Abstract
This paper deals with the issue of stability in determining the absorption and the diffusion coefficients in quantitative photoacoustic imaging. We establish a global conditional Hölder stability inequality from the knowledge of two internal data obtained from optical waves, generated by two point sources in a region where the optical coefficients are known.
Similar content being viewed by others
References
Alessandrini, G., Di Cristo, M., Francini, E., Vessella, S.: Stability for quantitative photoacoustic tomography with well chosen illuminations. Ann. Mat. Pura e Appl. 196(2), 395–406 (2017)
Ammari, H., Bossy, E., Jugnon, V., Kang, H.: Mathematical modeling in photoacoustic imaging of small absorbers. SIAM Rev. 52, 677–695 (2010)
Ammari, H., Garnier, J., Kang, H., Nguyen, L., Seppecher, L.: Multi-Wave Medical Imaging, Modeling and Simulation in Medical Imaging. World Scientific, London (2017)
Ammari, H., Kang, H., Kim, S.: Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media. J. Differ. Equ. 253(1), 41–72 (2012)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation, Massachusetts (1965)
Auscher, P., Tchamitchian, P.H.: Square root problem for divergence operators and related topics. Astérisque 249, viii+172 (1998)
Bal, G., Ren, K.: Multi-source quantitative photoacoustic tomography in a diffusive regime. Inverse Probl. 27(7), 075003 (2011)
Bal, G., Uhlmann, G.: Inverse diffusion theory of photoacoustics. Inverse Prob. 26, 085010 (2010)
Bellassoued, M., Choulli, M.: Global logarithmic stability of a Cauchy problem for anisotropic wave equations (2019). arXiv:1902.05878
Choulli, M.: Boundary Value Problems for Elliptic Partial Differential Equations, Graduate Course. arXiv:1912.05497
Choulli, M.: Some stability inequalities for hybrid inverse problems. CR. Math. Acad. Sci. Paris 359(10), 1251–1265 (2021)
Cox, B.T., Laufer, J.G., Beard, P.C., Arridge, S.R.: Quantitative spectroscopic photoacoustic imaging: a review. J. Biomed. Opt. 17(6), 061202 (2012)
Fabes, E., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rat. Mech. Anal. 96, 327–338 (1986)
Friedman, A.: Partial Differential Equations of Parabolic Type, p. xiv+347. Prentice-Hall Inc, Englewood Cliffs, N.J. (1964)
Garofalo, N., Lin, F.-H.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35(2), 245–268 (1986)
Garofalo, N., Lin, F.-H.: Unique continuation for elliptic operators: a geometric-variational approach. Commun. Pure Appl. Math. 40(3), 347–366 (1987)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin (1983)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24, p. xiv+410. Pitman (Advanced Publishing Program), Boston, MA (1985)
Kukavica, I.: Quantitative uniqueness for second-order elliptic operators. Duke Math. J. 91(2), 225–240 (1998)
Ladyzenskaja, O.A., Solonnikov, V. A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. (Russian) Translated from the Russian by S. Smith, Translations of Mathematical Monographs, vol. 23, p. xi+648 . AMS, Providence, R.I. (1968)
Naetar, W., Scherzer, O.: Quantitative photoacoustic tomography with piecewise constant material parameters. SIAM J. Imag. Sci. 7, 1755–1774 (2014)
Ren, K., Triki, F.: A Global stability estimate for the photo-acoustic inverse problem in layered media. Eur. J. Appl. Math. 30(3), 505–528 (2019)
ter Elst, A. F.M., Wong, M.F.: Hölder Kernel Estimates for Robin operators and Dirichlet-to-Neumann operators (2019). arXiv:1910.07431
Wang, L.V.: Photoacoustic Tomography: High-resolution Imaging of Optical Contrast In Vivo at Superdepths, pp. 1201–1201. In: From Nano to Macro, IEEE International Symposium on Biomedical Imaging (2009)
Wang, L.V. (ed.): Photoacoustic Imaging and Spectroscopy. CRC Press, Taylor & Francis, Boca Raton (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors were supported by the Grant ANR-17-CE40-0029 of the French National Research Agency ANR (Project MultiOnde)
Appendix A: Proof of technical lemmas
Appendix A: Proof of technical lemmas
Proof of Lemma 2
In this proof, \(C=C(n,\mu ,\nu )>1\) is a generic constant.
It is well known that \(G_{1,\nu }\), \(\nu >0\), the fundamental solution of the operator \(-\Delta +\nu \), is given by \(G_{1,\nu }(x,\xi )={\mathcal {G}}_{1,\nu }(x-\xi )\), \(x,\xi \in {\mathbb {R}}^n\), with
In the particular case \(n=3\), we have \(K_{1/2}(z)=\sqrt{\pi /(2z)}e^{-z}\) and therefore
Let \(f\in C_0^\infty ({\mathbb {R}}^n)\), \(\mu >0\) and \(\nu >0\) be two constants, and denote by u the solution of the equation
Then,
We remark that \(v(x)=u(\sqrt{\mu }x)\), \(x\in {\mathbb {R}}^n\) satisfies \((-\Delta +\nu )v=f(\sqrt{\mu }\; \cdot )\). Whence
Hence,
Comparing (A.1) and (A.2), we find
Consequently, \(G_{\mu ,\nu }(x,\xi )={\mathcal {G}}_{\mu ,\nu }(x-\xi )\) with
By the usual asymptotic formula for modified Bessel functions of the second kind (see for instance [5, 9.7.2, page 378]), we have, when \(|x|\rightarrow \infty \),
where O(1/|x|) only depends on n, \(\mu \) and \(\nu \).
Consequently, there exists \(R=R(n,\mu ,\nu )>0\) so that
Substituting if necessary R by \(\max (R,1)\), we have
Moreover, we have
Since the function \(x\rightarrow |x|^{(n-3)/2}e^{-\sqrt{\nu }|x|/(2\sqrt{\mu })}\) is bounded in \({\mathbb {R}}^n\), we deduce
Using (A.5) and (A.6) in (A.4) in order to obtain
We now establish a similar estimate when \(|x|\rightarrow 0\). To this end, we recall that according to formula [5, 9.6.9, p. 375] we have
from which we deduce in a straightforward manner that there exists \(0<r\le R\) so that
The expected two-sided inequality (2.10) follows by combining (A.4), (A.7) and (A.8). \(\square \)
Proof of Lemma 3
Let \({\mathcal {Q}}\) be an open subset of \({\mathbb {R}}^n\), set \(d=\text{ diam }({\mathcal {Q}})\), \(d_x=\text{ dist }(x,\partial {\mathcal {Q}})\) and \(d_{x,y}=\min (d_x,d_y)\).
We introduce the following weighted Hölder semi-norms and Hölder norms, where \(\sigma \in {\mathbb {R}}\), \(0<\gamma \le 1\), and k is nonnegative integer,
In terms of these notations, we have
In consequence,
Following [17], we define also
From [17, Lemma 6.32, page 130] and its proof, we have the following interpolation inequalities: Suppose that j and k, nonnegative integers, and \(0\le \beta ,\gamma \le 1\) are so that \(j+\beta <k+\gamma \). Then, there exist \(C=C(n,\alpha ,\beta )>0\) and \(\vartheta =\vartheta (\alpha ,\beta )\) so that, for any \(w\in C^{k,\alpha }({\mathcal {Q}})\) and \(\epsilon >0\), we have
Here, \(|w|_{0;{\mathcal {Q}}}=\sup _{x\in {\mathcal {Q}}}|w(x)|\).
Checking carefully the proof of interior Schauder estimates in [17, Theorem 6.2, page 90], we get, taking into account inequalities (A.9)-(A.11), the following result: There exist a constant \(C=C(n)>0\) and \(\tau =\tau (\alpha )\) so that, for any \(0<\mu \le 1/2\) and \(w\in C^{k,\alpha }({\mathcal {Q}})\) satisfying \(L_{a,b}w=0\) in \({\mathcal {Q}}\), we have
Substituting in (A.12) C by \(\max (C,2^{\alpha -1})\), we may assume in (A.12) that \(C=C(n,\alpha )\ge 2^{\alpha -1}\). Bearing in mind that \(\Lambda ({\mathbf {d}})>1\), we can take in (A.12), \(\mu =(2C\Lambda ({\mathbf {d}}))^{-1/\alpha } \). We find
for some constants \(C=C(n,\alpha )>0\) and \(\varkappa =\varkappa (\alpha )>1\).
Using again interpolation inequalities (A.10) and (A.11), we deduce that
Let \(\delta >0\) be so that \({\mathcal {Q}}_\delta =\{x\in {\mathcal {Q}};\; \text{ dist }(x,\partial {\mathcal {Q}})>\delta \}\) is non-empty. If \({\mathcal {Q}}'\) is an open subset of \({\mathcal {Q}}_\delta \), then (A.14) yields in a straightforward manner
This is the expected inequality. \(\square \)
Lemma 14
Let K be a compact subset of \({\mathbb {R}}^n\) and \(f\in C^{2,\alpha }(K)\) satisfying \(\min _K|f|\ge c_->0\). Then,
where \(c_+=\max (1,c_-^{-1})\) and \(C=C(\mathrm {diam}(K))\) is a constant.
Proof
Let \(x,y\in K\). Using \(|1/f|_{0;K}\le c_+\) and the following identities
we easily get
Also, we have
In light of (A.16), this identity yields
On the other hand, since
we find, by using again (A.16),
Inequalities (A.17), (A.18), the identity \(\partial _{ij}^2(1/f)=2\partial _if\partial _jf/f^3-\partial _{ij}^2f/f^2\) and the interpolation inequality [17, Lemma 6.35, p. 135] (by proceeding as in Corollary 2) imply
where \(C=C(\text{ diam }(K))\) is a constant.
The other terms for 1/f appearing in the norms \(\Vert \cdot \Vert _{C^{2,\alpha }(K)}\) can be estimated similarly to the semi-norm in (A.19). Inequality (A.15) then follows. \(\square \)
Recall that \(0<\theta<\alpha <1\).
Lemma 15
\(C^{2,\alpha }(\overline{{\mathcal {O}}})\) is continuously embedded in \(H^{2+\theta }({\mathcal {O}})\). Furthermore, there exists \(C=C(n,\alpha -\theta )\) so that, for any \(w\in C^{2,\alpha }(\overline{{\mathcal {O}}})\), we have
where \({\mathbf {d}}=\mathrm {diam}({\mathcal {O}})\).
Proof
Let \(w\in C^{2,\alpha }(\overline{{\mathcal {O}}})\) and, for fixed \(1\le i,j\le n\), set \(g=\partial _{ij}^2w\). Then,
In light of [10, Lemma A3, p. 246], this inequality yields
But \(|{\mathcal {O}}|\le |B(0,{\mathbf {d}})|\). Hence,
Using (A.21) and the inequality
we get from the definition of the norm of \(H^s\)-spaces in [18, formula (1.3.2.2), page 17]
for some constant \(C=C(n,\alpha -\theta )>0\). This is the expected inequality \(\square \)
Rights and permissions
About this article
Cite this article
Bonnetier, E., Choulli, M. & Triki, F. Stability for quantitative photoacoustic tomography revisited. Res Math Sci 9, 24 (2022). https://doi.org/10.1007/s40687-022-00322-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40687-022-00322-6