We investigate the application of the common circle method for estimating sample motion in optical diffraction tomography (ODT) of sub-millimeter sized biological tissue. When samples are confined via contact-free acoustical force fields, their motion must be estimated from the captured images. The common circle method identifies intersections of Ewald spheres in Fourier space to determine rotational motion. This paper presents a practical implementation, incorporating temporal consistency constraints to achieve stable reconstructions. Our results on both simulated and real-world data demonstrate that the common circle method provides a computationally efficient alternative to full optimization methods for motion detection.
We consider the tomographic imaging of sub-millimeter sized biological samples by means of optical diffraction tomography (ODT) [7,15,37]. In ODT, the object is illuminated from different angles to reconstruct its 3D refractive index. While traditional sample immobilization, e.g. in gel, may restrict biological processes, contact-free methods utilizing optical [14] or acoustical tweezers [8,38] allow imaging of cells in their natural environment. However, the motion parameters are not known exactly and must be estimated from the captured images.
In standard computed tomography, optical diffraction is negligible because the wavelength of x-rays is much smaller than object features, allowing for a straight-line light propagation model. In contrast, our ODT setup involves µm-sized biological cells imaged with visible light, where diffraction effects require more sophisticated wave propagation models.
Therefore, the common line method [9,31,36,39,41] for orientation detection in x-ray tomography, which uses that each x-ray image becomes a plane in Fourier space, is not applicable. Under the Born approximation, each ODT measurement corresponds to an Ewald sphere in Fourier space, cf. [28]. The common circle method [34], see also [4], reconstructs the motion of the imaged object by identifying the intersections (common circles) of these sphere, see Figure 1 left.
While the theoretical foundations of the common circle method were established in [34], this paper focuses on the application to real-world data. We obtain stable reconstructions by adding additional regularization and ensuring temporal consistency. Finally, we provide a quantitative comparison with the full optimization approach [27] for ODT motion detection, which simultaneously reconstructs object and motion based on the beam propagation method, see also [23]. In comparison, the common circle method is less accurate, but it is considerably faster and does not require an initialization of the rotation parameters, therefore providing a good initial guess for more computationally intensive techniques. This paper is structured as follows. Section 2 outlines the theoretical basis of ODT and the common circle method. Section 3 describes the reconstruction approach. Section 4 presents our numerical data processing and reconstruction. Conclusions are drawn in Section 5.
In this section, we describe the setup of diffraction tomography with a moving object and the common circle method for reconstructing the rotations, whose theoretical foundations were derived in [34].
We start with the model of optical diffraction tomography for a fixed object, see more detail in [18,19,42]. The unknown object is illuminated by a plane wave u inc (x) := e ik 0 x 3 , x ∈ R 3 , (
which propagates in direction e 3 = (0, 0, 1) ⊤ with wave number k 0 = 2πn 0 /λ 0 , where n 0 is the constant refractive index of the surrounding medium and λ 0 is the wavelength of the incident field. Let n(x) denote the refractive index at position x ∈ R 3 . We have n(x) = n 0 outside the object. The scattering potential or object function
vanishes outside the object. We assume f is piecewise continuous and compactly supported. The incident wave u inc induces a scattered wave u sca that solves the partial differential equation
More specifically, u sca is the outgoing solution, which fulfills the Sommerfeld radiation condition
If f is sufficiently small, we neglect f u sca on the right-hand side of (2.3) to obtain the Born approximation u of the scattered field u sca , determined by the Helmholtz equation
In the following, we assume the Born approximation to be valid, which holds for small objects which mildly scatter, in particular that total phase shift through the object being much less than 2π, cf. [11,15,43].
We denote the measurements of the scattered wave at the plane {x ∈ R 3 :
We define the d-dimensional Fourier transform of an integrable function g : R d → C by
and the ball of radius r > 0 by
The Fourier diffraction theorem relates the 2D Fourier transform of the measurements m to the 3D Fourier transform of the scattering potential f , see [15,18,29,42]. We have
where h :
, and κ(k
The left-hand side of (2.5) is the Fourier transform of the measured 2D image m, and the right-hand side is the 3D Fourier transform of f evaluated on a hemisphere whose north pole is the origin 0, see Figure 1.
The object is exposed to a rigid motion depending on time t ∈ [0, T ], such that the scattering potential of the moving object is
with a rotation matrix
and a translation vector d t ∈ R 3 . The incident wave (2.1) and the measurement plane {x ∈ R 3 :
x 3 = r M } stay the same as above. The scattered wave u t is the solution of (2.4) with f replaced by f t . Denoting the measurements by
the Fourier diffraction theorem (2.5) becomes [34] F
The scaled squared energy
depends only on the measurements m t , t ∈ [0, T ], and is related to the scattering potential f via
For every t, we see
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