Research ArticleOPTICS

Distortion matrix concept for deep optical imaging in scattering media

See allHide authors and affiliations

Science Advances  22 Jul 2020:
Vol. 6, no. 30, eaay7170
DOI: 10.1126/sciadv.aay7170
  • Fig. 1 Principle of the distortion matrix approach.

    (A) A resolution target (USAF 1951) is positioned underneath an 800-μm-thick sample of rat intestine (A1). In scanning microscopy, raster scanning in the focal plane is obtained using a set of plane-wave illuminations in the input pupil (A2). In the presence of sample-induced aberrations, the detected intensity will exhibit a much larger extent compared to the ideal PSF (A3). The resulting full-field image displays a low contrast and a reduced resolution (A4). (B) In the output pupil plane, the phase of the reflected wave field (B1) can be split into a diffraction (B2) and a distortion (B3) term. (C and D) The reflected distorted wave fields can be stored along column vectors to form the reflection and distortion matrices, R and D, respectively. The phase of R and D is displayed in (C1) and (D1), respectively. The autocorrelations of the complex reflected/distorted wave fields are computed in the focal (C2/D2; see section S2) and in the pupil (C3/D3; see section S1) planes, both in dB. All the data shown here are extracted from the rat intestine imaging experiment. Photo credit: Amaury Badon, CNRS. NA, numerical aperture.

  • Fig. 2 Extracting the aberration transmittance from the distortion matrix D.

    (A) The recording of the R-matrix consists in scanning the objects with a moving input focusing beam. (B) The removal of the geometric component in each reflected wavefront (Eq. 4) amounts to recenter each incident focal spot at the origin. The D-matrix is equivalent to the reflection matrix for a moving object. (C) The SVD of D leads to a coherent sum of the distorted wavefronts in the pupil plane. A coherent reflector is virtually synthesized in the focal plane, and the corresponding wavefront emerges along the output singular vector U1. The corresponding image of the object is provided by the first input singular vector V1, but its resolution is dictated by the width δin of the input focusing beam. (D) A normalization of U1 in the pupil plane makes the virtual scatterer point like. The corresponding input singular vector V̂1 yields a diffraction-limited image of the object in the focal plane.

  • Fig. 3 Imaging through a thick layer of rat intestinal tissue.

    (A) Experimental configuration. (B and C) Modulus of the first input singular vector V1 of D in the focal plane. (D) Modulus and phase of the first output singular vector Up in the pupil plane. (E) Example of PSF deduced from the central column (rin = 0) of the raw focused matrix R0. (F) Corresponding corrected PSF deduced from the central column of the focused matrix R1 (Eq. 12). (G and H) Comparison of the full-field images ℱ0 and ℱ1 (Eq. 14) before and after aberration correction. Photo credit: Amaury Badon, CNRS.

  • Fig. 4 Matrix imaging over multiple IPs.

    (A) Schematic of the experiment. A resolution target (USAF 1951) is positioned at a distance d = 1 mm underneath a rough plastic film (see inset). (B) Original full-field image ℱ0 (Eq. 14). (C) Example of PSF deduced from a column of the raw focused matrix R0. (D) Plot of the normalized singular values σ˜ι of D. The red circles correspond to the eight first singular values (signal subspace), while the noisy singular values are displayed in blue. (E) Matrix image constructed from the eight first eigenstates of D (Eq. 20). (F) Example of PSF deduced from a column of the corrected focused matrix R1. (G) Phase of the four first singular vectors Up. (H) Confocal images deduced from the focused reflection matrices Rp (Eq. 18). Photo credit: Amaury Badon, CNRS.

  • Fig. 5 Imaging through corneal tissue with deteriorated transparency.

    (A) Schematic of the experiment. A resolution target (USAF 1951) is positioned below an edematous nonhuman primate cornea (see inset). (B) Plot of the normalized singular values σ˜p of D. The red circles correspond to the 11 first singular values (signal subspace), while the noisy singular values are displayed in blue. (C) Original confocal image deduced from the focused reflection matrix R0 (Eq. 18). (D) Final matrix image constructed from the 11 first eigenstates of D (Eq. 20). (E) Real parts of U1, U6, and U11. (F) Corresponding confocal images deduced from the focused reflection matrices Rp (Eq. 18).

Supplementary Materials

  • Supplementary Materials

    Distortion matrix concept for deep optical imaging in scattering media

    Amaury Badon, Victor Barolle, Kristina Irsch, A. Claude Boccara, Mathias Fink, Alexandre Aubry

    Download Supplement

    This PDF file includes:

    • Sections S1 to S3
    • Figs. S1 to S5
    • Tables S1 and S2
    • References

    Files in this Data Supplement:

Stay Connected to Science Advances

Navigate This Article