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Recovery of continuous 3D refractive index maps from discrete intensity-only measurements using neural fields

A preprint version of the article is available at arXiv.


Intensity diffraction tomography (IDT) refers to a class of optical microscopy techniques for imaging the three-dimensional refractive index (RI) distribution of a sample from a set of two-dimensional intensity-only measurements. The reconstruction of artefact-free RI maps is a fundamental challenge in IDT due to the loss of phase information and the missing-cone problem. Neural fields has recently emerged as a new deep learning approach for learning continuous representations of physical fields. The technique uses a coordinate-based neural network to represent the field by mapping the spatial coordinates to the corresponding physical quantities, in our case the complex-valued refractive index values. We present Deep Continuous Artefact-free RI Field (DeCAF) as a neural-fields-based IDT method that can learn a high-quality continuous representation of a RI volume from its intensity-only and limited-angle measurements. The representation in DeCAF is learned directly from the measurements of the test sample by using the IDT forward model without any ground-truth RI maps. We qualitatively and quantitatively evaluate DeCAF on the simulated and experimental biological samples. Our results show that DeCAF can generate high-contrast and artefact-free RI maps and lead to an up to 2.1-fold reduction in the mean squared error over existing methods.

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Fig. 1: Conceptual illustration of DeCAF for IDT.
Fig. 2: Reconstruction of Spirogyra algae acquired by dIDT.
Fig. 3: Reconstruction of diatom algae acquired by aIDT.
Fig. 4: Reconstruction of a human buccal epithelial cell cluster acquired by aIDT.
Fig. 5: Reconstruction of C. elegans acquired by mIDT.
Fig. 6: Quantitative and visual comparison of DeCAF, SIMBA and Tikhonov for the reconstruction of the granulocyte phantom from simulated data.

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Data availability

The data used for reproducing the results in the manuscript are available at We visualized the pre-processed raw intensity images of the relevant samples in Figs. 1 and 4.

Code availability

The code used for reproducing the results in the manuscript is available at


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This work was supported by the NSF award nos. CCF-1813910 and CCF-2043134 (to U.K.), and CCF-1813848 and EPMD-1846784 (to L.T.).

Author information

Authors and Affiliations



The project was conceived by Y.S., R.L. and U.S.K. The code of the model was implemented by R.L. and Y.S. The experiments were designed by R.L. and Y.S. The numerical results were collected by R.L. Data acquisition and preparation were conducted by J.Z. and L.T. The manuscript was drafted by Y.S. with assistance from R.L. and J.Z. and revised by U.S.K and L.T. All of the authors reviewed manuscript.

Corresponding author

Correspondence to Ulugbek S. Kamilov.

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The authors declare no competing interests.

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Nature Machine Intelligence thanks Jan Funke, Jaejun Yoo and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Reconstruction of Diatom Algae acquired by mIDT.

(a) 2D rendering obtained by accumulating all the z slices from DeCAF. Scale bar 10 μm. (b) & (d) Lateral views corresponding to the colored lines in (a). (c) & (e) Axial views at z  {11, 0, 11} μm reconstructed by using DeCAF and Tikhonov, respectively. This figure illustrates the ability of DeCAF to reconstruct high-contrast RI maps for a relatively thin sample acquired by mIDT. Note how DeCAF successfully recovers the folding structure of the sample with two clear separate layers, which are barely recognizable in the Tikhonov reconstruction. Additional examples are shown in Supplementary Videos and

Extended Data Fig. 2 Quantitative Illustration of the scalability of DeCAF due to its off-the-grid feature using the C. elegans specimen as an example.

Note how the space required to store the reconstructed sample in DeCAF is independent of the reconstruction grid.

Extended Data Fig. 3 Reconstruction of the 3D Granulocyte Phantom using DeCAF, SIMBA, and Tikhonov.

(a) From left to right, 3D volumes correspond to Groundtruth, DeCAF, SIMBA, and Tikhonov, respectively. (b) Close-up views of the reconstructions at the location shown in (a). Note how DeCAF reconstructs sharper and better quality cell images compared to both SIMBA and Tikhonov.

Extended Data Fig. 4 Visual illustration of the network structure and the encoding strategy used in DeCAF.

(a) The overall structure of network Mφ. (b) Illustration of positional encoding for z coordinate. (c) Illustration of radial encoding for the coordinates in the (x, z) plane.

Extended Data Fig. 5 List of algorithmic hyperparameters used by DeCAF for different biological samples.

List of algorithmic hyperparameters used by DeCAF for different biological samples.

Supplementary information

Supplementary Information

Supplementary Discussion, Figs. 1–7 and Tables 1 and 2.

Reporting Summary

Supplementary Video 1

Spirogyra algae video reconstructed by DeCAF.

Supplementary Video 2

Spirogyra algae video reconstructed by SIMBA.

Supplementary Video 3

Spirogyra algae video reconstructed by Tikhonov.

Supplementary Video 4

Diatom algae video reconstructed by DeCAF under aIDT set-up.

Supplementary Video 5

Diatom algae video reconstructed by Tikhonov under aIDT set-up.

Supplementary Video 6

Diatom algae video reconstructed by DeCAF under mIDT set-up.

Supplementary Video 7

Diatom algae video reconstructed by Tikhonov under mIDT set-up.

Supplementary Video 8

Cell video reconstructed by DeCAF (Fig. 4b).

Supplementary Video 9

Cell video reconstructed by Tikhonov (Fig. 4b).

Supplementary Video 10

Cell video reconstructed by DeCAF (Fig. 4c).

Supplementary Video 11

Cell video reconstructed by Tikhonov (Fig. 4c).

Supplementary Video 12

C. elegans video reconstructed by DeCAF (body).

Supplementary Video 13

C. elegans video reconstructed by Tikhonov (body).

Supplementary Video 14

C. elegans video reconstructed by DeCAF (head).

Supplementary Video 15

C. elegans video reconstructed by Tikhonov (head).

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Liu, R., Sun, Y., Zhu, J. et al. Recovery of continuous 3D refractive index maps from discrete intensity-only measurements using neural fields. Nat Mach Intell 4, 781–791 (2022).

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