## 2 Materials and Methods
### Analysis Methods  
Optical systems generally are not free of aberrations or imperfections that affect the quality of the recorded images. Consequently, the image of a point object is an extended distribution of intensity called point spread function (PSF). During data analysis, we firstly sought to remove from the corrected images the distortions of observation due to the optical acquisition system and the contributions from out-of-focus planes for the sample. In this approach, we introduced the model of an appropriate PSF required for a deconvolution, which should remove any artefact. At the second time, we used the TIE method to extract the refractive index and the size of the specimen. Deconvolution requires a realistic model of the phenomena that induce distortions in the image formation process. The term \"deconvolution microscopy\" generally assumes fluorescence microscopy [14; 15], and PSF deconvolution imaging theory has traditionally been linked to fluorescence. A main reason behind this association is that for the case of self-luminous objects, one needs only to consider signal intensity, leading to a unique PSF and making deconvolution a linear process. In general, this is not the case in BF microscopy, where two point spread functions are needed to describe image formation [16]. In fluorescence microscopy, the 3D image \(\mathbf{i}(x,\,y,\,z)\) of an object corresponds to the convolution of the object \(\mathbf{o}(x,\,y,\,z)\) with the \(PSF(x,y,z)\)[17]:  
\[\mathbf{i}=\mathbf{o}\otimes PSF+\mathbf{B} \tag{2}\]  
where \(\mathbf{B}\) models noise, and the symbol \(\otimes\) denotes the convolution operation. BF microscopy is presented as an alternative to fluorescence in deconvolution image processing, because of the possibility for observing unstained objects. However, its corresponding PSF consists of two separate components, one for phase (\(PSF_{p}\)) and one for absorption (\(PSF_{a}\)). Therefore, the BF-3D image \(\mathbf{i}(x,\,y,\,z)\) is generally described as the sum of the convolutions of the real (\(\mathbf{P}\)) and imaginary (\(\mathbf{A}\)) parts of the object scattering potential with the corresponding \(PSF_{s}\)[18]:  
\[\mathbf{i}=\mathbf{P}\otimes PSF_{p}+\mathbf{A}\otimes PSF_{a}+\mathbf{B} \tag{3}\]  
This excludes the application of linear deconvolution processing to remove distortion. However, in the presence of the pure phase object as it is the case in this work, BF imaging essentially reduces to Equation (2), making it possible to perform linear deconvolution [16]. Therefore, knowing the PSF leads to the determination of the object through deconvolution. The PSF can be:  
- computed from the optical properties of the microscope system,  
- estimated from the measurements of the microspheres. The experimental PSF takes into account all of the aberrations introduced by the whole image acquisition system [19]. It is determined by recording images of a specimen (assumed to be a point object) in different planes of defocus above and below the focal plane. A point object does not exist in practical applications; the object necessarily has a particular size. Therefore, for the measured data to correspond to the microscope's PSF, it is very important that the object size is smaller than the diffraction limit. If this is not the case, the measured data will instead correspond to the convolution of the PSF with the object shape, according to Equation (2) [20]. Unfortunately, using a very small object introduces the problem of signal detection, due to a very low signal-to-noise ratio. This requires detectors of high sensitivity. The signal to noise ratio is so low that often a bead is used whose size is not smaller than diffraction limit, but of the order of the microscope's resolution [21]. This leads to an overestimated PSF. The experimental PSF is noisy and needs to be devoid of noise, which is a limitation in a deconvolution algorithm. The method of denoising sometimes distorts the real PSF of the microscope. [PERSON] et al.