Abstract

We propose and study a reconstruction method for photoacoustic tomography (PAT) based on total generalized variation (TGV) regularization for the inversion of the slice-wise 2D-Radon transform in 3D. The latter problem occurs for recently-developed PAT imaging techniques with parallelized integrating ultrasound detection where projection data from various directions is sequentially acquired. As the imaging speed is presently limited to 20 seconds per 3D image, the reconstruction of temporally-resolved 3D sequences of, e.g., one heartbeat or breathing cycle, is very challenging and currently, the presence of motion artifacts in the reconstructions obstructs the applicability for biomedical research. In order to push these techniques forward towards real time, it thus becomes necessary to reconstruct from less measured data such as few-projection data and consequently, to employ sophisticated reconstruction methods in order to avoid typical artifacts. The proposed TGV-regularized Radon inversion is a variational method that is shown to be capable of such artifact-free inversion. It is validated by numerical simulations, compared to filtered back projection (FBP), and performance-tested on real data from phantom as well as in-vivo mouse experiments. The results indicate that a speed-up factor of four is possible without compromising reconstruction quality.

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2019 (5)

I. Steinberg, D. M. Huland, O. Vermesh, H. E. Frostig, W. S. Tummers, and S. S. Gambhir, “Photoacoustic clinical imaging,” Photoacoustics 14, 77–98 (2019).
[Crossref]

R. Nuster and G. Paltauf, “Comparison of piezoelectric and optical projection imaging for three-dimensional in vivo photoacoustic tomography,” J. Imaging 5(1), 15 (2019).
[Crossref]

P. Theeda, P. U. P. Kumar, C. S. Sastry, and P. V. Jampana, “Reconstruction of sparse-view tomography via preconditioned Radon sensing matrix,” J. Appl. Math. Comput. 59(1-2), 285–303 (2019).
[Crossref]

J. Schwab, S. Antholzer, and M. Haltmeier, “Deep null space learning for inverse problems: convergence analysis and rates,” Inverse Problems 35(2), 025008 (2019).
[Crossref]

R. Huber, G. Haberfehlner, M. Holler, G. Kothleitner, and K. Bredies, “Total generalized variation regularization for multi-modal electron tomography,” Nanoscale 11(12), 5617–5632 (2019).
[Crossref]

2018 (4)

Y. E. Boink, M. J. Lagerwerf, W. Steenbergen, S. A. van Gils, S. Manohar, and C. Brune, “A framework for directional and higher-order reconstruction in photoacoustic tomography,” Phys. Med. Biol. 63(4), 045018 (2018).
[Crossref]

Y. Wang, T. Lu, J. Li, W. Wan, W. Ma, L. Zhang, Z. Zhou, J. Jiang, H. Zhao, and F. Gao, “Enhancing sparse-view photoacoustic tomography with combined virtually parallel projecting and spatially adaptive filtering,” Biomed. Opt. Express 9(9), 4569–4587 (2018).
[Crossref]

G. Wissmeyer, M. A. Pleitez, A. Rosenthal, and V. Ntziachristos, “Looking at sound: optoacoustics with all-optical ultrasound detection,” Light: Sci. Appl. 7(1), 53 (2018).
[Crossref]

M. W. Schellenberg and H. K. Hunt, “Hand-held optoacoustic imaging: A review,” Photoacoustics 11, 14–27 (2018).
[Crossref]

2017 (6)

G. Paltauf, P. Hartmair, G. Kovachev, and R. Nuster, “Piezoelectric line detector array for photoacoustic tomography,” Photoacoustics 8, 28–36 (2017).
[Crossref]

J. Bauer-Marschallinger, K. Felbermayer, and T. Berer, “All-optical photoacoustic projection imaging,” Biomed. Opt. Express 8(9), 3938 (2017).
[Crossref]

J. Adler and O. Oktem, “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Problems 33(12), 124007 (2017).
[Crossref]

F. Knoll, M. Holler, T. Koesters, R. Otazo, K. Bredies, and D. K. Sodickson, “Joint MR-PET reconstruction using a multi-channel image regularizer,” IEEE Trans. Med. Imaging 36(1), 1–16 (2017).
[Crossref]

M. Schloegl, M. Holler, A. Schwarzl, K. Bredies, and R. Stollberger, “Infimal convolution of total generalized variation functionals for dynamic MRI,” Magn. Reson. Med. 78(1), 142–155 (2017).
[Crossref]

S. M. Spann, K. S. Kazimierski, C. S. Aigner, M. Kraiger, K. Bredies, and R. Stollberger, “Spatio-temporal TGV denoising for ASL perfusion imaging,” NeuroImage 157, 81–96 (2017).
[Crossref]

2016 (3)

J. Duan, W. Lu, C. Tench, I. Gottlob, F. Proudlock, N. N. Samani, and L. Bai, “Denoising optical coherence tomography using second order total generalized variation decomposition,” Biomed. Signal Proces. 24, 120–127 (2016).
[Crossref]

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908–8940 (2016).
[Crossref]

L. V. Wang and J. Yao, “A practical guide to photoacoustic tomography in the life sciences,” Nat. Methods 13(8), 627–638 (2016).
[Crossref]

2015 (2)

Y. Dong, T. Görner, and S. Kunis, “An algorithm for total variation regularized photoacoustic imaging,” Adv. Comput. Math. 41(2), 423–438 (2015).
[Crossref]

C. Langkammer, K. Bredies, B. A. Poser, M. Barth, G. Reishofer, A. P. Fan, B. Bilgic, F. Fazekas, C. Mainero, and S. Ropele, “Fast quantitative susceptibility mapping using 3D EPI and total generalized variation,” NeuroImage 111, 622–630 (2015).
[Crossref]

2014 (5)

S. Niu, Y. Gao, Z. Bian, J. Huang, W. Chen, G. Yu, Z. Liang, and J. Ma, “Sparse-view x-ray CT reconstruction via total generalized variation regularization,” Phys. Med. Biol. 59(12), 2997–3017 (2014).
[Crossref]

K. Bredies and M. Holler, “Regularization of linear inverse problems with total generalized variation,” J. Inverse Ill-pose P. 22(6), 871–913 (2014).
[Crossref]

C. Zhang, Y. Zhang, and Y. Wang, “A photoacoustic image reconstruction method using total variation and nonconvex optimization,” BioMed. Eng. OnLine 13(1), 117 (2014).
[Crossref]

R. Nuster, P. Slezak, and G. Paltauf, “High resolution three-dimensional photoacoustic tomography with CCD-camera based ultrasound detection,” Biomed. Opt. Express 5(8), 2635 (2014).
[Crossref]

J. Xia, J. Yao, and L. V. Wang, “Photoacoustic tomography: principles and advances,” Electromagn. Waves (Camb.) 147, 1–22 (2014).

2013 (3)

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” SIAM J. Sci. Comput. 35(3), B644–B665 (2013).
[Crossref]

T. Valkonen, K. Bredies, and F. Knoll, “Total generalized variation in diffusion tensor imaging,” SIAM J. Imaging Sci. 6(1), 487–525 (2013).
[Crossref]

M. Kerschnitzki, P. Kollmannsberger, M. Burghammer, G. N. Duda, R. Weinkamer, W. Wagermaier, and P. Fratzl, “Architecture of the osteocyte network correlates with bone material quality,” J. Bone Miner. Res. 28(8), 1837–1845 (2013).
[Crossref]

2012 (1)

L. V. Wang and S. Hu, “Photoacoustic tomography: In vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012).
[Crossref]

2011 (7)

P. Beard, “Biomedical photoacoustic imaging,” Interface Focus 1(4), 602–631 (2011).
[Crossref]

L. Yao and H. Jiang, “Photoacoustic image reconstruction from few-detector and limited-angle data,” Biomed. Opt. Express 2(9), 2649–2654 (2011).
[Crossref]

F. Knoll, K. Bredies, T. Pock, and R. Stollberger, “Second order total generalized variation (TGV) for MRI,” Magn. Reson. Med. 65(2), 480–491 (2011).
[Crossref]

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys. 38(S1), S117–S125 (2011).
[Crossref]

L. Ritschl, F. Bergner, C. Fleischmann, and M. Kachelrieß, “Improved total variation-based CT image reconstruction applied to clinical data,” Phys. Med. Biol. 56(6), 1545–1561 (2011).
[Crossref]

Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Phys. Med. Biol. 56(18), 5949–5967 (2011).
[Crossref]

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40(1), 120–145 (2011).
[Crossref]

2010 (2)

K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM J. Imaging Sci. 3(3), 492–526 (2010).
[Crossref]

R. Nuster, G. Zangerl, M. Haltmeier, and G. Paltauf, “Full field detection in photoacoustic tomography,” Opt. Express 18(6), 6288 (2010).
[Crossref]

2009 (3)

M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imaging 28(11), 1727–1735 (2009).
[Crossref]

J. Provost and F. Lesage, “The application of compressed sensing for photo-acoustic tomography,” IEEE Trans. Med. Imaging 28(4), 585–594 (2009).
[Crossref]

G. Paltauf, R. Nuster, and P. Burgholzer, “Weight factors for limited angle photoacoustic tomography,” Phys. Med. Biol. 54(11), 3303–3314 (2009).
[Crossref]

2008 (1)

P. Burgholzer, H. Gruen, R. Nuster, G. Paltauf, and M. Haltmeier, “Model-based time reversal method for photoacoustic imaging of heterogeneous media,” J. Acoust. Soc. Am. 123(5), 3184 (2008).
[Crossref]

2007 (3)

G. Paltauf, R. Nuster, M. Haltmeier, and P. Burgholzer, “Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors,” Inverse Problems 23(6), S81–S94 (2007).
[Crossref]

P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Problems 23(6), S65–S80 (2007).
[Crossref]

P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E 75(4), 046706 (2007).
[Crossref]

2006 (1)

E. Y. Sidky, C.-M. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

2005 (2)

P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier, and O. Scherzer, “Thermoacoustic tomography with integrating area and line detectors,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 52(9), 1577–1583 (2005).
[Crossref]

B. T. Cox and P. C. Beard, “Fast calculation of pulsed photoacoustic fields in fluids using k-space methods,” J. Acoust. Soc. Am. 117(6), 3616–3627 (2005).
[Crossref]

2001 (1)

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46(7), 1863–1872 (2001).
[Crossref]

2000 (2)

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22(2), 503–516 (2000).
[Crossref]

G. H. Golub and H. A. van der Vorst, “Eigenvalue computation in the 20th century,” J. Comput. Appl. Math. 123(1-2), 35–65 (2000).
[Crossref]

1997 (1)

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik 76(2), 167–188 (1997).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1-4), 259–268 (1992).
[Crossref]

Adler, J.

J. Adler and O. Oktem, “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Problems 33(12), 124007 (2017).
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Figures (9)

Fig. 1.
Fig. 1. 3D model of synthetic pressure distribution.
Fig. 2.
Fig. 2. (a)–(f) Maximum amplitude projections along the $z$ axis of the reconstructions of numerical phantom data containing some thin vessel-like structures, two ellipsoids and a ramp. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$. All images share the same colormap, see (g).
Fig. 3.
Fig. 3. (a)–(h) Single slice of the reconstructions of numerical phantom data containing some thin vessel-like structures, two ellipsoids and a ramp. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$. All images share the same colormap, see (i).
Fig. 4.
Fig. 4. (a) Schematic showing the coordinate systems relative to the sample $(x,y,z)$ and relative to the experimental setup $(s,r,z)$. During wave pattern image acquisition the sample is rotated by an angle $\varphi$ about the $z$-axis. (b) Image acquisition with the camera-based setup. The sample is located above the field of view (FOV) of the camera. The data structure is a snap-shot of the acoustic field at time $T$ and sample orientation $\varphi$.
Fig. 5.
Fig. 5. (a)–(d) Maximum amplitude projections along the $z$-axis of the phantom sample containing black human hair loops and black polystyrene microspheres. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$. All images share the same colormap, see (e).
Fig. 6.
Fig. 6. (a)–(d) Single slice of the phantom sample containing black human hair loops and black polystyrene microspheres. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$. All images share the same colormap, see (e).
Fig. 7.
Fig. 7. (a)–(d) Histograms of the 3D reconstructions of the phantom sample containing black hair loops and black polystyrene microspheres. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$.
Fig. 8.
Fig. 8. (a)–(d) Maximum amplitude projections along the $z$-axis of the reconstruction of the in vivo measurements of the hind leg of a mouse. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$. All images share the same colormap, see (e).
Fig. 9.
Fig. 9. (a)–(d) Close up of a single slice of the reconstruction of the in vivo measurements of the hind leg of a mouse. The reconstruction method is indicated below each image, together with the choice of the regularization parameter $\mu$. All images share the same colormap, see (e). The detail visible in this images is visible in the upper center of the images in Fig. 8.

Tables (4)

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Table 1. Computation times for direct and variational Radon transform inversion of numerical phantom data containing some thin vessel-like structures, two ellipsoids and a ramp. The size of the computation domain is 281 × 281 × 40 voxels.

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Table 2. PSNR and SSIM values for the reconstructions obtained from numerical phantom data containing some thin vessel-like structures, two ellipsoids and a ramp, with respect to the ground truth.

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Table 3. Computation times for direct and variational Radon transform inversion of phantom sample data containing black human hair loops and black polystyrene microspheres. The reconstruction space P has the dimensions 571 × 571 × 91 .

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Table 4. Mean value and standard deviations for various reconstructions of the phantom sample.

Equations (44)

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R [ p T ] ( s , φ , z ) = 2 π λ P B d n d p R [ p T ] ( s , φ , z )
R [ p T ] ( s , φ , z ) := L L p T ( s ω ( φ ) + r ω ( φ ) + ( 0 , 0 , z ) ) d r .
2 p t 2 ( t , x ) = c s 2 Δ p ( t , x ) ,
R [ Δ p ] ( s , φ , z ) = Δ s , z R [ p ] ( s , φ , z ) ,
2 t 2 R [ p ] ( t , s , φ , z ) = c s 2 Δ s , z R [ p ] ( t , s , φ , z ) ,
f ( s , φ , z ) R [ p 0 ] ( s , φ , z ) = F s , z 1 [ F s , z [ 2 R [ p T ] ] cos ( c s | | T ) ] ( s , φ , z ) ,
min p 0 μ 2 R [ p 0 ] f 2 + R α ( p 0 ) ,
TV ( p ) = Ω | p | d x .
TGV α 2 ( p ) = inf v α 1 Ω | p v | d x + α 0 Ω | E ( v ) | d x .
min p 0 μ 2 R [ p 0 ] f 2 + TGV α 2 ( p 0 ) .
min u H 1 F ( u ) + G ( A u ) .
min u dom F sup ξ dom G L ( u , ξ ) , L ( u , ξ ) := ξ , A u + F ( u ) G ( ξ ) ,
G ( ξ ) = sup u H ξ , u G ( u ) ,
prox σ F ( u 0 ) = arg min u H 1 u u 0 2 2 + σ F ( u ) ,
P := R N x × N x × N z , V := ( R 3 ) N x × N x × N z , W := ( R 6 ) N x × N x × N z .
p P 2 = x = 0 N x 1 y = 0 N x 1 z = 0 N z 1 | p x , y , z | 2 , f S 2 = s = 0 N s 1 φ = 0 N φ 1 z = 0 N z 1 | f s , φ , z | 2 ,
min p P μ 2 R p f 2 + TGV α 2 ( p ) ,
v 1 := x = 0 N x 1 y = 0 N x 1 z = 0 N z 1 | v x , y , z | ,
TGV α 2 ( p ) = min q V α 1 p q 1 + α 0 E q 1  for all  p P .
min p P , q V μ 2 R p f 2 + α 1 p q 1 + α 0 E q 1 .
F ( p , q ) := 0 , G ( g , v , w ) := μ 2 g f 2 + α 1 v 1 + α 0 w 1 .
A ( p , q ) := ( R p , p q , E q ) .
A ( g , v , w ) = ( R g div v , div w v ) .
prox τ F ( p , q ) = ( p , q ) ,
G ( g , v , w ) = ( μ 2 f 2 ) ( g ) + ( α 1 1 ) ( v ) + ( α 0 1 ) ( w ) .
χ B ( ζ ) = { 0 , if  ζ B , , otherwise ,
( α 1 1 ) ( v ) = χ { α 1 } ( v ) , ( α 0 1 ) ( w ) = χ { α 0 } ( w ) ,
( μ 2 f 2 ) ( g ) = 1 2 μ g 2 + g , f .
prox σ G ( g , v , w ) = ( prox σ ( ( μ / 2 ) f 2 ) ( g ) , prox σ ( α 1 1 ) ( v ) , prox σ ( α 0 1 ) ( w ) ) .
prox σ ( ( μ / 2 ) f 2 ) ( g ) = μ μ + σ ( g σ f ) ,
prox σ ( α 1 1 ) ( v ) = proj { v ¯ V :   v ¯ α 1 } ( v ) ,
prox σ ( α 0 1 ) ( w ) = proj { w ¯ W :   w ¯ α 0 } ( w ) ,
( proj { v ¯ V :   v ¯ α 1 } ( v ) ) x , y , z = { v x , y , z if  | v x , y , z | α 1 , α 1 | v x , y , z | v x , y , z if  | v x , y , z | > α 1 ,
( δ x + p ) x , y , z := { p x + 1 , y , z p x , y , z , if  x { 0 , , N x 2 } , 0 , otherwise ,
( δ x p ) x , y , z := { p x , y , z p x 1 , y , z , if  x { 1 , , N x 1 } , 0 , otherwise .
p := ( δ x + p , δ y + p , δ z + p ) ,
E q := ( δ x q 1 , δ y q 2 , δ z q 3 , δ x q 2 + δ y q 1 2 , δ x q 3 + δ z q 1 2 , δ y q 3 + δ z q 2 2 ) .
( δ x + , p ) x , y , z := { p x + 1 , y , z p x , y , z if  x { 1 , , N x 2 } , p 1 , y , z , if  x = 0 , p N x 1 , y , z , if  x = N x 1 ,
( δ x , p ) x , y , z := { p x , y , z p x 1 , y , z , if  x { 1 , , N x 2 } , p 0 , y , z , if  x = 0 , p N x 2 , y , z , if  x = N x 1.
div v = δ x , v 1 + δ y , v 2 + δ z , v 3 , div w = ( δ x + , w 1 + δ y + , w 4 + δ z + , w 5 , δ x + , w 4 + δ y + , w 2 + δ z + , w 6 , δ x + , w 5 + δ y + , w 6 + δ z + , w 3 ) ,
min p P μ 2 R p f 2 + α p 1 .
F ( p ) = 0 , G ( g , v ) = μ 2 g f 2 + α v 1 , A = ( R p , p ) ,
A ( g , v ) = R g div v ,
prox τ F ( p ) = p , prox σ G ( g , v ) = ( μ μ + σ ( g σ f ) , proj { v ¯ V :   v ¯ α } ( v ) ) ,