The decoupling problem

Quantitative phase imaging (QPI) measures the phase retardation \(\phi(x,y)\) introduced by an object, such as a cell, to a light field, usually a plane wave. Under the assumption that light travels along straight lines, the measured phase \(\phi(x,y)\) can be described as the projection of the refractive index (RI) \(n(x,y,z)\) of the imaged object onto the detector plane

\[\phi(x,y) = \frac{2 \pi}{\lambda} \int (n(x,y,z) - n_\text{med}) \, dz,\]

with the vacuum wavelength \(\lambda\) of the imaging light and the RI of the medium surrounding the object \(n_\text{med}\). This equation describes a coupling between the RI of the imaged object and its shape: From a single phase image, it is not possible to compute the RI of an object without knowing its shape and vice versa. Moreover, it is not possible to infer the RI of an object from its shape if the RI is not constant. Thus, in general, it is not possible to solve the coupling problem in QPI. However, if the object has a spherical shape and if its RI is constant, it is possible to infer the radius \(R\) and the RI \(n\) from a single phase measurement. The equation above then reduces to

\[\phi(x,y) = \frac{4 \pi}{\lambda} (n - n_\text{med}) \cdot \sqrt{R^2 - (x-x_0)^2 - (y-y_0)^2}\]

with the lateral position of the sphere \((x_0, y_0)\). Note that this approach is often applied to suspended (spherical) cells. Even though cells are complex structured objects, this approach yields good estimates for the average RI and radius. Qpsphere offers several approaches to address this decoupling problem, ranging from simple edge-detection to image fits with a 2D Mie model.

Why qpsphere?

The purpose of qpsphere is to make our QPI image analysis tools ([SSM+15] [SSM+16] [MSG+18]) available to other research groups. Qpsphere makes heavy use of qpimage, a resourceful QPI image manager and is a key component of our QPI analysis software DryMass.

Citing qpsphere

If you are using qpsphere in a scientific publication, please cite it with:

(...) using qpsphere version X.X.X (available at

or in a bibliography

Paul Müller (2017), qpsphere version X.X.X: Phase image analysis
[Software]. Available at

and replace X.X.X with the version of qpsphere that you used.

Furthermore, several ideas implemented in qpsphere have been described and published in scientific journals:

  • Retrieval of RI and radius using the OPD edge-detection approach is described in [SSM+15] and [SSM+16] (method="edge" in qpsphere.analyze()).

  • Retrieval of RI and radius by fitting 2D models (OPD projection, Rytov approximation, systematically corrected Rytov approximation, Mie) to phase images is described in [MSG+18]. (method="image" in qpsphere.analyze() and the corresponding scattering models in qpsphere.models).