Abstract

A total variation minimization (TVM)-based finite element reconstruction algorithm for photoacoustic (PA) tomography is described in this paper. This algorithm is used to enhance the quality of reconstructed PA images with time-domain data. Simulations are conducted where different contrast levels between the target and the background, different noise levels, and different sizes and shapes of the target are studied in a 30mm diameter circular heterogeneous background. These simulated results show that the quality of the reconstructed images can be improved significantly due to the decreased sensitivity to noise effect when the TVM is included in the reconstruction algorithm. The enhancement is further confirmed using experimental data obtained from several phantom experiments and an in vivo animal experiment.

© 2011 Optical Society of America

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2011 (1)

K. Wang, E. Y. Sidky, M. A. Anastasio, A. A. Oraevsky, and X. Pan, “Limited data image reconstruction in optoacoustic tomography by constrained total variation minimization,” Proc. SPIE 7899, 78993U (2011).
[CrossRef]

2010 (1)

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010).
[CrossRef] [PubMed]

2009 (2)

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

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time-domain,” J. Opt. A 11, 085301 (2009).
[CrossRef]

2008 (1)

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (2)

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: recovery of optical absorption coefficient maps of heterogenous media,” Appl. Phys. Lett. 88, 231101 (2006).
[CrossRef]

H. Jiang, Z. Yuan, and X. Gu, “Spatially varying optical and acoustic property reconstruction using finite-element-based photoacoustic tomography,” J. Opt. Soc. Am. A 23, 878–888(2006).
[CrossRef]

2003 (1)

2002 (2)

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

G. Paltauf, J. Viator, S. Prahl, and S. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112, 1536–1544 (2002).
[CrossRef] [PubMed]

2001 (1)

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

2000 (1)

1996 (3)

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35, 3447–3458 (1996).
[CrossRef] [PubMed]

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227–238(1996).
[CrossRef]

D. C. Dobson and F. Santosa, “Recovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181–1198(1996).
[CrossRef]

1995 (2)

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[CrossRef] [PubMed]

P. M. van den Berg and R. E. Kleinmann, “A total variation enhanced modified gradient algorithm for profile reconstruction,” Inverse Probl. 11, L5–L10 (1995).
[CrossRef]

1994 (2)

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

D. C. Dobson and F. Santosa, “An image-enhancement technique for electrical impedance tomography,” Inverse Probl. 10, 317–334 (1994).
[CrossRef]

1993 (1)

D. C. Dobson, “Exploiting ill-posedness in the design of diffractive optical structures,” Proc. SPIE 1919, 248–257(1993).
[CrossRef]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

1991 (1)

S. R. Arridge, P. van der Zee, M. Cope, and D. Delpy, “Reconstruction methods for infrared absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

1986 (1)

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58381–430 (1986)
[CrossRef]

1980 (1)

A. Bayliss and E. Turkel, “Radiation boundary conditions for wave-like equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

Acar, R.

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

Anastasio, M. A.

K. Wang, E. Y. Sidky, M. A. Anastasio, A. A. Oraevsky, and X. Pan, “Limited data image reconstruction in optoacoustic tomography by constrained total variation minimization,” Proc. SPIE 7899, 78993U (2011).
[CrossRef]

Andreev, V. A.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Arridge, S. R.

S. R. Arridge, P. van der Zee, M. Cope, and D. Delpy, “Reconstruction methods for infrared absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

S. R. Arridge, “Forward and inverse problems in time-resolved infrared imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G.J.Mueller, B.Chance, R.R.Alfano, S.B.Arridge, J.Beuthen, E.Gratton, M.Kaschke, B.R.Masters, S.Svanberg, and P.van der Zee, eds., SPIE Institute Series (SPIE, 1993), Vol.  IS11, pp. 35–64.

Bayliss, A.

A. Bayliss and E. Turkel, “Radiation boundary conditions for wave-like equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

Bovik, A.

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

Carney, P. R.

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

Chen, H.

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, and D. Delpy, “Reconstruction methods for infrared absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Delpy, D.

S. R. Arridge, P. van der Zee, M. Cope, and D. Delpy, “Reconstruction methods for infrared absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Dobson, D. C.

D. C. Dobson and F. Santosa, “Recovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181–1198(1996).
[CrossRef]

D. C. Dobson and F. Santosa, “An image-enhancement technique for electrical impedance tomography,” Inverse Probl. 10, 317–334 (1994).
[CrossRef]

D. C. Dobson, “Exploiting ill-posedness in the design of diffractive optical structures,” Proc. SPIE 1919, 248–257(1993).
[CrossRef]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

Fleming, R. D.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Gatalica, Z.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Griffiths, D. V.

I. M. Smith and D. V. Griffiths, Programming the Finite Element Method (Wiley, 2004).

Gu, X.

Guo, Z.

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010).
[CrossRef] [PubMed]

Iftimia, N.

Jacques, S.

G. Paltauf, J. Viator, S. Prahl, and S. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112, 1536–1544 (2002).
[CrossRef] [PubMed]

Jiang, H.

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time-domain,” J. Opt. A 11, 085301 (2009).
[CrossRef]

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett. 32, 2556–2558 (2007).
[CrossRef] [PubMed]

H. Jiang, Z. Yuan, and X. Gu, “Spatially varying optical and acoustic property reconstruction using finite-element-based photoacoustic tomography,” J. Opt. Soc. Am. A 23, 878–888(2006).
[CrossRef]

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: recovery of optical absorption coefficient maps of heterogenous media,” Appl. Phys. Lett. 88, 231101 (2006).
[CrossRef]

N. Iftimia and H. Jiang, “Quantitative optical image reconstruction of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
[CrossRef]

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35, 3447–3458 (1996).
[CrossRef] [PubMed]

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[CrossRef] [PubMed]

Karabutov, A. A.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Kleinmann, R. E.

P. M. van den Berg and R. E. Kleinmann, “A total variation enhanced modified gradient algorithm for profile reconstruction,” Inverse Probl. 11, L5–L10 (1995).
[CrossRef]

Lesage, F.

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

Li, C.

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010).
[CrossRef] [PubMed]

Liu, Z.

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

Norton, S. J.

Oman, M. E.

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227–238(1996).
[CrossRef]

Oraevsky, A. A.

K. Wang, E. Y. Sidky, M. A. Anastasio, A. A. Oraevsky, and X. Pan, “Limited data image reconstruction in optoacoustic tomography by constrained total variation minimization,” Proc. SPIE 7899, 78993U (2011).
[CrossRef]

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

Paltauf, G.

G. Paltauf, J. Viator, S. Prahl, and S. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112, 1536–1544 (2002).
[CrossRef] [PubMed]

Pan, X.

K. Wang, E. Y. Sidky, M. A. Anastasio, A. A. Oraevsky, and X. Pan, “Limited data image reconstruction in optoacoustic tomography by constrained total variation minimization,” Proc. SPIE 7899, 78993U (2011).
[CrossRef]

Paulsen, K. D.

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35, 3447–3458 (1996).
[CrossRef] [PubMed]

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[CrossRef] [PubMed]

Prahl, S.

G. Paltauf, J. Viator, S. Prahl, and S. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112, 1536–1544 (2002).
[CrossRef] [PubMed]

Provost, J.

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

Roper, S. N.

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithm,” Physica D 60, 259–268 (1992).
[CrossRef]

Santosa, F.

D. C. Dobson and F. Santosa, “Recovery of blocky images from noisy and blurred data,” SIAM J. Appl. Math. 56, 1181–1198(1996).
[CrossRef]

D. C. Dobson and F. Santosa, “An image-enhancement technique for electrical impedance tomography,” Inverse Probl. 10, 317–334 (1994).
[CrossRef]

Savateeva, E. V.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Sidky, E. Y.

K. Wang, E. Y. Sidky, M. A. Anastasio, A. A. Oraevsky, and X. Pan, “Limited data image reconstruction in optoacoustic tomography by constrained total variation minimization,” Proc. SPIE 7899, 78993U (2011).
[CrossRef]

Singh, H.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Smith, I. M.

I. M. Smith and D. V. Griffiths, Programming the Finite Element Method (Wiley, 2004).

Solomatin, S. V.

A. A. Oraevsky, A. A. Karabutov, S. V. Solomatin, E. V. Savateeva, V. A. Andreev, Z. Gatalica, H. Singh, and R. D. Fleming, “Laser optoacoustic imaging of breast cancer in vivo,” Proc. SPIE 4256, 6–15 (2001).
[CrossRef]

Song, L.

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010).
[CrossRef] [PubMed]

Tam, A. C.

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58381–430 (1986)
[CrossRef]

Taylor, R. L.

O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, Finite Element Method: Its Basis and Fundamentals (Butterworth, 2005).

Turkel, E.

A. Bayliss and E. Turkel, “Radiation boundary conditions for wave-like equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

van den Berg, P. M.

P. M. van den Berg and R. E. Kleinmann, “A total variation enhanced modified gradient algorithm for profile reconstruction,” Inverse Probl. 11, L5–L10 (1995).
[CrossRef]

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, and D. Delpy, “Reconstruction methods for infrared absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Viator, J.

G. Paltauf, J. Viator, S. Prahl, and S. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112, 1536–1544 (2002).
[CrossRef] [PubMed]

Vo-Dinh, T.

Vogel, C. R.

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227–238(1996).
[CrossRef]

R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl. 10, 1217–1229 (1994).
[CrossRef]

Wang, K.

K. Wang, E. Y. Sidky, M. A. Anastasio, A. A. Oraevsky, and X. Pan, “Limited data image reconstruction in optoacoustic tomography by constrained total variation minimization,” Proc. SPIE 7899, 78993U (2011).
[CrossRef]

Wang, L. V.

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010).
[CrossRef] [PubMed]

Wang, Q.

Wang, Z.

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

Yao, L.

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time-domain,” J. Opt. A 11, 085301 (2009).
[CrossRef]

Yin, L.

Yuan, Z.

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

H. Jiang, Z. Yuan, and X. Gu, “Spatially varying optical and acoustic property reconstruction using finite-element-based photoacoustic tomography,” J. Opt. Soc. Am. A 23, 878–888(2006).
[CrossRef]

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: recovery of optical absorption coefficient maps of heterogenous media,” Appl. Phys. Lett. 88, 231101 (2006).
[CrossRef]

Zhang, Q.

Q. Zhang, Z. Liu, P. R. Carney, Z. Yuan, H. Chen, S. N. Roper, and H. Jiang, “Non-invasive imaging of epileptic seizures in vivo using photoacoustic tomography,” Phys. Med. Biol. 53, 1921–1931 (2008).
[CrossRef] [PubMed]

L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett. 32, 2556–2558 (2007).
[CrossRef] [PubMed]

Zhu, J. Z.

O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, Finite Element Method: Its Basis and Fundamentals (Butterworth, 2005).

Zienkiewicz, O. C.

O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, Finite Element Method: Its Basis and Fundamentals (Butterworth, 2005).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: recovery of optical absorption coefficient maps of heterogenous media,” Appl. Phys. Lett. 88, 231101 (2006).
[CrossRef]

Commun. Pure Appl. Math. (1)

A. Bayliss and E. Turkel, “Radiation boundary conditions for wave-like equations,” Commun. Pure Appl. Math. 33, 707–725 (1980).
[CrossRef]

IEEE Signal Process. Lett. (1)

Z. Wang and A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Reconstructed absorbed energy density images from simulated data with and without the TVM enhancement under different noise levels (case 1). (a) Without the TVM, 0% noise, (b) with the TVM, 0% noise, (c) without the TVM, 10% noise, (d) with the TVM, 10% noise, (e) without the TVM, 25% noise, (f) with the TVM, 25% noise. The axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Fig. 2
Fig. 2

Comparison of the exact and reconstructed absorbed energy density profiles along transect y = 0 mm for the images appearing in Fig. 1. (a) 0% noise, (b) 10% noise, (c) 25% noise.

Fig. 3
Fig. 3

Reconstructed absorbed energy density images from simulated data with and without the TVM enhancement with different target sizes (case 2). (a) Without the TVM, 2 mm diameter target; (b) with the TVM, 2 mm diameter target; (c) absorbed energy density profiles along the transect y = 0 mm for the images appearing in Figs. 1a, 1b ( 4 mm diameter target); (d) absorbed energy density property profiles along the transect y = 0 mm for the images appearing in Figs. 3a, 3b ( 2 mm diameter target). In (a) and (b), the axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Fig. 4
Fig. 4

Reconstructed absorbed energy density images from simulated data with and without the TVM enhancement with different contrast levels between the target and the background (case 3). (a) Without the TVM, 1.5 1 contrast; (b) with the TVM, 1.5 1 contrast; (c) absorbed energy density profiles along the transect y = 0 mm for the images appearing in Figs. 1a, 1b ( 2 1 contrast); (d) absorbed energy density profiles along the transect y = 0 mm for the images appearing in Figs. 4a, 4b ( 1.5 1 contrast). In Figs. 4a, 4b, the axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Fig. 5
Fig. 5

Reconstructed absorbed energy density images from simulated data with and without the TVM enhancement for three targets having different shapes (case 4). (a) Exact image, (b) without the TVM, (c) with the TVM, (d) absorbed energy density profiles along the transect y = 0 mm . In (a)–(c), the axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Fig. 6
Fig. 6

UQI calculated from the recovered images with and without TVM enhancement from simulated data. (a) Case 1 when different noise levels are considered, (b) cases 2–4.

Fig. 7
Fig. 7

Reconstructed absorbed energy density images from the three phantom experiments. (a) Case 1 without the TVM, (b) case 1 with the TVM, (c) case 2 without the TVM, (d) case 2 with the TVM, (e) case 3 without the TVM, (f) case 3 with the TVM. The axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Fig. 8
Fig. 8

Recovered absorbed energy density profiles along (a)  y = 7.0 mm crossing the 3 mm diameter target for experimental case 1, (b)  y = 8.0 mm crossing the 2 mm diameter target for experimental case 1, (c)  y = 1.0 mm for experimental case 2, and (d)  y = 6.5 mm for experimental case 3.

Fig. 9
Fig. 9

CNR calculated for the recovered images using the method with and without TVM. (a)–(c) Images shown in Figs. 7b, 7d, 7f with the ROIs marked (four pairs of ROIs: solid line circle, t-ROI; dashed line circle, b-ROI). (d) CNR computed for the four pairs of ROIs shown in (a)–(c). In Figs. 9a, 9b, 9c, the axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Fig. 10
Fig. 10

Recovered absorbed energy density images from the rat brain (a) without the TVM, (b) with the TVM. The axes (left and bottom) illustrate the spatial scale in millimeters, whereas the color scale (right) records the absorbed energy density in millijoules per cubed millimeter.

Equations (19)

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2 p ( r , t ) 1 v 0 2 2 p ( r , t ) t 2 = Φ ( r ) β C p J ( t ) t ,
j = 1 N p j [ S ψ i · ψ j d S ] + j = 1 N p ¨ j [ S 1 v 0 2 ψ i ψ j d S ] l ψ i p · n ^ d l = S β Φ C P J t ψ i d S .
p · n ^ = 1 v 0 p t p 2 r ,
[ K ] { p } + [ C ] { p ˙ } + [ M ] { p ¨ } = { B } ,
K i j = S ψ i · ψ j d S + 1 2 r l ψ i ψ j d l ; C i j = 1 v 0 l ψ i ψ j d l ; M i j = 1 v 0 2 S ψ i ψ j d S ;
B i = β C p S ψ i ( k ψ k Φ k ) d S · J t { p } = { p 1 , p 2 , p N } T ; { p ˙ } = { p ˙ 1 , p ˙ 2 , p ˙ N } T ; { p ¨ } = { p ¨ 1 , p ¨ 2 , p ¨ N } T .
F ( p , Φ ) = j = 1 M ( p j 0 p j c ) 2 ,
( I T I + λ I ) Δ χ = I T ( p 0 p c ) ,
F ˜ ( p , Φ ) = F ( p , Φ ) + L ( Φ ) .
L ( Φ ) = ω Φ 2 | Φ | 2 + δ 2 d x d y
F ˜ Φ i = j = 1 M ( p j o p j c ) p j c Φ i + V i , ( i = 1 , 2 N ) ,
V i = L Φ i = ω Φ 2 [ ( k = 1 N Φ k ψ k x ) ψ i x + ( k = 1 N Φ k ψ k y ) ψ i y ] ω Φ 2 [ ( k = 1 N Φ k ψ k x ) 2 + ( k = 1 N Φ k ψ k y ) 2 ] + δ 2 d x d y .
( I T I + R + λ I ) Δ χ = I T ( p 0 p c ) V ,
R = [ V 1 Φ 1 V 1 Φ 2 V 1 Φ N V 2 Φ 1 V 2 Φ 2 V 2 Φ N V N Φ 1 V N Φ 2 V N Φ N ]
f ¯ j = 1 N k = 1 N f k j ,
σ j = 1 N k = 1 N ( f k j f ¯ j ) 2 ,
Cov { f 1 , f 0 } = 1 N k = 1 N ( f k 1 f ¯ 1 ) ( f k 0 f ¯ 0 ) .
UQI { f 1 , f 0 } = 2 Cov { f 1 , f 0 } ( σ 1 ) 2 + ( σ 0 ) 2 2 f ¯ 1 f ¯ 0 ( f ¯ 1 ) 2 + ( f ¯ 0 ) 2 .
CNR = | f ¯ ( t ) f ¯ ( b ) | σ ( b ) ,

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