Source Reconstruction for 3D Bioluminescence Tomography with Sparse regularization

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1 1/33 Source Reconstruction for 3D Bioluminescence Tomography with Sparse regularization Xiaoqun Zhang Department of Mathematics/Institute of Natural Sciences, Shanghai Jiao Tong University International Conference on Immage Processing: Theory, Methods and Applications

2 Outline Problem Description Reconstruction based on Gaussian Noise model 1 Reconstruction based on Poisson Noise model 2 Conclusions and future work 1 Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, Source Reconstruction for Spectrally-resolved Bioluminescence Tomography with Sparse A priori Information, Optical express, X. Zhang, Y. Lu, T. F. Chan, A novel sparsity reconstruction method from Poisson data for bioluminescence tomography, submit 2/33

3 Problem description Bioluminescent Imaging system Powerful tumor detection/monitoring technique for in vivo Imaging. Extremely low backgrounds, high sensitivity and relatively simple instrumentation Tissue is a medium exhibiting both scattering and absorption properties. Amount of light is proportional to number of cells producing it. BLT reconstruction: reconstruct sources from observed photons intensity on the surfaces Surface intensity depends on: Source depth Source shape and brightness Surface shape (curvature) Wavelength Figure: By the Courtesy of XENOGEN 3/33

4 Problem description 4/33 Photons Diffusion Model For x Ω R 3, λ: wavelength, let Φ(x, λ): photon flux density S(x, λ): source energy density Steady-state diffusion equation (γ Φ ) µ a Φ+S = 0, ( x Ω) where γ(x,λ) is diffusion coefficient and µ a (x,λ) is absorbtion coefficient Robin boundary condition Φ+cγ ( v(x) Φ ) =0 (x Ω) where v: unit outer normal on Ω; c: parameter Measurements: the photon density on the body surface Ω for discretized λ i Q(x,λ i ) = Φ(x,λ i) c = γ ( v Φ(x,λ i ) ) (x Ω)

5 Problem description 5/33 Inverse Problem: Linear Relationship establishment Weak solution for Φ(x, λ) Using finite elements method to discretize the domain Ω, Φ(x, λ) and S(x, λ) Linear equation Au = f where u: unknown nodes vector for S in Ω, f is the measurable nodes vector for Φ on the boundary Ω. A is undetermined, highly ill-posed, depending on mesh sizes and shape functions.

6 Problem description 6/33 Noisy Model Gaussian Noise Model f = Au+N N: noise modeled as a Gaussian distribution Poisson Noise Model f i = Poisson((Au) i ) P(f i (Au) i ) = (Au)f i i e(au) i f i!

7 Gaussian Noise model 7/33 Classical Methods for Gaussian noise model Classical Maximum Likelihood method (Least square) Tikhonov regularization 1 min u 2 Au f 2 s. t. D = {0 u C} 1 min u 2 Au f 2 + δ 2 u 2 s. t. D = {0 u < C}

8 Gaussian Noise model Sparsity as a priori information l 1 regularization: sparsity 1 min S 2 Au f 2 +µ u 1 s. t. D = {0 u C} where S 1 = i S i denotes the l 1 norm of the vector S. Algorithm: Bound constrained quasi-newton method (BLMVM) 3 using smoothed l 1 norm. 3 Reference: S. J. Benson and J. More, A limited-memory variable-metric algorithm for bound-constrained minimization, /33

Gaussian Noise model 9/33 Simulations settings Data simulated by Monte Carlo Simulation of photon diffusion Cube domain with a width of 15mm 15mm 15mm and

9 Gaussian Noise model 9/33 Simulations settings Data simulated by Monte Carlo Simulation of photon diffusion Cube domain with a width of 15mm 15mm 15mm and discretized by hexahedra based FEM Photon distribution observed only on the top surface of the cubic domain at three wavelengths λ 1 = 600nm,λ 2 = 650nm,λ 3 = 700nm

Gaussian Noise model 10/33 Observed photon distribution on one surface Photons 600nm 650nm 700nm 10 6

10 Gaussian Noise model 10/33 Observed photon distribution on one surface Photons 600nm 650nm 700nm Figure: Observed photon distribution of single source (located at ((0, 0, 0)) at the top surface.

11 Gaussian Noise model 11/33 Single Source Reconstruction No regularization l 2 regularization l 1 regularization Figure: Reconstruction for single source located at (0,0,0). First row: 10 6 photons; Second row: 10 4 photons

12 Gaussian Noise model 12/33 Dual sources with homogeneous media No regularization l 2 regularization l 1 regularization Figure: Dual source BLT reconstructions when the real source central positions are at ( 3.0,0.0,3.0) and (3.0,0.0,3.0) with 10 4 photons for 600nm.

Dual source BLT reconstructions when the real source central

13 Gaussian Noise model 13/33 Dual sources with heterogeneous media No regularization l 2 regularization l 1 regularization Figure: Dual source BLT reconstructions when the real source central positions are at ( 3.0,0.0,3.0) and (3.0,0.0,3.0) with 10 4 photons for 600nm.

Gaussian Noise model Figure: Experimental BLT reconstructions with

0) (CT scanning), No regularization: (111.7,132.6,2.

14 Gaussian Noise model Figure: Experimental BLT reconstructions with mouse-shaped phantom. The actual source position is at (114.5, 131.0, 3.0) (CT scanning), No regularization: (111.7,132.6,2.7), l 2 regularization: (115.1,131.7,2.4), l 1 regularization: (114.7, 131.7, 2.9). 14/33 Experimental data Reconstruction GFP filter observation DsRed filter observation Volumetric mesh No regularization l 2 regularization l 1 regularization

15 Poisson Noise model 15/33 Poisson Noise model Random observation: f i Poisson(Au) i Maximum a-posteriori (MAP) estimation max u 0 p(u f) min u 0 (Au f logau) i Ω min u 0 D KL(f,Au) where D KL is the Kullback-Leihbler distance. Optimality condition KKT : F(u) λ = 0 λ i u i = 0 for i = 1,,m λ i 0 for i = 1,,m where F(u) = A 1 A ( f Au ) (1)

16 Poisson Noise model 16/33 Reconstruction models EM algorithm (Richardson-Lucy algorithm): fixed point algorithm on (A 1 A ( f ))u = 0 (2) Au u k+1 = u k A A 1 ( f Au k ) Sparsity A priori: l 1 norm regularization: l 0 norm regularization: min u 0 Φ(u) = D KL(f,Au)+µ u 1 min D KL(f,Au) s.t 0 < u 0 K u 0

17 Poisson Noise model Algorithms 17/33 Poisson l 1 minimization algorithm Forward backward splitting+em-l 1 Algorithm 4 u k+1 2 = (1 ω k )u k +ω k u k A A ( f 1 Ω Au k) u k+1 1 (A 1 Ω ) i = argmin u 0 2 (u k (u u k+1 2) 2 i +ω k µ u 1 ) i Ω 4 Brune, Sawatzky, Wubbeling, Losters, Burger, 2009

18 Poisson Noise model Algorithms SPIRAL-TAP 5 Let F(u) = D KL (f,au), F k (u) = F(u k )+(u u k ) T F(u k )+ α k 2 u u k 2. SPIRAL-TAP u k+1 = argmin u 0 F k (u)+µ u 1 Initialize Choose η > 1,σ (0,1),M Z +, 0 < α (α min,αmax), and initial solution u 0. Start iteration counter k = 0. Repeat choose α (α min,αmax) by Barzilai-Borwein method: α k = (uk u k 1 ) T 2 F(u k )(u k u k 1 ) u k u k u k+1 = argmin u>=0 2 u sk + µ α l u 1 α k ηα k, until u k+1 satisfies acceptance criteria : Φ(u k+1 ) max [k M]+,,kΦ(u k ) σα k 2 uk+1 u k 2 k k +1, until stopping criterion is satisfied 5 Harmany, Marcia, Willett, This is SPIRAL-TAP: Sparse Poisson Intensity 18/33

19 Poisson Noise model Algorithms 19/33 l 0 minimization with Orthogonal Matching Pursuit (OMP) Gaussian Noise model: OMP algorithm min u Au f 2 s. t. 0 < u 0 K (1) Initialize the residual r 0 = f, the index set Γ 0 = [ ]; (2) At step k, find the index j that solves the easy optimization problem j k = arg max i=1,...,n r k,a j (3) Add in the index set Γ k+1 = Γ k {jk } (4) Solve a least-squares problem to obtain a new signal estimate: (5) Repeat (2) until k = K. u k+1 = argmin u A Γk+1 u f 2

20 Poisson Noise model Algorithms 20/33 Iterative Poisson OMP Algorithm Iterative Poisson OMP min D KL(Au,f) s. t. 0 < u 0 K u 0 Initialize the index set Γ 0 = [ ], the set of all index I = {1,,n}; Outer iteration:t = 0 to t = T Set ˆΓ 0 = Γ t Inner iteration:k = 0 to k = K 1 Find û k+1 and i k+1 by solving the subproblem: [i k+1,û k+1 ] = arg min mind KL(f,A ˆΓk i {1,,n} u 0 {i} u) (3) Merge the new index: ˆΓ k+1 = ˆΓ k {i k+1 }. Find the K largest elements of û K and set the corresponding index set as Γ t+1. If Γ t+1 is the same as Γ t, stop; otherwise continue.

21 Poisson Noise model Numerical simulations Compressive sensing simulations Tests (m, n, d, K) Photons EM-L1 SPIRAL IterPOMP 1e6 K-L Dist RelErr 4.61e e e-3 (256,1024,40,4) Time e4 K-L Dist RelErr 4.70e e e-2 Time (256, 1024, 400, 4) (256, 1024, 40, 10) 1e6 K-L Dist RelErr 2.36e e e-3 Time e4 K-L Dist RelErr 2.55e e e-2 Time e6 K-L Dist RelErr 7.21e e e-3 Time e4 K-L Dist RelErr 5.96e e e-2 Time Table: Compressive sensing reconstruction based on 5 trials. For each test setting, we generate a Bernoulli matrix of size m n with d non zeros in each row. The randomly generated signals have K nonzeros of intensity 1e6 or 1e4. 21/33

22 Poisson Noise model Numerical simulations 22/33 BLT Reconstruction results with FEM simualted data True EM-l 1 SPIRAL POMP IterPOMP

23 Poisson Noise model Numerical simulations 23/33 Dual sources True EM-l 1 SPIRAL POMP IterPOMP

24 Poisson Noise model Numerical simulations 24/33 More sources setting tests True Sources Reconstructed Sources RelErr Time (0.5, 0.5, 0.5, 5.0e+6) (0.5, 0.5, 0.5, 5.0e+6) (3.5, -3.5, 2.5, 3.0e+6) (3.5, -3.5, 2.5, 3.0e+6) (-2.5, 2.5, 4.5, 1.0e+6) (-2.5, 2.5, 4.5, 1.0e+6) 5.50e (2.5, 1.5, 5.5, 2.0e+6) (2.5, 1.5, 5.5, 2.0e+6) (0.5, 0.5, 0.5, 5.0e+6) (0.5, 0.5, 0.5, 5.0e+6) (3.5, -3.5, 2.5, 3.0e+6) (3.5, -3.5, 2.5, 3.0e+6) (0.5, 0.5, 2.5, 2.0e+6) (0.5, 0.5, 1.5, 4.4e+6) 8.52e (-2.5, 2.5, 4.5, 1.0e+6) (-2.5, 2.5, 4.5, 1.0e+6) (0.5, 0.5, 0.5, 5.0e+4) (0.5, 0.5, 0.5, 5.0e+4) (3.5, -3.5, 2.5, 3.0e+4) (3.5, -3.5, 2.5, 3.0e+4) (-2.5, 2.5, 4.5, 1.0e+4) (-2.5, 2.5, 4.5, 9.9e+3) (2.5, 1.5, 5.5, 2.0e+4) (2.5, 1.5, 5.5, 2.0e+4) (0.5, 0.5, 0.5, 5.0e+4) (0.5, 0.5, 1.5, 5.6e+4) (3.5, -3.5, 2.5, 3.0e+4) (3.5, -3.5, 2.5, 3.0e+4) (0.5, 0.5, 2.5, 2.0e+4) (0.5, 0.5, -0.5, 1.4e+4) (-2.5, 2.5, 4.5, 1.0e+4) (-2.5, 2.5, 4.5, 1.0e+4) 2.72e e Table: Synthesized test examples with iterpomp algorithm for l 0 regularization model

25 Poisson Noise model Numerical simulations Test with larger K True Sources Reconstructed Sources RelErr Time (0.5, 0.5, 0.5, 5.0e+006) (0.5, 0.5, 0.5, 4.8e+006) (3.5, -3.5, 2.5, 3.0e+006) (3.5, -3.5, 2.5, 3.0e+006) (-2.5, 2.5, 4.5, 1.0e+006) (-2.5, 2.5, 4.5, 1.0e+006) (2.5, 1.5, 5.5, 2.0e+006) (2.5, 1.5, 5.5, 2.0e+006) (1.5, -0.5, 2.5, 1.2e+005) (0.5, 0.5, -1.5, 5.6e+004) (0.5, 0.5, 0.5, 5.0e+6) (0.5, 0.5, 1.5, 4.9e+6) (3.5, -3.5, 2.5, 3.0e+6) (3.5, -3.5, 2.5, 2.9e+6) (0.5, 0.5, 2.5, 2.0e+6) (0.5, 0.5, -0.5, 1.2e+6) (-2.5, 2.5, 4.5, 1.0e+6) (-2.5, 2.5, 4.5, 1.0e+6) (-0.5, 1.5, 0.5, 2.9e+5) (1.5, -0.5, 1.5, 7.1e+5) 3.32e e (0.5, 0.5, 0.5, 5.0e+4) (0.5, 0.5, 0.5, 5.0e+4) (3.5, -3.5, 2.5, 3.0e+4) (3.5, -3.5, 2.5, 3.0e+4) (4.5, -2.5, 2.5, 5.5e+2) (1.5, -0.5, 2.5, 2.2e+2) (-2.5, 2.5, 4.5, 1.0e+4) (-2.5, 2.5, 4.5, 9.9e+3) (2.5, 1.5, 5.5, 2.0e+4) (2.5, 1.5, 5.5, 2.0e+4) 1.21e (0.5, 0.5, 0.5, 5.0e+4) (0.5, 0.5, 1.5, 5.2e+4) (3.5, -3.5, 2.5, 3.0e+4) (3.5, -3.5, 2.5, 2.9e+4) (-0.5, 1.5, 0.5, 4.1e+3) (0.5, 0.5, 2.5, 2.0e+4) (0.5, 0.5, -1.5, 6.2e+3) (-2.5, 2.5, 4.5, 1.0e+4) (-2.5, 2.5, 4.5, 9.9e+3) 1.21e (1.5, -0.5, 1.5, 7.8e+3) 25/33

26 Poisson Noise model Numerical simulations 26/33 Monte-Carlo simulation recovery Figure: Triple sources with MC generated data. Left: true sources. Right: IterPOMP Recovered. True location and intensity: (-2.5, 2.5, 4.5, 1e+4), (0.5, 0.5, 0.5, 5e+4), (3.5, -3.5, 2.5, 3e+4). The l 0 algorithm has faithfully reconstructed the locations.

27 Poisson Noise model Numerical simulations 27/33 Monte-Carlo simulation recovery 7 x 104 True True Reconstructed Reconstructed Figure: Triple sources with MC generated data. True location and intensity: (-2.5, 2.5, 4.5, 1e+4), (0.5, 0.5, 0.5, 5e+4), (3.5, -3.5, 2.5, 3e+4). The l 0 algorithm has faithfully reconstructed the locations.

28 Poisson Noise model Numerical simulations 28/33 More test True sources Reconstructed RelErr Time (0.5, 0.5, 0.5, 5.0e+6) (0.5, 0.5, 0.5, 1.7e+6) (3.5, -3.5, 2.5, 3.0e+6) (3.5, -3.5, 2.5, 1.0e+6) 6.64e (-2.5, 2.5, 4.5, 1.0e+6) (-2.5, 2.5, 4.5, 3.4e+5) (0.5, 0.5, 0.5, 5.0e+4) (0.5, 0.5, 0.5, 1.7e+4) (3.5, -3.5, 2.5, 3.0e+4) (3.5, -3.5, 2.5, 1.0e+4) 6.63e (-2.5, 2.5, 4.5, 1.0e+4) (-2.5, 2.5, 4.5, 3.3e+3) (0.5, 0.5, 0.5, 5.0e+6) (0.5, 0.5, 0.5, 1.9e+6) (3.5, -3.5, 2.5, 3.0e+6) (3.5, -3.5, 2.5, 1.0e+6) (0.5, 0.5, 2.5, 2.0e+6) (0.5, 0.5, 3.5, 3.7e+5) 6.78e (-2.5, 2.5, 4.5, 1.0e+6) (-2.5, 2.5, 4.5, 3.4e+5) (0.5, 0.5, 0.5, 5.0e+4) (0.5, 0.5, 0.5, 1.6e+4) (3.5, -3.5, 2.5, 3.0e+4) (3.5, -3.5, 2.5, 9.9e+3) (0.5, 0.5, 2.5, 2.0e+4) (0.5, 0.5, 2.5, 7.4e+3) 6.68e (-2.5, 2.5, 4.5, 1.0e+4) (-2.5, 2.5, 4.5, 3.3e+3) Table: MC recovery examples with iterpomp algorithm.

29 Conclusions and future work 29/33 Conclusions and future work Sparsity improves location accuracy l 0 regularization is more robust than l 1 regularization. l 1 fails due to high correlation (ill-condition) of forward system matrix: higher order minimization algorithm for l 1 poisson model? Poisson OMP matching pursuit is efficient for small number of sources reconstruction problem. Parallelizable and need less parameters compared to l 1 minimization. Poisson OMP is a greedy algorithm. Need to improve the efficiency by a faster proximal solution Theoretical study on l 1 and l 0 reconstruction model for BLT problem. Test on real data, incomplete data with more complicated 3D body. Extension to total variation or framelets regularization together with better FEM discretization and more complicated 3D shapes.

GREEDY SIGNAL RECOVERY REVIEW

GREEDY SIGNAL RECOVERY REVIEW DEANNA NEEDELL, JOEL A. TROPP, ROMAN VERSHYNIN Abstract. The two major approaches to sparse recovery are L 1-minimization and greedy methods. Recently, Needell and Vershynin