases.

Key points for RPCA-4DCT View spatio-temporal images as a matrix with the row dimension in space and the column dimension in time. Decompose the matrix as low rank + sparse + noise +… A dynamic scanning strategy that acquire complementary data in different phases minimizing redundant measurements of the common background structure.

The formulation (1) Form the spatio-temporal images as a matrix: j is the temporal (column) index each x j is an image. The measurement data Y is X-Ray transform of X with noise, where the transform matrix A j can be dynamically changing.

The formulation (2) Decompose the full 4DCT matrix into three parts, where X 1 is low rank, X 2 is sparse (in transformed domain), and N is the noise. If the noise is Gaussian, our RPCA-4DCT model where W is the framelet analysis operator W T W=I,

Dynamic scanning Reduce the radiation dose by acquiring complementary data in different phases minimizing redundant measurements of the common background structure.

Wavelet frame transform Function is represented by a tight frame in multi-resolution framework. Redundancy makes it robust to noise. Easy decomposition and synthesis, W T W=I Our framelet transform is composed of low pass filter: + high pass filters: which can capture sparsity in function, its first and second derivatives.

The optimization algorithm The optimization problem: Where Convex optimization Difficulties: ◦ non-smooth due to and ◦ non-separable due to and Key points: reduce the problem to a series of easy separable subproblems We use the following split Bregman/augmented Lagrangian iterative method:

Easy separable optimizations The solution is given by simple shrinkage: The solution is given by singular value thresholding (SVT):

The optimization algorithm Transform into separable subproblems by introducing intermediate variable: augmented Lagragian/split Bregmen (Goldstein and Osher, 09)

The optimization algorithm The second step can be solved by singular value thresholding (SVT): where The third step is given by the shrinkage formula with

Related work Regularization in space and time independently and locally, Local coherence in space and time is used. Nonlocal regularization. X. Jia,et al, “4D computed tomography reconstruction from few-projection data via temporal non-local regularization”, (2010) Robust PCA for video, face recognition … E. Candès, et al, “Robust principal component analysis?”, (2009) H. Ji,et al, “Robust video denoising using low rank matrix completion”, (2010). G. Liu,et al, “Robust subspace segmentation by low-rank representation”, (2010).

Low rank + Sparse Decomposition Low rank + Sparse Decomposition ◦ Matrix is of low rank. ◦ Observation matrix so that where is a sparse matrix supported on ◦ Recover both and from ◦ Principle component analysis (PCA) with many outliers --- Robust PCA [Candes, Li, Ma, Wright]. Low rank + Sparse Decomposition via Convex Optimization: Compute and by where

Differences Decomposition is in X rather than directly in the measurement data space Y, the tomographic data of X generated by some system matrix A which is ill-posed. Sparsity is enforced in transformed domain. Dynamic scanning with reduced measurements (radiation dose).

Tests Phantom 1 mimics a half respiratory cycle. The temporal variations consist of (1) the intensity increase of the top circle, (2) the vertical movement of two central circles, and (3) the horizontal movement of two ellipses (with a low contrast) apart of each other. Phantom 2 is to model the case with small temporal variations, which is even hardly seen by human eyes. It is based on a MRI brain image and the temporal variations consist of the horizontal movement of two ellipses (with a low contrast).

4D CT: Phantom 1 16 Caption. Phantom 1 for 4D CT. (a), (b) and (c) are the image X, the background of the image X 1 and the motion/change of the image X 2 respectively at Phase 1, i.e., X=X 1 +X 2. Similarly, (d), (e) and (f) correspond to X, X 1 and X 2 at Phase 16, and (g), (h) and (i) correspond to X, X 1 and X 2 at Phase 32.

4D CT: Phantom 1 (RPCA-4DCT Model) 17 Caption. Reconstructed images with RPCA-4DCT model for Phantom 1. (a), (b) and (c) are the total image X, the low-rank component X1 and the sparse component (in tight frames) X2 respectively at Phase 1, i.e., X=X1+X2. Similarly, (d), (e) and (f) correspond to X, X1 and X2 at Phase 16, and (g), (h) and (i) correspond to X, X1 and X2 at Phase 32.

4D CT: Phantom 1 (Other Models) 18 Caption. Reconstructed images with other various models for Phantom 1. (a), (b) and (c) are from “L2”, “TV” and “TV+TVt” respectively at Phase 1. Similarly, (d), (e) and (f) correspond to the above models at Phase 16, and (g), (h) and (i) correspond to the above models at Phase 32.

4D CT: Phantom 1 (Quantitative Comparison) 19

4D CT: Phantom 2 (Motion Detection) 20 Caption. Phantom 2 for 4D CT. (a), (b) and (c) are the image X, the background of the image X 1 and the motion/change of the image X 2 respectively at Phase 1, i.e., X=X 1 +X 2. Similarly, (d), (e) and (f) correspond to X, X 1 and X 2 at Phase 16, and (g), (h) and (i) correspond to X, X 1 and X 2 at Phase 32.

4D CT: Phantom 2 (RPCA-4DCT Model) 21 Caption. Reconstructed images with RPCA-4DCT model for Phantom 2. (a), (b) and (c) are the total image X, the low-rank component X1 and the sparse component (in tight frames) X2 respectively at Phase 1, i.e., X=X1+X2. Similarly, (d), (e) and (f) correspond to X, X1 and X2 at Phase 16, and (g), (h) and (i) correspond to X, X1 and X2 at Phase 32.

4D CT: Full Views 22 Caption. Reconstructed images with the RPCA-4DCT model for Phantom 2 with full 256 projections.

4D CT: Partial Views 23 Caption. Reconstructed images with the RPCA-4DCT model for Phantom 2 with stationary 32 projections.

4D CT: Dynamic Views 24 Caption. Reconstructed images with the RPCA-4DCT model for Phantom 2 with dynamic 32 projections.

25 4D CT: Full, Partial, and Dynamics Views

4D CT: Scanning Schemes (Quantitative Comparison) 26

More general model Key point: make low rank assumption valid in more general setup. sparsity in appropriate transform domain/basis.

Incorporate deformation/motion RASL model for small deformation (Peng, et al) where, and is the deformation field. For small deformation, one can linearize and still get a convex minimization problem

4D CT If has a left inverse, we have for small deformation The convex minimization problem becomes

Challenges Under-determinedness: does not have a left inverse. More ill-posed, i.e., decomposition is more non-unique. Large deformation.

Further improvement Physics and prior based constrained/regularized deformation field, e.g., rigid motion, incompressibility,.. Knowledge based decomposition by designing a proper weighting matrix Time coherence + Spatial coherence

References References: 1. H. Gao and H. Zhao, A fast forward solver of radiative transport equation, Transport Theory and Statistical Physics 38, H. Gao and H. Zhao, A multilevel and multigrid optical tomography based on radiative transfer equation, Proceedings of SPIE (Munich, Germany, 2009). 3. H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization, Optics Express 18, H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation Part 2: total variation and l1 data fidelity, Optics Express 18, (2010). 5. H. Gao, Y. Lin, G. Gulsen and H. Zhao, Fully linear reconstruction method of fluorescence yield and lifetime through inverse complex-source formulation, Optics Letter, 35, H. Gao, H. Zhao, W. Cong and G. Wang, Bioluminescence tomography with Guassian Prior, Optics Express, 1, G. Hao, J. Cai, Z. Shen and H. Zhao, Robust principal component analysis- based four-dimensional computed tomography, Physics in Medicine and Biology, 56, (2011) RTE-MG webpage:

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