COMPRESSIVE OPTICAL DEFLECTOMETRIC TOMOGRAPHY

Transcription

1 COMPRESSIVE OPTICAL DEFLECTOMETRIC TOMOGRAPHY Adriana González ICTEAM/UCL March 26th, 214 1

2 ISPGroup - ICTEAM - UCL Université catholique de Louvain, Louvain-la-Neuve, Belgium. ISP Group 4 Professors 17 researchers 12 PhD students Compressed Sensing Group Prof. Laurent Jacques Prof. Christophe De Vleeschouwer Dr. Prasad Sudhakar Kévin Degraux 2

3 Outline 1 Optical Deflectometric Tomography 2 Compressiveness in RIM Reconstruction 3 Compressiveness in Acquisition 3

4 Outline 1 Optical Deflectometric Tomography 2 Compressiveness in RIM Reconstruction 3 Compressiveness in Acquisition 4

5 Optical Deflectometric Tomography Interest ODT Optical characterization of (transparent) objects Tomographic Imaging Modality Measures light deviation caused by the difference in the object refractive index p θ e 2 α(τ, θ) Deviated Light Rays τ Incident Light Rays n(r) θ t θ e 1 5

6 Schlieren Deflectometer SLM (modulation by ' i ) Uniform Light Source I s p x = f tan Optical axis x Lens 1 (f) Rotating Lens 2 Lens 3 object Intensity change Pinhole e 2 e 3 e 1 p Telecentric system y p CCD yp = sp, ϕi 6

7 Schlieren Deflectometer s p y p = s p, ϕ i 1 Compressiveness in RIM reconstruction ϕ sinusoidal pattern 4 shifted patterns ϕ 1, ϕ 2, ϕ 3, ϕ 4 4 measurements to recover α Assuming deflections at one point Objects RIM variation only on e 1 e 2 α, 2-D slices 7

8 Schlieren Deflectometer s p yp = sp, ϕi 2 Compressiveness in acquisition Deflections produced by several points Objects RIM variation also on e 3 α and β, 3-D volume M binary modulation patterns ϕ i to eliminate nonlinearities 8

9 Outline 1 Optical Deflectometric Tomography 2 Compressiveness in RIM Reconstruction 3 Compressiveness in Acquisition 9

10 Framework Joint work with Prof. Laurent Jacques and Prof. Christophe De Vleeschouwer from UCL and Dr. Philippe Antoine from Lambda-X Problem To reconstruct the refractive index map of transparent materials from light deflection measurements (α) under few orientations (θ) Assumption Objects are constant along the e 3 direction Deflections at only one point [1] A. González et al. itwist12 [2] P. Antoine et al. OPTIMESS 212 [3] A. González et al. IPI Journal (214) 1

11 Continuous facts Mathematical Model Eikonal equation R curved : r(s) d ds ( n d ds r(s)) = n Approximation small α R straight : r p θ = τ error < 1% (τ, θ) = sin(α) (τ, θ) = 1 n r R 2 ( n(r) pθ ) δ(τ r pθ ) d 2 r τ p θ Incident Light Rays e 2 α(τ, θ) Deviated Light Rays n(r) θ t θ e 1 Deflectometric Central Slice Theorem y(ω, θ) := R (τ, θ) e 2πiτω dτ = 2πiω n r n ( ω p θ ) n ( ω p θ ) : 2-D Fourier transform of n in Polar grid 11

12 Discrete Forward Model y = 2πi(δr)2 n r diag(ω (1),, ω (M) ) n y = DFn + η n R N ; Cartesian grid of N = N 2 pixels; sampling: δr y R M ; Polar grid of M = N τ N θ pixels; sampling: δτ, δθ D : 2πi(δr)2 n r diag(ω (1),, ω (M) ) C M M F C M N : Non-equispaced Fourier Transform (NFFT) [4] η C M : numerical computations, model discretization, model discrepancy, observation noise [4] J. Keiner et al. (29) 12

13 ODT vs. AT y = DFn + η Main difference: Operator D Without noise η classical tomographic model ỹ = D 1 y = Fn For η Not a classical tomographic model - η: AWGN D 1 η not homoscedastic 13

14 ODT vs. AT Observation: 1-D FT of sinograms along the τ direction Absorption Tomography ω y(ω,θ) θ ω 3 x x 1 3 Optical Deflection Tomography ω y(ω,θ) θ ω 14

15 Naive Reconstruction Methods Filtered Back Projection y = Φn + η = DFn + η Analytical method Solution ñ FBP : - Filtering the tomographic projections AT: ramp filter; ODT: Hilbert filter - Backprojecting in the spatial domain by angular integration Minimum Energy Reconstruction ñ ME = Φ y = Φ (ΦΦ ) 1 y ñ ME = arg min u 2 s.t. y = Φu u R N Problems: Noise Compressiveness M(N θ ) < N ill-posed problem Solution: Regularization 15

16 Sparsity prior Heterogeneous transparent materials with slowly varying refractive index separated by sharp interfaces TV and BV promote the perfect cartoon shape model Sparse gradient Small Total Variation norm n TV := n 2,1 16

17 Other priors Positive RIM n The object is completely contained in the image. Pixels in the border are set to zero in order to guarantee uniqueness of the solution. n δω = SOLUTION UNIQUENESS 17

18 TV-l 2 reconstruction and Noise TV-l 2 Reconstruction ñ TV l2 y = Φn + η = DFn + η = arg min u TV s.t. y Φu 2 ε, u, u Ω = u R N Noise Observation noise σ 2 obs Modeling error ray tracing with Snell law 1% Interpolation noise NFFT error (very small) 18

19 TV-l 2 reconstruction TV-l 2 Reconstruction ñ TV l2 ñ TV l2 y = Φn + η = DFn + η = arg min u TV s.t. y Φu 2 ε, u, u Ω = u R N = arg min u TV + ı C (Φu) + ı P (u) u R N Indicator function: ı X (x) = if x X ; + otherwise ı C and ı P are the indicator functions into the following convex sets: - C = {v C M : y v ε} - P = {u R N : u i if i int Ω; u i = if i Ω} Sum of 3 proper, lower semicontinuous, convex functions Reconstruction using CP algorithm [5] expanded in a product space [5] A. Chambolle and T. Pock. Journal of Mathematical Imaging and Vision. (211) 19

20 Reconstruction Algorithm Chambolle-Pock (CP) min x X F (Kx) + G(x) F, G: proper, lsc, convex; τσ K 2 < 1 v (k+1) = prox σf (v (k) + σk x (k) ) x (k+1) = prox τg (x (k) τk v (k+1) ) x (k+1) = 2x (k+1) x (k) min x X Proximal mapping f : proper, lsc, convex prox σf z = arg min z σf ( z) z z 2 2 e.g., prox σl1 z = SoftTh(z, σ) Conjugate function F (v) = max v v, v F ( v) t F j (K j x) + G(x) j=1 K = diag(k 1,, K t ) CP Product-Space Expansion v (k+1) ( (k) j = prox σf j v j + σk j x (k)), j {1, t} x (k+1) = prox τ t H(x(k) τ t t j=1 K i v (k+1) j ) x (k+1) = 2 x (k+1) x (k) 2

21 Results: TV l 2 vs. ME & FBP Compressiveness and noise robustness RSNR [db] TV L2 ME FBP N θ / 36 [%] RSNR [db] TV L2 2dB 5 TV L2 1dB ME 2dB ME 1dB 5 FBP 2dB FBP 1dB N θ / 36 [%] MSNR = 2 log 1 2 η 2 RSNR = 2 log 1 n 2 n ñ 2 21

22 Results: TV-l 2 vs. ME.12 No measurement noise (MSNR = ) N θ /36 = 25% ñ TV l2 : RSNR = 71dB ñ ME : RSNR = 13dB x

23 Results: TV-l 2 vs. ME.12 No measurement noise (MSNR = ) N θ /36 = 5% x ñ TV l2 : RSNR = 67dB ñ ME : RSNR = 5dB 2 23

24 Results: TV-l 2 vs. ME.12 MSNR = 2dB N θ /36 = 25% ñ TV l2 : RSNR = 39dB x ñ ME : RSNR = 13dB x

25 Results: TV l 2 Convergence Non-Adaptive: step-sizes constant along the iterations σ = τ =.9 K Adaptive: step-sizes σ and τ are updated according to the residuals of the algorithm [6] Adapt Non Adapt 5 45 x (k+1) x (k) / x (k) RSNR [db] # iter 15 Adapt Non Adapt # iter [6] T. Goldstein et al. Preprint (213) 25

26 Experimental Results Bundle of 1 fibers immersed in an optical fluid MSNR 1dB N θ = 6 N θ /36 = 17% Slices τ.4 3 x x TV L2 Expected y (mm) x (mm) x (mm) 26

27 Outline 1 Optical Deflectometric Tomography 2 Compressiveness in RIM Reconstruction 3 Compressiveness in Acquisition 27

28 Schlieren Deflectometer s p yp = sp, ϕi 28

29 Framework Work by Dr. Prasad Sudhakar and Prof. Laurent Jacques from UCL; Xavier Dubois, Dr. Luc Joannes and Dr. Philippe Antoine from Lambda-X Problem Objects RIM variation on e 1, e 2, e 3 Local curvature at every p is characterized by the deflection spectra s p (tan α, β) = s p (α, β) Deflection Spectra Only measured indirectly Sparse : smooth objects controlled distortions Objective To reconstruct the deflection map at all p using relatively few measurements (M) per orientation (θ) [7] P. Sudhakar et al. IEEE ICASSP 213 [8] P. Sudhakar et al. SampTA

30 Forward model Location p in object s p R l l = R L pixel p in CCD camera y i,p = s p, ϕ i M Modulation patterns ϕ i R l l = R L Φ = [ϕ T 1 ϕ T M ]T R M L y p = Φs p + n Challenges : - Find a sparse s p such that y p Φs p 2 ε; n 2 ε - Design of Φ for M < L 3

31 Sparsity k-sparse signals and sparsity basis 31

32 Sparse Recovery Sensing Basis Φ = Γ T Ω Γ R L L : sensing basis Γ Ω R L M : random (uniform) selection of M columns of Γ Objective : Find a sparse s p such that y p Φs p 2 ε Basis Pursuit Denoising (P1) Ω [L] := {1,, L} α p = arg min α p 1 s.t. y p ΦΨα p 2 ε ŝ p = Ψ α p α p R L (P1) succeeds if M = O ( µ 2 k log 4 (L) ) µ = L max 1 i,j L Γ j, ψ i Coherence between Γ and Ψ - 1 µ L : µ (Γ and Ψ less coherent) M - e.g. Fourier-Dirac are maximally incoherent µ = 1 32

33 Sensing Basis Compressiveness (M < L) Sensing basis Γ incoherent with sparsity basis Ψ Spread Spectrum Compressive Sensing [9]: random phase modulation of s p to make sensing and sparsity bases incoherent - Spread Spectrum matrix: M = diag(m) R L L, m i = 1 (e.g. Rademacher) Φ = Γ T ΩM y p = Γ T ΩMΨα p + n - For universal sensing bases ( Γ ij = const., e.g., Fourier, Hadamard) Successful recovery if M C ρk log 5 (L) with probability 1 O(N ρ ) Sensing matrix Γ needs to satisfy 3 criteria: - Randomness for optimal measurements - Binary sensing matrix entries to avoid non-linearities - Structured measurements for fast computations Γ = H : Hadamard basis H T M {±1} L L Physical constraints non-negative, real-valued sensing matrix entries Φ = 1 2 ( ) H T ΩM + 1 L 1 T L {, 1} M L [9] G. Puy et al. Journal on Adv. in Sig. Proc. (212) 33

34 Deflection spectrum reconstruction y = Φs p + n = 1 2 ( ) H T ΩM + 1 L 1 T L Ψα p + n Ψ : Daubechies 9 wavelet basis α p = arg min α p 1 s.t. y p ΦΨα p 2 ε; Ψα p ŝ p = Ψ α p α p R L α p = arg min α p 1 + ı C (ΦΨα p ) + ı P (Ψα p ) ŝ p = Ψ α p α p R L Indicator function: ı X (x) = if x X ; + otherwise ı C and ı P are the indicator functions into the following convex sets: - C = {v R M : y p v ε} - P = {u R L : u +} Sum of 3 proper, lower semicontinuous, convex functions Reconstruction using CP algorithm expanded in a product space (non-adaptive) 34

35 Results Lambda-X NIMO system (SLM 64 64) Compressiveness 35

36 Information from Deflection Spectra Knowing the center of the spectral figure we can: - Recover deflections β and α = atan( r f 1 ) for each θ RIM reconstruction 36

37 Information from Deflection Spectra Knowing the center of the spectral figure we can: - Describe an object surface 1% B A B A 4% 37

38 References 1 A. González et al. itwist12 2 P. Antoine et al. Proceedings of OPTIMESS A. González et al. To appear in IPI Journal (214) 4 J. Keiner et al. ACM TOMS (29) 5 A. Chambolle et al. Journal of Mathematical Imaging and Vision (211) 6 T. Goldstein et al. Preprint, arxiv: v1 (213) 7 P. Sudhakar et al. IEEE ICASSP P. Sudhakar et al. SampTA G. Puy et al. Journal on Adv. in Sig. Proc. (212) Thank you! 38