2D X-Ray Tomographic Reconstruction From Few Projections

Transcription

1 2D X-Ray Tomographic Reconstruction From Few Projections Application of Compressed Sensing Theory CEA-LID, Thalès, UJF 6 octobre 2009

2 Introduction Plan 1 Introduction 2 Overview of Compressed Sensing Theory 3 Formulations of SR problem 4 Numerical results

3 Introduction Motivations and Objectives Applications of X-Ray imaging systems : Industrial non-destructive control Medical imaging Security control Main interests by using few X-ray projections : Acceleration of data aquisation Reduction of dose

4 Introduction Radon transform Radon Transform : Rf(ρ, θ) = L f(x, y)dl Analytical tomographic reconstruction : f = R 1 Rf

5 Introduction Projection-slice theorem Frequential intepretation of Radon transform : F ρ (Rf)(ω, θ) Rf(ρ, θ)e 2πiρω dρ = ˆf(ω cos θ, ω sin θ) IR Starshaped incomplete frequency domain Reconstruction is an ill-posed problem!

6 Introduction Formulation of reconstruction problem Representation of image by discret basis : f(x) = j f jφ j (x), then : Rf(ρ, θ) = Rφ j (ρ, θ)f j j In real applications, Rf(ρ, θ) is known only for a limited position of (ρ i, θ i ). This gives : A f = b where, Projection matrix A i,j = Rφ j (ρ i, θ i ), A P M N 2 Discret image f = [f 1... f N ] Sinogram b i = Rf(ρ i, θ i ), i = 1... P M Tomographic reconstruction becomes a linear inverse problem.

7 Introduction Regularization Classical approaches : Reconstruction without regularization : Examples : FBP, EM, ART. min f ker(a) Af b 2 f = A b (1) Introducing regularization term g(f) and and limit the convex admissible set of f : 1 min f C 2 Af b λg(f) (2) Examples : Edge-preserving reconstruction, POCS New approach : Regularization by sparsity

8 Overview of Compressed Sensing Theory Plan 1 Introduction 2 Overview of Compressed Sensing Theory 3 Formulations of SR problem 4 Numerical results

9 Overview of Compressed Sensing Theory Notion of sparsity Definition (S-sparsity) A signal x IR N is S-sparse if it has only S non zero entries. Definition (S-biggest term approximation) x S is the S-biggest term approximation of x by retaining only first S biggest (in absolute value) entries. Hypothesis (Fundamental) A good solution is a sparse solution, the best solution is the sparsest solution.

10 Overview of Compressed Sensing Theory Sparsest solution : l 0 and l 1 norms Find the sparsest solution by l 0 semi-norm : (P 0 ) is NP hard, convex relaxation : Important questions : Uniqueness of (P 0 )? min x x 0 s.t. Ax = b (P 0 ) min x x 1 s.t. Ax = b (P 1 ) Equivalence between (P 0 ) and (P 1 )? Answered by Compressed Sensing (CS) Theory

11 Overview of Compressed Sensing Theory System matrix characterization Definition (Restricted Isometry Property) RIP of A is the smallest δ S 0 s.t. for all S-sparse signal x : (1 δ S ) x 2 2 Ax 2 2 (1 + δ S ) x 2 2 (3) i.e., all S-submatrix of A act like isometry. Observations : δ s bounds the singular values of S-submatrix Small δ s, nice behaviour of A If δ 2S = 1, then there exists some 2S submatrix A 2S noninjective.

12 Overview of Compressed Sensing Theory P 0 and P 1 equivalence Theorem (Perfect reconstruction) Given that 2S-RIP of A is δ 2S 2 1, for a S-sparse true signal x, P 1 solution x is exactly x. Theorem (Almost Perfect reconstruction) Under the same hypothesis, for a general true signal x, P 1 solution x obeys : x x 1 C x x S 1 and x x 2 C x x S 1 / S where x S is the S biggest term approximation of x.

13 Overview of Compressed Sensing Theory Application of CS to X-Ray CT State-of-art CS theory has two parts : Sensing matrix design (satisfying RIP) High precision recovery algorithm (P 1 ) X-Ray CT system matrix is deterministic X-Ray transform! No sensing matrix design problem Image reconstruction by solving (P 1 ) Sparse Reconstruction (SR) approach

14 Formulations of SR problem Plan 1 Introduction 2 Overview of Compressed Sensing Theory 3 Formulations of SR problem 4 Numerical results

15 Formulations of SR problem Extentions of CS Prompts more sparsity by sparsifying transform Ψ : Synthesis SR : Analysis SR : min α α 1 s.t. AΨ α = b (P S ) min x Ψ(x) 1 s.t. Ax = b (P A )

16 Formulations of SR problem Synthesis SR Represent x in a sparse way under basis Ψ : min α α 1 s.t. AΨ α = b (4) Reconstruction of representation coefficient : α, synthesis by x = Ψ α. Frequential basis : DCT Multi-resolution basis : Wavelet Spatial basis : Gradient, Laplacian

17 Formulations of SR problem Analysis SR Use a (maybe noninvertible) (non)linear transform Ψ(x) : min x Ψ(x) 1 s.t. Ax = b (P A ) Examples : T V transform : T V (x) = i,j (x i+1,j x i,j ) 2 + (x i,j+1 x i,j ) 2 (5) Gradient transform (IR N IR 2N ) : Ψ(x) 1 = i,j x i,j+1 x i,j + x i+1,j x i,j (6)

18 Numerical results Plan 1 Introduction 2 Overview of Compressed Sensing Theory 3 Formulations of SR problem 4 Numerical results

19 Numerical results Test phantoms (d) (e) Fig.: test images. (a) Shepp-Logan phantom, (b) a brain slice Objective : Reduce the number of projections. IPM(Interior Point Method) solver is used for Synthesis and Analysis SR.

20 Numerical results (P S ) and (P A ) reconstructions (a) (b) Fig.: Reconstruction using : (a) 20 projections by (P A ) with gradient transform, (b) 45 projections by (P S ) with Wavelet dictionary

21 Numerical results (P S ) and (P A ) reconstructions 1.0 Analysis by Gradient base Original image 0.8 Profile Fig.: Central profile of reconstruction (a)

22 Numerical results (P S ) and (P A ) reconstructions 1.0 Synthesis by Wavelet Original image Fig.: Central profile of reconstruction (b)

23 Numerical results Monte-Carlo test of projection matrix 64 Relative reconstruction error 0.42 Number of projections Sparsity 0.00 Fig.: Monte-Carlo test of Fan-beam siddon projection matrix

24 Numerical results Conclusions and perspectives Sparse reconstruction is feasible for both piece-wise constant image and natural image Linear gradient transform is as efficient as the non linear TV transform, and wavelet basis can be used to reduce projections on natural image. Study the stability wrt noise Use more efficient sparsifying transforms